PERSIFLAGE: MATH BLOG POSTSFRANK CALEGARI

I though your post on swans wasyour best post ever.

The nilmanifold fibres spread thecohomology around by a Künneth
type formula like a Frenchman
expectorating over-oaked California
Chardonnay into a spittoon

§50

§24

1. Introduction
It does not seem entirely impossible that wordpress might disappear atsome point. It would be a shame to lose all my old blog posts. Already, some of
my older posts suffer both from link rot and from latex issues related to changes in
how wordpress interacts with latex. There are a number of posts which I wrote that
contain arguments and remarks (and even Theorems) not proved anywhere else, and(as much for me as anyone else) I thought I would try to preserve them by collatingthem into a single file. It also makes it easier to search. I’ve restricted myself toposts with some mathematical content, with the exception of § 50 (OK, perhaps § 96
is not about mathematics either). I’ve included some of the comments, and I also
occasionally added notes which reflect any particularly relevant updates. I’ve madesome latex modifications (such as using theorem and conjecture environments) andI’ve also included a bibliography and put in citations (inconsistently, sorry) where
the blog simply referenced the paper either by name or by link, but otherwise
the posts are unmodified. (The conversion to latex may even have introduced newerrors.) My rate of posting has slowed over the years. There are two reasons. One,ironically, is that I started collaborating more and doing more mathematics, whichmeant more time was spent writing papers and less writing blog posts. The second
reason is explained later (at least obliquely). Still, I have every intention to continue
posting. This is intended to be an organic document which gets updated from time
to time as I keep posting, so please free free to point out errors or updates that canbe included in future notes.

Contents
1. Introduction
List of Figures
2. Even Galois Representations mod p
3. Hilbert modular forms of partial weight one, Part I
4. Why it is good to be pureThe author was supported in part by NSF Grant DMS-2001097.
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4
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F. CALEGARI

5. Remarks on Buzzard–Taylor
6. Jacobi by pure thought
7. There are no unramified abelian extensions of Q (almost)
8. Hilbert modular forms of partial weight one, Part II
9. The two cultures of mathematics: a rebuttal
10. Number theory and 3-manifolds
11. NT seminar: a haruspicy
12. Random p-adic matrices
13. Small cyclotomic integers
14. Torsion in the cohomology of co-compact arithmetic lattices
15. Galois representations for non self-dual forms, Part I
16. Galois representations for non self-dual forms, Part II
17. Inverse Galois problems I
18. Galois representations for non-self dual forms, Part III
19. Catalan’s constant and periods
20. Exercise concerning quaternion algebras
21. Equidistribution of Heegner points
22. Finiteness of the global deformation ring over local deformation rings
23. Scholze on torsion 0
24. Scholze on torsion, Part I
25. Scholze on torsion, Part II
26. Scholze on torsion, Part III
27. Scholze on torsion, Part IV
28. Effective motives
29. Life on the modular curve
30. Virtual congruence betti numbers
31. Abelian varieties
32. Local representations occurring in cohomology
33. Daleks
34. The mystery of the primes
35. Gross Fugue
36. Local crystalline deformation rings
37. The thick diagonal
38. The congruence subgroup property for thin groups.
39. Robert Coleman
40. Are Galois deformation rings Cohen–Macaulay?
41. A Preview of Barbados/Bellairs
42. A postview of Bellairs/Barbados
43. Thurston, Selberg, and random polynomials, Part I.
44. Thurston, Selberg, and random polynomials, Part II.
45. I don’t know how to prove Serre’s conjecture.
46. There are non-liftable weight one forms modulo p for any p
47. An obvious claim
48. Report from Luminy
49. A public service announcement concerning Fontaine–Mazur for GL(1)50. 100 Posts
51. The distribution of Hecke eigenvalues, Part I
52. The Abelian house is not closed
53. The distribution of Hecke eigenvalues, part II
54. Horizontal vanishing conjectures.
55. Is Serre’s conjecture still open?
56. K2 (OF ) for number fields F
57. The Artin conjecture is rubbish
58. The nearly ordinary deformation ring is (usually) torsion over weight space
59. Applying for an NSF grant
60. In brief
61. Mysterious formulae
62. Harris 60

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63. Derived Langlands
64. Stable completed homology without Quillen–Lichtenbaum
65. Higher direct images of canonical extensions
66. Abelian spiders
67. Inverse Galois problems II
68. H2 (ΓN (p), Z)
69. Review of Buzzard–Gee
70. Chenevier on the Eigencurve
71. 144169
72. Counting solutions to ap = λ73. Hilbert modular forms of partial weight one, Part III
74. Ventotene, Part II
75. Tensor products
76. Report from Berkeley
77. Central extensions and weight one forms
78. Prime divisors of polynomials
79. Serre 1: Calegari 0
80. Zp -extensions of number fields, Part I
81. Zp -extensions of number fields, Part II
82. Artin no-go Lemma
83. Correspondance Serre–Tate, Part I
84. Central extensions, updated
85. The class number 100 problem
86. Virtual coherent cohomology
87. A non-liftable weight one form modulo p2
88. Pseudo-representations and the Eisenstein Ideal
89. Who proved it first?
90. Elementary class groups updated
91. New results In modularity, Part I
92. New results in modularity, Part II
93. Schaefer and Stubley on class groups
94. Mathieu magic
95. Abelian surfaces are potentially modular
96. The ABC conjecture has (still) not been proved
97. Abandonware
98. The paramodular conjecture is false for trivial reasons
99. The boundaries of Sato–Tate, part I
100. Chicago seminar roundup
101. Update on Sato–Tate for abelian surfaces
102. Mazur’s program B on abelian surfaces
103. More or less OPAQUE
104. Irregular lifts, Part I
105. Irregular lifts, Part II
106. A strange continuity
107. Local-global compatibility for imaginary quadratic fields
108. Jacquet–Langlands and a new R = T conjecture
109. Jacquet–Langlands and an old R = T conjecture110. Jean-Marc Fontaine, 1944-2019
111. The stable cohomology of SL(Fp )112. I asked. . . and you responded!
113. Read my NSF proposal
114. A homework exercise for Oaxaca
115. Appropriate citations
116. En Passant VI
117. New results in modularity, Christmas update II118. The last seven words of Kedlaya–Medvedovsky
119. Vesselin Dimitrov on Schinzel–Zassenhaus120. Counting solutions to ap = λ, Part II

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121. NSF proposal, graduate fellowship edition242
122. More on Lehmer’s conjecture
243
123. Chidambaram on genus two curves, I
245
124. Chidambaram on genus two curves, II
249
125. Picard groups of moduli stacks
251
126. Picard groups of moduli stacks update
254127. Families of Hilbert modular forms of partial weight one.
255
128. Chidambaram on Galois representations (not) associated to abelian varieties257
129. Hire my students!
259
130. Ramanujan machine redux
260
131. Test Your Intuition: p-adic local Langlands edition
263
132. Potential automorphy for GL(n)264√133. Divisors near n
265134. 59281
267
135. Polymath proposal: 4-folds of Mumford’s type269
136. Schur–Siegel–Smyth–Serre–Smith
271
137. ArXiv × 3
275138. A random curve over Q
277
139. What would Deuring do?
279
140. Murphy’s law for Galois deformation rings and Ozaki’s theorem279141. Joël Bellaïche
281
142. Locally induced representations
283
143. The future is now; recap from Cetraro
284
144. Potential modularity of K3 surfaces
286
145. Check the arXiv regularly!
287
146. What the slopes are
288
147. Deciphering Quanta291
148. Quadratic reciprocity
295
149. Clozel 70, Part I
295
150. Clozel 70, Part II
297
151. Magma instability
300
152. The horizontal Breuil–Mezard conjecture
301
153. Unramified Fontaine–Mazur for representations coming from abelian varieties302
154. A talk on my new work with Vesselin Dimitrov and Yunqing Tang on irrationality 303
155. SLn versus GLn
306
References
310
List of Figures
1

Three consecutive frames: the daleks are all destroyed60

2

The occludation of P = (5, 3)

61

3

The chance of survival w∞ for dalek density ρ

624Upper and lower bounds: t5 ≥ t∞ = w∞ ≥ w563

5

Davros enjoying a cuppa

65

6

The chance d∞ of surviving with a TARDIS65

7The change of winning with the sonic screwdriver

66

8

The vanilla game at the optimal value ρ ∼ 0.517 — the Doctor lives!

66

9σα/α ∈ B(1) for Perron α ∈ [1, 5] which are roots of random monic polynomials withcoefficients in [−5, 5] ∩ Z

80

10σα/α ∈ B(1) for Perron α ∈ [1, 2] which are roots of random monic polynomials withcoefficients in [−5, 5] ∩ Z

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11

The distance of the Moon

215

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Number boxes

266

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The function Q(x) is positive

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5

14

Congruence modular forms (left) have additional structure that noncongruence modular
forms (right) lack291

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Geodesics through i with fixed cusp width

293

16

Which one is congruence?

293

2. Even Galois Representations mod p
Sun, 7 Oct 2012
Suppose that ρ : GQ :→ GL2 (Fp ) is a continuous irreducible Galois representation. What does the Langlands program say about such ρ? When ρ is odd, thesituation is quite satisfactory, the answer being given by Serre’s conjecture. For
example, having fixed a Serre weight k ≥ 2 and a Serre level N , one knows that
there will only be finitely many such representations and they will all come fromclassical modular forms for GL(2)/Q.
When ρ is even, however, there is an equally good (if conjectural) descriptionof such representations. First, the dihedral representations are well understood byclass field theory, so let us assume we are not in this case. Then, replacing ρ by
the adjoint representation ad0 ρ and restricting to some (any) imaginary quadratic
field, one obtains an irreducible (conjugate) self-dual representation, which, by thegeneralization of Serre’s conjecture [Ser87], should come from an automorphic representation for U (3). It follows that, as in the odd case, there will (conjecturally)
only be finitely many such ρ for a fixed pair (N, k). However, things are even better in the even case. Namely, if one fixes (N, k) but allows p to vary, then there
will still only be finitely many even representations, in contrast to the odd casewhere (for (N, k) = (1, 12) for example) such representations occur for infinitely
many p. The reason is that all such representations will have to arise from a fixed
finite dimensional space of automorphic forms determined by N and k, and thus
(by the pigeonhole principle) there will exist an automorphic Π for U (3) whose
mod p representation extends to an even representation of Q for infinitely many p.
By multiplicity one, it would follow that Π ≃ Πc ≃ Π∨ and hence Π itself mustcome from the adjoint representation of a form from GL(2) over Q, which would
imply (since we are in regular weight) that the representations are odd. Note that
it is important in the definition of Serre weight here that k ≥ 2; if one allows
k = 1 then there exist representations in characteristic zero which give rise to modp representations for all p.
Here’s a specific example in which one can prove finiteness. Suppose that we
consider representations with k = 2 and N = 1. Then there are no such even ρ
for a stupid reason, because the determinant will be cyclotomic (Tate deals withthe case p = 2.) Now consider the case when k = 2 and N = 4. In the evencase, the determinant must be the cyclotomic character times the unique (odd)character of conductor 4. Let’s prove that there are no such representations. Tate
like arguments reduce to the case when the representation has image containing√
SL2 (Fp ) and p > 7. Now take the auxiliary imaginary quadratic field to be Q( −1).
The corresponding adjoint representation now is unramified outside primes above p(the quadratic extension eliminating the ramification at 2) and is Fontaine–Laffaillewith weights [−1, 0, 1] at primes dividing p. Using the lifting results of [BLGGT14],
we may lift this to a compatible family of self-dual representations of level one

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and weight zero which is potentially modular. Because these representations arepotentially modular and are not CM, we know that they are all irreducible byBlasius–Rogawski. We now specialize these representations to√ p = 5, and because
the Hodge–Tate weights are sufficiently small ([0, 1, 2]) and Q( −1) is also small, we
can use results of Fontaine [Fon85] and Abrashkin to deduce that the corresponding
5-adic representation is reducible, which is a contradiction. We thus deduce (using
Khare–Wintenberger [KW09a, KW09b] for the odd case) that there√do not exist
any irreducible finite flat group schemes G of type (p, p) over Spec(Z[ −1]) whose
generic fibre admits descent data to Q. This entire argument is really just a versionof the Khare–Wintenberger proof of Serre’s conjecture for U (3). Unfortunately,
one doesn’t quite have enough modularity lifting theorems at this point to deduce
Serre’s conjecture completely for U (3).
These arguments are quite general. For example, there should only exist finitely
many even representations ρ : GQ :→ GLn (Fp ) (whose image contains SLn (Fp ))of fixed Serre weight and level, even when one ranges over all primes p, providingn ≥ 2.

3. Hilbert modular forms of partial weight one, Part I
Sat, 13 Oct 2012
Let π be an algebraic Hilbert modular cuspform for some totally real field F + .
Then, associated to π, one has a compatible family of Galois representations:rλ (π) : GF + → GL2 (Oλ )which are unramified outside finitely many primes (this is the work of many people).
The expectation is that this representation should satisfy local global compatibility
at all primes. This is known if π has regular weight, and also if π has parallel weight
one. However, this is not known, even for the case p ̸= ℓ (Here ℓ is the characteristicof O/λ). The problem is that these representations are constructed via congruences,
not from geometry. Deforming in families does give some control, and indeed one
can prove that, for v|p and p ̸= ℓ,
WD(rλ (π)|Gv )F-ss ≺ rec(πv )which is a way of saying you get the correct answer up to the monodromy operator N , and moreover the monodromy operator on the Galois side can only be moredegenerate than the automorphic side. In English, if (for example) πv is Steinberg,
then one may deduce (as expected) that the image of inertia on the Galois side is
unipotent, but not necessarily that it is non-trivial. In fact, by solvable base change,
this is really the only problem one has to worry about (so we shall assume we arein this case below).
The usual methods for computing the monodromy N are all geometric (nearby
cycles), and, as it seems hopeless to try to construct any (conjectural) motive associated to π, there doesn’t seem to be much one can do.
One does, however, have the following strategy, which I learnt from Martin Luu,
which should suffice for all but finitely many primes λ for which rλ (π) is ordinary.
Namely, take the λ-adic Galois representation associated to π, and prove that it
is potentially automorphic using extensions of the Buzzard–Taylor idea (which has
been employed by Sasaki, Kassaei, Pilloni and others in the case of Hilbert modular

PERSIFLAGE: MATH BLOG POSTS

7

forms of parallel weight one, but should also apply in this context). The result is thatone shows that rλ (π)|GE + for some totally real extension E + /F + is now associatedto a cuspidal automorphic form Π of the right level. How does this help? Well, now
using what we know from local global compatibility (which is ok in the unramifiedcase), we deduce that Πw for some w|v is associated to the corresponding local
Galois representation rλ (π)|Gw . Now this representation has the property that it
looks unipotent on inertia mod λn for all n, but, assuming local-global compatibility
fails, is actually unramified at p. In particular, the semi-simplification is given by
two characters whose ratio is the cyclotomic character, whereas Πw is an unramified
principal series. This implies that the Satake parameters {αw , βw } satisfy αw /βw =N (w), which contradicts Ramanujan. We are not done yet, because one doesn’t
have purity in partial weight one. However, one can appeal to bounds coming from
Rankin-Selberg, and this is enough to obtain a contradiction.
The only obvious examples of partial weight one HMF (which are not of parallel
weight one) are CM, and since those are potentially unramified, the monodromy
operator will always be trivial on the autormophic side (and hence also on the Galoisside). So this suggests (but does not beg) the question: do there actually exist any
partial weight one (but not parallel weight one) Hilbert modular forms which are
not CM? Stay tuned for part II!

4. Why it is good to be pure
Mon, 15 Oct 2012
There do not exist any regular pure motives M over Q which are not essentiallyself dual. Here is why. M gives rise to a compatible family of Galois representations
for each rational prime v such that the characteristic polynomial R(X) of Frobenius
is independent of this choice. By purity, the eigenvalues α of R(X) are algebraic
integers lying in a CM-field such that |ια|2 = pw for some integral weight w andany complex embedding ι. In particular, if α is a root of R(X), then αc = pw /α isa root of X n R(pw /X). Since R(X) has coefficients in Z, it follows that αc is also
a root of R(X), from which one may deduce that R(X) = X n R(pw /X) (up to theappropriate constant which makes the RHS monic, this doesn’t affect any of thearguments). Yet this implies that M ∨ (w) ≃ M , by the Cebotarev density theorem.
(Caveat: it really says that the p-adic avatars of M are essentially self-dual. Perhaps
deducing the result for M actually requires the standard conjectures.)This argument no longer applies if one relaxes the conditions slightly; there doexist non-self dual motives of rank three with coefficients; Bert van Geemen andJaap Top [vGT94] found some explicit examples with coefficients in an imaginary
quadratic extension of Q. The point where the argument above fails is that it identifies the polynomial X n R(pw /X) with the complex conjugate polynomial Rc (X),
which need not equal R(X) anymore.
Stefan Patrikis and Richard Taylor use a similar argument in their recent paper [PT15] to prove a nice result. Start with a regular pure motive M over Q (soby the above remarks, it is essentially self dual). Suppose that the corresponding
v-adic Galois representation:
rv : GQ → GLn (Qv )8

F. CALEGARI

is not absolutely irreducible. One may ask: are the irreducible constituents sv themselves essentially-self dual? They show that the answer is yes. Let S(X) denote thecorresponding characteristic polynomials. If S(X) lies in Z[X], then the same argument above applies to sv . But it may be the case that the representation rv
only decomposes over an extension of Q. By looking at the eigenvalues, it triviallyfollows that each of the S(X) may be defined over some CM field F/F + . More importantly, by a technical argument which I will omit but which is not too difficult,one may find a fixed CM field M/M + which contains all the polynomials S(X)(one may even do this [in some sense] independently of v, although we won’t usethat here). Consider the Galois representation (sv )c , where c is acting on the coefficients. Let α be a root of S(X). Then αc = pw /α is now a root of X m S(pw /X),and so S c (X) and X m S(pw /X) coincide. Since X n R(pw /X) = R(X), we deducethat (sv )c is a sub-representation of (rv )c = rv . In particular, (sv )c and sv areboth sub-representations of rv . But the Hodge–Tate weights of sv and (sv )c are thesame! (Literally, the Hodge–Tate weights of (sv )c are the Hodge–Tate weights ofc(sv ) where c (sv )(g) = sv (cgc−1 ), but since sv is a representation of Q, conjugationby c is conjugation by a matrix, so there is an isomorphism sv ≃ c (sv ).) It follows(from the regularity assumption) that sv = (sv )c , and then the argument above
implies that sv is self-dual.
One may use this argument as follows. As in [BLGGT14], one may find a prime v
such that all of the sv are residually irreducible, and so (if v is sufficiently large) arealso potentially modular (by [BLGGT14] again). In particular, either all of the rv
are reducible or they are irreducible for a set of density one set of primes. Moreover,any regular motive over Q is potentially modular, which is only three adjectives
away from the complete reciprocity conjecture!
Patrikis and Taylor do something slightly more general, instead of pure regularmotives over Q, they consider essentially self-conjugate regular compatible systems(with coefficients) of GF for some CM field F/F + . For reasons alluded to above,the coefficients live in some CM-field M . This extra generality (mostly) adds somenotational complexity to the argument above. (To see the type of complications
that arise, consider an elliptic curve E with CM and then restrict to the CM field
F . Then any reducible constituent sv = χv is related not to its complex conjugate
χcv acting on M , but the complex conjugate c χcv of this where complex conjugation
is now acting on the coefficients M and on the Galois group F .) As expected, one
obtains (using [BLGGT14]) some nice consequences, like potential automorphy of
regular polarizable compatible systems, as well as irreducibility (for a density one
set of primes) of Galois representations associated to RAESDC automorphic formΠ.
Comment 4.1 (Persiflage). Regularity is not used anywhere in the first result, so
the argument applies to all motives with coefficients in Q. As a sanity check, if χ
is a character of a finite group with values in Q, then the dual character is χ = χ,so χ is self-dual. Indeed, this is basically the same argument (in weight w = 0.)

5. Remarks on Buzzard–Taylor
Thu, 18 Oct 2012

PERSIFLAGE: MATH BLOG POSTS

9Let ρ : GQ,S → GL2 (Qp ) be continuous and unramified at p. The Fontaine–
Mazur conjecture predicts that ρ has finite image and is automorphic. Buzzard
and Taylor proved this result under the assumption the natural assumption that
ρ is odd, that ρ is modular (now uneccessary), but also under the further two
assumptions:(1) ρ is irreducible,(2) ρ is p-distinguished.
(caveat: there are mild extra assumptions required when p = 2.) The point of
this post is to note that is seems possible to remove either of these conditions andto wonder whether both can be removed simultaneously.
Suppose the second condition fails. One may enlarge the ordinary Hecke algebrae After appropriate localizaT to include the operator Up , call the resulting ring T.
tions, this is different from T exactly when ρ(Frobp ) is a scalar. On the Galois side,
let Rp denote the universal (framed) deformation ring. Then, for any lift of Frobeep = Rp [α] where α is an eigenvalue ofnius, one can define the quadratic extension R
ord
Frobenius. Fix a weight k ≥ 2. If Rp denotes the ordinary deformation ring, thenep which records ordinary deformations
epord of R
there is a corresponding quotient R
together with the action of Frobenius on the unramified quotient. The Qp -pointsof these local deformation rings are the same (since k ≥ 2). The usual Taylor–eeWiles-Kisin method produces an isomorphism of the type R[1/p]= T[1/p],wheree
R is the global deformation ring which takes into account the extra data at p. This
isomorphism holds for all weights k ≥ 2, which is enough to get an isomorphism on
the corresponding ordinary families.
If ρ(Frobp ) has distinct eigenvalues, one may now deduce Buzzard–Taylor (bythe same argument as BT). If ρ(Frobp ) is scalar, then one has to make a slightadjustment. To see what to do, note that if f (τ ) is the desired weight one form,
then the old space f (τ ), f (pτ ) can no longer be diagonalized with respect to Up .
e → O[ϵ]/ϵ2 such that the image
Instead, it should give rise to a surjective map ψ : T
of T is O. Conversely, if there is such a map ψ, this produces the ordinary old forms
necessary to recover f (both forms have the same Hecke eigenvalues away from p,
so one can determine the q-expansions). From the modularity result, it suffices toconstruct such a map on the Galois side. Yet this exists precisely because ρ comeswith two distinct unramified quotients.
David Geraghty and I used these these flavours of deformations rings for a somewhat different purpose (although we required more precise integral information
epord coming out of a very nice paper of Snowden [Sno18]). On the otherconcerning R
hand, as far as the argument above goes, it was apparently known to Richard many
years ago (as I learnt by chatting with Toby Gee whilst drinking an $8 can of
Boddingtons in Toronto).
Suppose one assumes instead that ρ is reducible. Recall that one has maps Rps →R and Rps → T for a suitable pseudo-deformation ring Rps . In higher weights,
Skinner-Wiles essentially prove that R[1/p]red = T[1/p], which should be sufficient
to construct the required overconvergent forms fα and fβ in weight one. While
chatting with Patrick Allen over espresso today, it also seems reasonable that (using
appropriate framings, as above) one may generalize this to the case where ρ is nolonger p-distinguished, as long as the characteristic zero eigenvalues α and β are
distinct. The problem, however, with the α = β case is that one needs to promote a

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F. CALEGARI

non-reduced quotient R → O[ϵ]/ϵ2 to a map from T, and the methods of SkinnerWiles in the reducible case only give information about the reduced quotients of
R. Is there any way around this? This seems (pretty close) to the only remaining
obstruction for a complete solution to the weight one odd case of Fontaine–Mazur.

6. Jacobi by pure thought
Fri, 26 Oct 2012Joël Bellaïche asks here whether there is a conceptual proof of Jacobi’s formula:∞Y∆=q(1 − q n )24
n=1

Here (to me) the best proof is one that requires the least calculation, not necessarily
the “easiest.” Here is my attempt. We use the following property of ∆, which follows
from its moduli theoretic definition: the only zero of ∆ is a simple zero at the cusp,
moreover, the evaluation of ∆ on the Tate curve is normalized so that the leading
coefficient is q. Let p be prime. I claim that
p−1
p−1
Y
Y τ + i.
∆(τ )p+1
ζ i = ∆(pτ )
∆
p
i=0
i=0
Observe that both these expressions are modular forms of level one and weight(p + 1) times the weight of ∆. One can prove this “by hand,” but also by notingthat the RHS is equal to the norm of ∆(pτ ) on X0 (p) down to X0 (1). On theother hand, the RHS also has a zero of order p + 1 at q = 0, from which the result
immediately follows, since the ratio will be holomorphic of weight zero. If one defines
Hecke operators on q-expansions in the usual way, it also immediately follows thatthe logarithmic derivative qd/dq log(∆) is (as a q-expansion) an eigenform for Tp
of weight two with eigenvalue p + 1 for all primes p. In fact, the same argument
as above (with X0 (p) replaced by X0 (n))
Ximplies that this derivative is also an
eigenform for Tn with eigenvalue σ1 (n) =d. This is almost enough to determined|nthe q-expansion uniquely: in particular, it implies that∞Xdq·log(∆) = 1 + m ·σ1 (n)q ndq
n=1
for some integer m, from which it follows that∞Y∆=q
(1 − q n )m .
n=1

To finish the argument, it suffices to check that m = 24, or that τ (2) = −24. One
way to do this is to note (by uniqueness) that ∆ is a Hecke eigenform, and then use
the equation τ (2)τ (3) = τ (6) which implies that m ∈ {0, 1, 2, 3, 24}; the cases m =1, 2, 3 are then ruled out by the equations τ (2)τ (7) = τ (14) and τ (2)τ (13) = τ (26),and m = 0 is ruled out by the fact that q is not a modular form. Curiously enough,this determines ∆ without ever using the fact that it has weight 12. Another (moretraditional way) is to show that 1728∆ = E43 − E62 = q − 24q 2 + . . .. Is there a way
to do this final step by pure thought?

PERSIFLAGE: MATH BLOG POSTS

11

Comment 6.1 (Emmanuel Kowalski). This argument is very similar to one by
Kohnen [Koh05].

7. There are no unramified abelian extensions of Q (almost)Tue, 27 Nov 2012
In my class on modularity, I decided to explain what Wiles’ argument (in theminimal case) would look like for GL(1)/F . There are two ways one can go withthis. On the one hand, one can try to prove (say) Kronecker-Weber using Selmer
groups, but avoiding any kind of circularity (by not assuming class field theory).
On the other hand, one can allow oneself to be completely circular in an effort toconcentrate on the technical details of Wiles’ arguments. This post concerns thelatter, and we “prove” the following:
Theorem 7.1. Let F be a number field which does not contain ζp . Then the Galois
group of the maximal abelian extension of F unramified everywhere is isomorphic
to the p-part of the class group.
To prove this, we will (only) assume the following:(1) Local class field theory.
(2) For any ray class group of F , there exists a corresponding abelian ray classfield whose Galois group is the ray class group (this is half of global classfield theory).
(3) The abelian extensions coming from the ray class group are compatible
with local class field theory (this is local-global compatibility).
(4) The Wiles-Greenberg Selmer group formula.
The first three assumptions for F = Q are equivalent to giving oneself thecyclotomic extensions and understanding their ramification properties. The lastassumption, of course, contains every part of global class field theory (making theargument circular).
Let Γ∅ denote the Galois group of the maximal pro-p abelian unramified extension of F . Let ΓQN denote the corresponding group where ramification is allowed
at some set of primes QN not containing p, where one also insists that the order ofinertia at primes in QN is at most pN . Formally, we have the universal deformationrings
R∅ = Zp [[Γ∅ ]],RQN = Zp [[ΓQN ]],and we also have the universal “modular” deformation ringsFT∅ = Zp [[(F × \A×
F /U ) ]],F
TQN = Zp [[(F × \A×
F /UQ ) ]].

Here M F denotes the biggest finite quotient of M , U is the obvious maximal open
compact, and UQ is the variant of U such that Uv = Ov× is replaced by UQN ,v =N
Ov×p for v ∈ QN . The half of global class field theory we are assuming gives us
a compatible diagram of maps RQN → TQN and R∅ → T∅ . The Wiles–Greenberg
formula gives us an equality:
dim |H∅1 (F, Fp )| − dim |H∅1∗ (F, Fp (1))| = −(r1 + r2 − 1),
where for this computation we use that ζp ∈/ F . In order to annihilate the dual
Selmer group, √we need to annihilate classes in H 1 (F, µp ), which come from extensions F (ζp , p α). We can do this in the usual way, but we have to assume

12

F. CALEGARI(again) that ζp p∈
/ F , since otherwise one cannot annihilate the class defined overF (ζp2 ) = F (ζp , p ζp ) using a prime q ≡ 1 mod p2 . We see that we can annihilate
the dual Selmer group with q := |QN | = dim H∅1 + (r1 + r2 − 1) primes. What are
the auxiliary rings SN here? As rings, they are
SN = Zp [(Z/pN Z)q ]the action on RQN is via the inertia group at the auxiliary primes. To make thiswork, one needs local class field theory; this shows that inertia at q is acting via
N
Oq× /Oq×p . The action of SN on TQN is given by the structure of TQN as a module
over Zp [U/UQ ] ≃ SN . The compatibility of these actions is given by the compatibility of local and global class field theory. Moreover, if aN is the augmentation
ideal of SN , then RQN /aN = R∅ by definition, and TQN /aN = T∅ by construction.
Thus, in the usual way, one ends up with a map R∞ → T∞ where:(1) R∞ is a quotient of a power series ring with q − (r1 + r2 − 1) variables.
(2) S∞ is a power series ring in q variables.
The final thing to understand is the structure of T∞ as a module over S∞ .
At level QN , the annihilator in SN ≃ Zp [U/UQ ] of TQN is given by the image ofthe global units. By Dirichlet’s Theorem, this is generated by at most r1 + r2 − 1generators (assuming again that ζp ∈/ F ). By patching, it follows (in the limit) thatT∞ has co-dimension at most r1 + r2 − 1, and thus (from dimension considerations)that T∞ = R∞ , and then (after taking the quotient by a∞ ) that R∅ ≃ T∅ , which
proves that Γ∅ is the p-part of the class group. Note that (as expected) whengluing, we need to take into account all the (finitely many) possible R/mN , T/mN ,the possible maps from the global units to SN , etc. etc.
In order to see the “circularity” more clearly, one may compute the Selmer groups
directly. The group H∅1 (F, Fp ) is equal to Γ∅ /p, by definition. On the other hand, thegroup H∅1∗ (F, µp ) by the Kummer sequence is equal to O× /O×p ⊕Pic0 (Spec(OF ))[p],and thus the Greenberg-Wiles formula is equivalent to the equality:|Pic0 (Spec(OF ))[p]| = |Γ∅ /pΓ∅ |or equivalently the claim that the maximal exponent p-quotient of the class group
captures all exponent p-unramified extensions. I guess this is very very slightly
weaker than Γ∅ ⊗ Zp ≃ Pic0 (Spec(OF )) ⊗ Zp .

8. Hilbert modular forms of partial weight one, Part II
Sat, 08 Dec 2012
Anyone who spends any time thinking about Hilbert modular forms of partial
weight one — see § 3 — should, at some point, wonder whether there actually exist
any examples, besides the “trivial” examples arising as inductions of Grossencharacters. Fred Diamond asked me this very question at Fontaine’s birthday conferencein March of 2010. There are various reasons why one should not expect to prove thisby pure thought, including the possibility that (for certain levels) there may exist
no such forms, and that at any level there may exist only finitely many such forms(more on these heuristics another time). Thus the only way I can really imagine
showing that such a beast exists is by explicitly finding an example.
As of today, my students Richard Moy and Joel Specter have found sucha form! Here is (roughly) the strategy they use. As with computing weight one

PERSIFLAGE: MATH BLOG POSTS

13

classical modular forms, one starts by computing a basis of q-expansions in someregular weight, divides by some Eisenstein series, takes the intersection of thatspace with its Hecke translates, and hopes that the resulting space has bigger dimension than the space of CM forms (which one can compute in advance). There
are a few hiccoughs which occur along the way, of course. How does one compute qexpansions of Hilbert modular forms? Since computing uniformizations of surfaces
is not realistic, they use the fact that (fortunately!) the q-expansion of a Hilbert
modular form can be recovered from its Hecke eigenvalues. On the other hand, by
Jacquet–Langlands, a Hilbert modular eigenform over (say) a real quadratic fieldcorresponds to an eigenform on the arithmetic manifold associated to a quaternion
algebra which is ramified at all infinite places, which then allows one to pass fromthe Hilbert modular variety to an adelic quotient which is now a finite set. Lassina
Demebele wrote a magma programme which computes the eigenvalues for Hilbert
modular eigenforms by this method, although for some reason the programme requires the level to be squarefree, and the character to be trivial. Using Atkin-Lehnertheory, one can construct the entire space of forms by √this method.
In practice, Richard and Joel worked with F = Q( 5), computed the forms oflevel Γ0 (N ) (with N squarefree) and weight [4, 2], then divided by an Eisenstein
Series of weight [1, 1] level Γ1 (N ) and character χ−1 , then computed the Hecke
operator T2 on this space and intersected away. Many (many) bugs later, and various
annoying steps overcome (to take a random example, magma can compute the Lvalues of Hecke characters necessary to find constant terms of Eisenstein series[nice] but only as a complex number, not as an algebraic number [not so nice] so“L-value recognition” had to be coded in), the progams finally worked, and after
much grinding away (for all squarefree N of norm less than 500) they didn’t findanything at all (or at least, anything besides CM forms).
So they started working in weight [6, 2], computed away, and eventually found aform π of weight [5, 1], level Γ1 (14), and character χ, where χ has √conductor√√7 andis of order 6. The coefficient field of the eigenform is, I believe, Q( 5, −3, −19)
(note that it must contain the base field as well as the field of the character). Notethat this automorphic form π is Steinberg at 2! In particular, it is not CM, and one
doesn’t know whether local-global compatibility holds for the corresponding p-adic
Galois representations even restricted to 2.
I should say that finding the form actually turned out to be easier than proving
the form exists rigorously. Theoretically, the proof should be easy: one has founda form F/E for some cuspform F and some Eisenstein series E which looks like
it is holomorphic. All one needs to do is square it (so it becomes regular), find a
candidate form G of weight [10, 2] such that GE 2 − F 2 = 0 (which one can provesince the spaces are finite dimension), and then E/F has no poles and is thus
holomorphic. The problem is that the form [10, 2] has non-trivial character, andLassima’s program only works with trivial character. One can take the 6th power
and work with a form of weight [30, 6], but this is way beyond what magma cancope with. In the end, Richard and Joel had to come up with a few tricks to do
this (which took about three months!), but the final computations are in, and theexistence has now been proven.
Comment 8.1 (Akshay Venkatesh). That’s a very nice computation indeed. Did
they look mod p? Is there really an obstruction to using Hecke action to verify thatthey have no poles (as in Schaeffer’s thesis), or it just seemed annoying to prove?

14

F. CALEGARI

Comment 8.2 (Persiflage). Dear AV, Some good questions. First, they did notcompute anything mod p, and one reason is that it is not clear whether one cancompute the integral structure of the module of Hilbert modular forms in regularweight (at least not obviously); the only thing one can compute are the eigenvaluesof eigenforms, and this only tells you about the rational structure.
As for Schaeffer’s approach, it’s not obvious how to prove that T F = λF inpractice (even for a fixed T ) since this runs in to the same computational issues
related to the fact that magma only computes spaces without character. There also
does seem to be a genuine issue with applying Schaeffer’s ideas for operators T
dividing the level; we thought about this for a while, but couldn’t quite make it
work.
Notes 8.3. The corresponding paper is [MS15]. One frustrating aspect of thisexample was that the base field was strictly bigger than the field generated by F
and the value of the character; it would have been nice to find a non-CM form
defined over F . However, my student Abhijit Mudigonda pointed out to me thatthe classical argument that any modular form of odd weight and coefficients in Q
also applies here to show that there are no such forms (more precisely, forms inweight [1, 2k + 1] defined over F must be CM).

9. The two cultures of mathematics: a rebuttal
Wed, 12 Dec 2012
Gowers writes thoughtfully about combinatorics here, in an essay which references Snow’s famous lectures (or famous amongst mathematicians — I’ve never
met anyone else who has ever heard of them). The trouble, however, starts (as it
often does) with the invocation of the word “obvious”:
It is equally obvious that different branches of mathematics requiredifferent aptitudes.
I do not think that this claim stands up to scrutiny. By “aptitude,” Gowers
specifically distinguishes the following two abilities: problem-solving and theorybuilding. Here algebraic number theory is singled out as area which is firmly tilted
towards theory-builders. Yet the vision of algebraic number theory as a rising sea
with progress signaled by the application of (to quote Gowers) deep theorems ofgreat generality is not, in my opinion, an accurate reflection on reality.
A good lemma is worth a thousand theorems. Gowers describes various principles
of combinatorics which (he suggests) play the role of (a direct quotation again)
precisely stated (general) theorems. Yet examples similar to his are readily available
in algebraic number theory. Consider, for example, the following Lemma of Ribet(modified from its original formulation):
If a reducible representation U ⊕ V of a group G deforms continuously into an irreducible representation of G, then either there
exists a non-trivial extension of U by V , or an extension of V byU.
As a mathematical result, this is not particularly deep. For example, if G is finite,
it relates two well known facts: there are no extensions between irreducible representations (Maschke’s theorem), and representations of finite groups are definedPERSIFLAGE: MATH BLOG POSTS

15

over number fields (and so do not deform). Yet this lemma is a crucial ingredientbehind many key results (Ribet’s construction of unramified extensions, the proofof the main conjecture of Iwasawa Theory by Mazur–Wiles, the non-triviality of
the Selmer group of an ordinary Elliptic curve with L(E, 1) = 0 by Skinner–Urban,
and many more). It seems to me (as in the examples Gowers discusses) that thevalue of this lemma is not in its difficulty, but in the principle it encapsulates: inorder to construct extensions of U by V , try to deform U ⊕ V .
It is a common graduate student error to imagine that mathematics consists
merely of judicious applications of highly technical machinery. But I am not accusing Gowers of making this mistake; I would expect him to argue that algebraic
number theory is not exclusively the domain of theory-builders, but rather only
strongly slanted in that direction, and to that end, he might point to Grothendieck.
A fascinating essay on Grothendieck may be found here. Grothendieck contrasts his
own way of thinking with that of Serre, whom he describes as using the hammer
and chisel approach, which might loosely be considered synonymous to “problemsolver” (and I would count Serre as someone who has worked in algebraic number
theory). Note that, despite the merits of Grothendieck’s work, he famously failed
to prove the Weil conjectures (by “failing” to prove the standard conjectures) andit required Deligne’s use of the tensor power trick (a problem solving techniquepar excellence) to finish the argument. Thus, while Grothendieck’s role in modern
number theory is significant, it would be an error to imagine that it constitutes thewhole subject.
Perhaps Gowers would instead argue that what set combinatorics apart from(say) algebraic number theory is not that it requires problem solvers while thelatter field does not, but that (in contrast) it is the exclusive domain of problem
solvers. There’s a hint of this opinion in the following quote:One will not get anywhere in graph theory by sitting in an armchair
and trying to understand graphs better.
Why is this claim any more convincing than the same statement with the word
“graphs” replaced by the word “rings”? I don’t see any a priori reasons why there
cannot be a Grothendieck of graphs. If the history of mathematics teaches us anything, it is that the nature of a subject can change quite radically over a relatively
short period of time (say 30 years). I am not claiming that there is no difference
between combinatorics an algebraic number theory. There may well be a differencein the overall structure of the field, the level of background, the need to understandideas in a broader conjectural framework, etc. And I might also consider agreeing
to the claim that these fields, as they are currently constituted, may well be better
suited to different personalities. But it is my opinion that the divide between thetype of mathematics required for either subject is not as great as Gowers claims it
is.
Gowers main point is that a significant part of the mathematical establishment
looks down on combinatorics as not being “deep”, and that this attitude is both
harmful and ignorant. On this point, I think that Gowers criticisms are fair, accurate, and valuable. It’s undeniably true that there are many graduate students
who fall in love with formalism to the detriment of content, and milder forms ofthis predujice are pervasive throughout mathematics. To this end, I think Gowers’
essay is timely and relevant. However, I can’t help but sense a little that, perhaps
after having spent a career defending combinatorics against ignorant snobs, Gowers

16

F. CALEGARI

suffers from the opposite prejudice, where “theory-builders” are a short distanceaway from empty formalists, sitting comfortably in their armchairs thinking deep
thoughts, studying questions so self referential that they no longer have any application to the original questions which motivated them (this sense also comes fromreading some of the remarks on the Langlands programme here).

10. Number theory and 3-manifolds
Sun, 13 Jan 2013
It used to be the case that the Langlands programme could be used to saysomething interesting about arithmetic 3-manifolds qua hyperbolic manifolds. Now,
after the work of Agol, Wise, and others (see [Ago13]) has blown the subject tosmithereens, this gravy train appears to be over. It seems to me, however, thatthe great advance in our knowledge of hyperbolic 3-manifolds has precious little
to say about arithmetic 3-manifolds qua lattices in semi-simple groups. As a basic
example, suppose that X is a maximal compact arithmetic three orbifold associatedto a quaternion algebra Q/F for some field F (with the appropriate behavior at
the infinite primes). Then one may ask whether X has positive Betti number aftere → X. Let’s call this the virtual congruence positive
some finite congruence cover X
Betti number conjecture. (This conjecture should be true — it is a consequence ofLangland’s conjectural base change for SL(2), which everyone believes but is probably very difficult.) AFAIK, there’s not really much one can say about this problem
from the geometric group theory/RAAG/LERF/etc perspective, where the arithmetic structure of the tautological SL(2)-representation does not seem to play somuch of a role. A related question is the extent to which arithmetic 3-manifolds
are intrinsically different from their non-arithmetic hyperbolic brethren. Is the virtual congruence Betti number conjecture (for arithmetic manifolds) something that
could plausibly answered using geometric group theory?
Notes 10.1. I think the short answer is that it seems unlikely.

11. NT seminar: a haruspicy
Fri, 18 Jan 2013
Following Jordan Ellenberg’s advice, I will blog on something that I know absolutely nothing about. Apologies in advance for mathematical errors!
Simon Marshall gave a number theory seminar this week about the first Betti
number of Γ(n) — as n varies — for certain lattices in SU(2, 1). In particular, he
proved an upper bound of the form:
dim H 1 (Γ(n), Q) ≪ [Γ : Γ(n)]3/8+ϵ ,which turns out (in certain cases) to be essentially the best possible estimate. As
was known to Rogawski, the forms contributing to H 1 all arise via endoscopy.
In particular, if Γ is simple in the sense of Kottwitz, then the first cohomology
vanishes (this also is due to Rogawski). So assume we are not in that case. The
argument proceeds mostly as one would expect: Rogawski classifies the endoscopic
forms which contribute to cohomology — they come from certain representations
ξ × µ for U (2) × U (1). Here I think the choice of Grossencharacter µ is almost

PERSIFLAGE: MATH BLOG POSTS

17

determined by ξ, so I will drop it from the notation below. The possible packets
can be described as follows:(1) Singletons for the split primes.
(2) A set {J + , D− } for the interesting infinite prime, where J + contributes (via(g, K) cohomology) to H 1 and another representation D− which doesn’t(although it contributes to H 2 , I think).
(3) A set {πs , πp } consisting of a supercuspidal representation and anotherrepresentation at the inert primes.
(4) Something similar to 3. for the ramified primes.
Using Matsushima’s formula, in order to count the contribution to cohomology
one has to deal with the following:(1) The global multiplicity: this is either 1 or 0 depending on certain signs
related to epsilon factors. As one varies n this should vanish half the time,
but one can ignore it as far as an upper bound goes.
(2) Suppose that p divides n, and let K be a hyperspecial maximal compact at
p. Then one has to bound the trace of the characteristic function of K(pk )on the representations πs and πp .
Let f be such a characteristic function. One would like to write down a corresponding transfer function f H on the endoscopic group such that:
Tr(πs , f ) + Tr(πt , f ) = Tr(ξ, f H )By the Fundamental Lemma, if f is the characteristic function of the hyperspecial K itself, then f H turns out to be the characteristic function on the maximal
compact of U (2). SML shows that (using some of the same computations required
for the fundamental lemma for U (3)) the same identity holds for the corresponding characteristic function for K(pn ), that is, the transfer f H is the characteristic
function of U (2)(pn ). Is this true for any deeper reason? More generally, to what
extent do characteristic functions transfer to characteristic functions?
Notes 11.1. Simon’s paper is [Mar14]; see [GG23] for more recent developments.

12. Random p-adic matrices
Wed, 23 Jan 2013
Does anyone know if the problem of random matrices over (say) Zp have beenstudied? Here I mean something quite specific. One could do the following, namely,
since Zp is compact with a natural measure, look at random elements in MN (Zp )and then ask about the distribution of several obvious quantities as N goes to∞. For example, one can consider the rank of M mod p, which translates into
an elementary counting problem over Fp . However, I don’t mean this, that would
just be rubbish for my purposes. What I am looking for is something that models
a random compact operator, and then I want to understand the behavior of thenormalized eigenvectors as the eigenvalue λ → 0. To be concrete, let B = Qp ⟨T ⟩be the Tate algebra corresponding to the open unit ball. Then consider a “random”
compact operator U acting on B. What does random mean? This is a good question,
to which I do not know the answer. But let me give several properties that it should
satisfy. Because the ball B is a disk, it is “dimension 2 as a real manifold”, and so

18

F. CALEGARI— imagining that our compact operator is a p-adic avatar of e−∇ for the Laplacian∇ — the eigenvalues of U should satisfy Weyl’s Law:
Vol(B)· T.
4π
Here v(λ) denotes the valuation of λ ∈ Qp . Ignoring the volume factor, this justmeans that the Fredholm determinant det(1 − U T ) has a Newton Polygon withcertain quadratic growth. I’m not sure exactly what ensembles one can come up withto define such operators, which is one of my questions. Let us also assume, although
this may not be necessary, that U is semi-simple and admits nice convergent spectral
expansions. We can’t quite insist that U is a self-adjoint operator, because one
doesn’t have p-adic Hilbert spaces. For such an operator, what behavior should one
expect of the normalized eigenvalues ϕj of U ? For example, suppose one knows thatthe number of zeros of ϕj goes to infinity. What limit distribution should the zerosof ϕj satisfy when λ → 0? (Somewhat troubling here is that the eigenvalues will lie
in Qp in general and Cp has compactness issues. . . )As you might guess, this is related to p-adic arithmetic quantum CHAOS, agroup of subjects which gets sexier every time an extra adjective is added, and will
form part of my student project at the Arizona Winter School (see here).
N (T ) := {#λ ∥ − v(λ) ≤ T } ∼13. Small cyclotomic integers
Sat, 26 Jan 2013
Julia Robinson is a famous mathematician responsible for fundamental work inlogic and in particular on Hilbert’s Tenth problem. Less well known nowadays is that
her husband, Raphael Robinson, was a number theorist at Berkeley. One question
R.Robinson asked (see [Rob65]) concerned small cyclotomic integers. Namely, let α
be a cyclotomic integer, and suppose that every conjugate of α has absolute value atmost R. Then what can one say about α? If R ≤ 1, then Kronecker’s theorem saysthat α is a root of unity (this statement only requires that α is an algebraic integer).
√
Robinson studied the problem of what happens when R ≤ 2 and also R ≤ 5. He
made five conjectures concerning these questions, four of which were solved in the
60’s by Jones, Cassels, and Schinzel. Five decades later, Frederick Robinson (norelation!) and Michael Wurtz proved the last of these conjectures (while working
with me as summer students), and their paper paper has just been accepted by
Acta Arithmetica (see [RW13]). In particular, they answer the following √problem:
if α is an algebraic integer the largest of whose absolute values is R ≤ 5, thenwhat are the possible values of R? Two such families of such numbers are those of
the formζ + ζ −1 ,i + ζ + ζ −1
for a root of unity ζ. These give all R of the formp
2 cos(π/N ),1 + 4 cos2 (π/N ).
√√Note that these sets have limit points at 4 and 5 respectively. It turns out that
there exactly two further exceptions, as follows:
s
√
√
√
3+ 7
5 + 13,
2
2

PERSIFLAGE: MATH BLOG POSTS

19

The first element is totally real and cyclotomic, and so manifestly occurs as such
4
an R. The second turns out to be the absolute value of 1 + ζ13 + ζ13. The proof by
Robinson and
Wurtz
actually
appliestoslightlylargervaluesofR,and after the√limit point 5 there is another gap, and the next smallest possible R is√1029
|1 + ζ70 + ζ70+ ζ70| ∼ 5.017655 . . .
The first two exceptional numbers turn up in relation to subfactors. How about thelast example?
Notes 13.1. Kiran Kedlaya has pointed out to me that one of Robinson’s fiveproblems remains open, but should also be solvable by the methods of the Robinson–
Wurtz paper [RW13]. (It relates to the difference between finding all possible valuesof α and finding all β with β = α .)14. Torsion in the cohomology of co-compact arithmetic lattices
Wed, 06 Feb 2013
Various authors (including Bergeron and Venkatesh) have shown that the cohomology of certain arithmetic groups have a lot of torsion. For example, if Γ is aco-compact arithmetic lattice in SL2 (C), and L is an acyclic local system, thenlog |H ∗ (Γ(N ), L)| ≫ [Γ : Γ(N )].
The proof relies on the fact that the difference l0 in ranks of SL2 (C) and SU2 (C)is one. As the invariant l0 grows, one expects there to be less torsion. How muchtorsion should one expect in general? I’m not sure I have an answer, but the point of
this post is that Poincare duality gives a non-trivial bound, at least if one restricts
to covers up a p-adic tower. Let G be a semi-simple
group over Q, Let G = G(R),L m
let K be a maximal compact, let H ∗ =H , let Γ be a co-compact lattice, andlet L be an acyclic local system. Suppose that n = dim(G) and d = dim(G/K).
Then, for a fixed prime p (for which G(Qp ) is split) and varying m, I claim that
one has the inequalitydlog |H ∗ (Γ(pm ), L)| ≫ [Γ : Γ(pm )]1− n .
An elementary exercise shows that L/pL is trivial as a local system for Γ(pm ) andlarge enough m. The inequality above can then be reduced to the following claim:
there is an inequality:
dim H∗ (Γ(pm ), Fp ) ≫ pm(n−d) .
Assume otherwise. The main point is as follows: taking the inverse limit over all m,e j over the Iwasawa algebra Λ. This algebra, by results of Lazard
we obtain modules H
and Venjakob [Laz65, Ven02], is essentially a regular local ring, in particular, it
makes sense to talk about the dimension of modules over that ring. If the inequality
above does not hold, then these modules will have small dimension, explicitly, codimension greater than d. This is so small that Poincare duality will, Ouroboros
like – swallow itself completely and collapse into nothingness. However, the only
way that could happen is if there was nothing to start with, which is nonsense.
More mathematically, consider the completed homology groupse ∗ = lim H∗ (Γ(pm ), Fp )
H
←

20

F. CALEGARIThe homology groups may be computed by a complex of free Λ-modules obtain bylifting an initial triangulation on the base. (Here one thinks of group cohomology as
the cohomology of the associated arithmetic quotients, of course.) Poincare duality
then explains what happens when one takes the dual of this sequence and considers
the corresponding homology groups, namely, there is a spectral sequence:e j , Λ) ⇒ H
e d−i−j .
Exti (HThis spectral sequence might be more familiar to some readers if one imagines Λ tobe a field, in which case the zeroth Ext group is a Hom and the higher Exts vanish,
and one obtains the duality isomorphisms between homology and cohomology overa field. Or, if Λ was the integers, then then zeroth Ext group is a Hom, the first
Ext group is torsion, the higher Ext groups vanish, and one obtains the usual short
exact sequence comparing the dual of homology to cohomology up to a torsion
error term.) The dimension assumption we made implies that the limits are small
e j , Λ) = 0 for all i ≤ d. The key hereas Λ-modules, in particular that Exti (His a Theorem of Ardakov and Brown relating the size of the cohomology growthunder towers to the codimension of the module. Yet putting this assumption into
the spectral sequence shows that all terms with i + j ≤ d vanish, and hence thate0 = He d−d = 0. Yet it is easy to see thatHe 0 = Fp ,Hand thus we have a contradiction.
In fact, this is the same argument that Matthew Emerton and I used to givelower bounds on torsion for p-adic analytic covers of 3-manifolds. There is someslack where the argument can be improved — since one only needs vanishing fora triangular portion of the spectral sequence, you are in good shape if you haveextra information about the lower rows. Of course, the real answer to the amountof mod p torsion in these towers (which is a different question to the original oneof torsion over the integers) should be:
dim H∗ (Γ(pm ), Fp ) ∼ pm(n−l0 ) ,where l0 was defined above.
In a previous version of this post, I confused the roles of dim(K) and d =dim(G/K). For complex groups one has n = 2d, and this is asymptotically thecorrect estimate for simple real groups. In general, one has n ≥ (3/2)d, with theworse case, ironically, corresponding to (any number of copies of) SL2 (R). So you
get a bound of the form:
log |H ∗ (Γ(N ), L)| ≫ [Γ : Γ(N )]1/3 .

15. Galois representations for non self-dual forms, Part I
Tue, 26 Mar 2013This is the first of a series of posts discussing the recent work of Harris, Lan,
Taylor, and Thorne [HLTT16] on constructing Galois representations associatedto regular algebraic automorphic forms for GL(n) over a CM field F/F + . I will
dispense with any niceties about why one should care, and try simply to decipher
the scribbles I made during a talk Richard gave at the Drinfeld seminar. I should

PERSIFLAGE: MATH BLOG POSTS

21

warn the reader of two difficulties: this paper does not exist as a public manuscript,
and it also involves technical details which I generally prefer not to avoid thinking
about. So caveat emptor.
First, some simplifying assumptions. Let’s assume that:(1) π∞ has trivial infinitesimal character.
(2) πp is unramified.
(3) F is an imaginary quadratic field in which p splits.
For examples, I will generally consider the case n = 1 and n = 2. The goal will
be to construct a Galois representation
Rp (π) = rp (π) ⊕ ϵ1−2n rp (π c,∨ )If one can do this for π and for π ⊗ χ for enough characters χ, then one can recoverrp (π). Naturally enough, Rp (π) will be associated to an automorphic form Π fora bigger group. Now π ⊞ ϵ1−2n π c,∨ is automorphic for GL(2n)/F ; it is, moreover,an essentially conjugate self-dual (RAESD) although no longer cuspidal. It does,
however, come from a smaller group, namely, the unitary similitude group G which
is ubiquitous in the papers of of Harris and Taylor. Over the complex numbers,
G looks like GL(2n) × GL(1), but over the real numbers I think it must look likeGU(n, n). Although it’s true that the natural — i.e. occurring in cohomology of
X(G) — Galois representations associated to RAESDC forms ϖ for G will actually
be nth exterior powers, I don’t think that matters so much, since once one has
congruences between ϖ and Π one gets Galois representations of the right degreefor Π.
OK. Now associated to G and an open compact U of G(Af ) one has threenatural objects: a smooth quasi-projective Shimura variety Y = YU , a (typically
non-smooth) normal minimal compactification X = XU , and a (family of) smoothtoroidal compactifications W = WU . The complement of Y in W is SNCD (smoothnormal crossing divisor). I’m using somewhat non-standard terminology as far as
the letters go because I don’t want too many subscripts. If n = 1, then Y is an open
modular curve, X = W is a smooth compactification, and the complement of Y in
W is a finite number of points (cusps). If n = 2, then Y has complex dimension 4.
More on that example later.
As usual, one has the Hodge bundle E = π∗ Ω1A/Y , from which one may build
automorphic bundles ξρ in the usual way for suitable algebraic representations ρ ofwhat I guess amounts to the levi of G(C). In my notes I have written:
ξst = stτ ⊕ st′ τ ′Here st means the standard n-dimensional representation of GLn , and st′ denotes
the complex conjugate representation. One must have has E = ξst , where thedecomposition into a direct sum of two rank n-modules comes from the action of
the auxiliary ring on the tangent space to the universal abelian variety (built into
the definition of G which I have omitted). I also have written:
KS = stτ ⊗ st′ τ ′This presumably relates to the Kodaira–Spencer isomorphism. It’s certainly consistent with a surjection:2^π∗ Ω1A/Y → Ω1Y /k

22

F. CALEGARI

Now it turns out that ξρ extends to W in two natural ways, there is the canonical extension ξρcan and the sub-canonical extension ξρsub ; they differ by the divisor
corresponding to the boundary. Just as in the case n = 1, the bundle ξ can shouldbe though of as having log-poles at the boundary. Last but not least, for the one
dimensional representation ∧2n (stτ ⊕ st′ τ ′ ), one has the line bundle ω on Y . Denotethe canonical extension of ω to W by ω. Then it turns out that ω is the pull-backof an ample line bundle ω on X. Of course, if n = 1, then ω is what you think it is— well, almost, since we are using GU (1, 1) Shimura varieties rather than GL(2).
However, for general n, things are a little trickier. For example, ω is ample on X,
but not (in general) on W .
If U is maximal at p, then the previous constructions also work over a finite field
k of characteristic p and the appropriate smoothness claims are still true. One has
the Hasse invariant H, which is a section of ω p−1 over X/k. Since ω is ample on
X, the complement of the zero divisor of H is affine, it is of course the ordinary
locus. In particular, one has Galois representations of the correct flavor associatedto forms in the infinite dimensional space
H 0 (X ord , ξρ )This follows in the “usual” way; Richard sketched an argument, it goes as expected,
although I think the Kocher principle must have slipped in at some point.
So far, I haven’t really said anything related to the actual argument, but I think
I will stop here for now. The next step is to connect Π in any way to classes in thep-adic modular forms arising in the cohomology group above.

16. Galois representations for non self-dual forms, Part II
Sun, 21 Apr 2013
Let’s recap from part I. We have a Shimura variety Y , a minimal projective
compactification X, and a (family of) smooth toroidal compactifications W . We
also have Galois representations of the correct shape associated to eigenclasses in
H 0 (X ord , ξρ ).
So at this point (well, not only at this point) there is some confusion. In the construction above, I am imagining that we are working with the rigid analytic spacecorresponding to the ordinary locus. But now there are some remarks in my notes
about dagger spaces. Here is what I am imagining is going on. For any sufficiently
small radius, we may consider the rigid analytic space Y [ν] which corresponds (onthe moduli level) to the appropriate abelian varieties A (with polarization and levelstructure and enomorphisms, blah blah) together with a canonical subgroup which(under some measure) is close to being ordinary. Then there is a “dagger space”
Y † which is the limit of all such spaces. The issue (for me) is that I don’t reallyknow anything about dagger spaces, but since this is probably not the main point,
I will (again) elide the issue here. Of course, the goal is to realize the eigenvaluesof the Eisenstein series Π inside this cohomology. Let’s assume that Π actually
has good reduction at p. Then it is probably going to be true that Π actually has
finite slope, and so it lives inside the cohomology of some overconvergent neighbourhood of X ord . So there’s some flexibility with exactly what spaces one is working

PERSIFLAGE: MATH BLOG POSTS

23

with. Perhaps working with finite slope eigenforms might help to get local-global
compatibility at p.
(It’s most natural to work with the dagger spaces (whose cohomology is as described above) since that most naturally corresponds to the rigid cohomology groups
occurring below.)OK, so, we may take the direct limit over all compact subgroups U of the cohomology above, and we want to realize the Eisenstein series Π as a p-adic cusp form
inside this space.
To this end, one introduces the following cohomology groups:
∗
Hc,∂
(X

ord

) := H∗ (W ord , Ω•W ord (log ∞) ⊗ L)

OK. So this is just a definition, it isn’t supposed to obviously be functorial: we aretaking the special fibre, lifting to characteristic zero, taking a toroidal compactification, then looking at the hypercohomology of the de Rham complex with log
poles at the boundary. Well I guess one can do whatever one wants, I suppose.
So what is this? The hycohomology of the de Rham complex of a smooth variety
M with log poles along some divisor D with normal crossings should just be theBetti cohomology of the complement of D in M . The factor L is the difference
between the sub-canonical and canonical extensions, not entirely sure why it isthere, presumably for some fundamentally important reason. So morally, I think
the RHS should be computing something like the Betti cohomology of Y ord , withthe proviso that these are dagger spaces, not smooth complex varieties. So one
should think of the LHS is some type of algebraic Betti cohomology of the ordinary
locus.
Update: the remark about the Betti cohomology of the complement of D is correct, but the presence of the boundary divisor L is exactly what, in the classicalsense, changes the answer from the cohomology of the open variety to the interior cohomology. So the cohomology is somehow compactly supported towards theboundary of W , but not the “other” part of the boundary (that is, the difference
between W and W ord . Let’s write down a spectral sequence:i+j
H i (W ord , Ω•W ord (log ∞) ⊗ L) ⇒ Hc,∂
(X

ord

),

The existence of this spectral sequence must be a formal consequence of the definition and properties of hypercohomology. Note that the ΩjW ord (log ∞) are canonicalautomorphic sheaves of the standard type, so with the boundary piece L the LHS
consists of terms of the form H i (W ord , ξ sub ). To compute these terms, one can pushfoward via the map π : W → X from the toroidal compactification to the minimal
one. Then one notes that:(1) The higher direct images Ri π∗ ξ sub vanish.
(2) Since X ord is affinoid, its higher cohomology also vanishes.
The second point seems reasonable, I have no idea why the first is true. It is
probably a really key point, which I might talk about in part III (note: Richard said
nothing about this and there is no pre-print, so I have no idea how to prove this at
the moment). Apparently it is important that one uses the subcanonical extension
here. This implies that every class which occurs in the RHS in this new cohomology
actually occurs in an H 0 term on the LHS. Now one has Galois representations ofterms of the form H 0 (W ord , ξ sub ), by the first construction - here it must be OK to24

F. CALEGARI

pass between W and X using the Kocher principle. So we are reduced to showing•that Π contributes to this new cohomology Hc,∂
(X ord ).
Update: here is some more about higher direct images. Let’s say a little bit about
what the toroidal compactifications look like. Let’s even imagine we are working
with A2 and are looking at a cusp where one has purely toric reduction. For the
purposes of computing the higher direct images all that matters is the formal completion of W , which at the boundary looks something like Z/Γ for some toric variety
Z which is not of finite type. One shows that H i (Z, OZ ) = 0 using Cech cohomology for i > 0, which allows one to think of Z as contractible. Then one would
like to say that H i (Z/Γ, OZ ) is also zero, which comes down to understanding theaction of Γ on H 0 (Z, OZ ). Roughly one would like to say that Γ acts with no fixed
points and and use Shapiro’s Lemma. Back to the specific example, one finds that
H 0 (Z, OZ ) corresponds to positive semi-definite 2 × 2 matrices, and Γ a finite index
subgroup of GL2 (Z). Here one should be reminded of the q-expansions of Siegel
modular forms at the cusp — recall that q-expansions are given in terms of such
matrices whose coefficients are invariant under M 7→ XM X T . This action is free aslong as det(X) ̸= 0; at the level of q-expansions this corresponds exactly to working
with cusp forms; this is why working with the sub-canonical extension allows one
to restrict the positive definite forms on which the action is indeed free. In thedegenerate case when n = 1, then Γ is trivial, and so it even acts freely on the
non-cusp form 1, which is why it doesn’t matter in that case.
Note: the spectral sequences above is, like the Hodge-de Rham spectral sequence,a 1-st page spectral sequence. Thus the vanishing above does not imply that itdegenerates. Moreover, it certainly won’t degenerate, since the RHS will turn out
to consist of finite dimensional vector spaces, whereas the terms on the LHS arecertainly not (as they are spaces of p-adic or overconvergent forms). (Note to self:
compare to work of Coleman.)
The next point is the following. Suppose one now simply replaces X ord by X.
•
is probably literally computing the Betti cohoThen the cohomology theory Hc,∂
mology of Y . The Betti cohomology of Y does indeed see the classes coming fromthe boundary that we would like to find.
Recall that W ∖ Y is a normal crossings divisor. Let ∂0 denote the variety, ∂1
the (disjoint) union of the irreducible components of the boundary divisor, ∂2 theunion of the intersection of these components, and so on. One now writes down
another 1st page spectral sequence as follows:i+j
Hj (∂i , Ω•W ord (log ∞)) ⇒ Hc,∂
(X ord ).

This is supposed to be an example of the following: in a nice geometric situation(normal crossings divisor) one may compute cohomology with compact supports in∗terms of the cohomology of the boundary strata. (I’m still a little confused why Hc,∂
is cohomology with compact supports rather than the cohomology of the interior,
but anyway. . . (update: this is explained above: the presence of L means it hascompact supports in the direction of W ∖ Y , but not W ∖ W ord )). Moreover, a key
point is that the LHS can be interpreted as the rigid cohomology of ∂i . This allows
one to use results of Berthelot and Chiarellotto to deduce that the terms of the
LHS are given in terms of the rigid cohomology of (open) varieties. In particular:(1) They admit a theory of weights,(2) H j is mixed of weight at least j.

PERSIFLAGE: MATH BLOG POSTS

25(3) They are finite dimensional.
We deduce that RHS is also mixed of weight at least i + j and finite dimensional.
We want our Π to occur in the RHS, so it certainly suffices to show it actuallyoccurs in H 0 . But then by weights it suffices to show that it is coming from theH 0 -terms in the LHS. These are simply given by component groups, and so thecomputation reduces to a problem concerning the combinatorics of the boundary,on which we shall say more in part III.

17. Inverse Galois problems I
Wed, 24 Apr 2013
My favourite group as far as the inverse Galois problem goes is G = SL2 (Fp ).
This is not known to be a Galois group over Q for any p ≥ 13, the difficulty ofcourse being that is must correspond to an even Galois representation. A more
tractable case is G = PSL2 (Fp ), and this was recently answered by David Zywina
here (see [Zyw15]). Here is a more elementary version of that construction.√Supposethat π is a classical modular form of weight three with coefficients in Z[ −1] andquadratic Nebentypus character χ. Note that there is an isomorphism π c := π ≃π ∨ ⊗ ∥ · ∥2 χ. For all primes v in Q(i), one obtains a representation:ϱ = ρ ⊗ ϵ−1 : Gal(Q/Q) → GL2 (Fv ).
with determinant χ. There are two cases, depending on whether v|p is split or not.
If p ≡ 1 mod 4 splits, then, assuming π is not CM, the image of ϱ restricted to thekernel of χ is SL2 (Fp ) for sufficiently large p which can be explicitly determinedin any specific case. Thus the image of ϱ is SL2 (Fp ) plus the image of complex
conjugation:


1 0
.
0 −1
Since p ≡ 1 mod 4, there exists an element α ∈ Fp of square −1, and hence anelement in SL2 (Fp ) equal to


α 0
.
0 −α
Hence the image of ϱ contains a scalar element of determinant −1, and thus it has
projective image PSL2 (Fp ).
If p ≡ −1 mod 4, then, from the isomorphism π c ≃ π ∨ ⊗ ∥ · ∥2 χ, there is
an isomorphism ϱc ≃ ϱ ⊗ χ, where ϱc is the Galois conjugate induced by complex
conjugation. It follows that the projective image of ϱ lands in PGL2 (Fp ). The imageof ϱ is thus, for sufficiently large p, a subgroup of F×p2 GL2 (Fp ) with projective image
containing PSL2 (Fp ). We first observe that this implies that ϱ contains SL2 (Fp ). It
suffices to show that it contains all the transvections; yet the lift of any transvection
in PSL2 (Fp ) is a transvection of order p times a scalar of order prime to p, which
one can remove by taking an appropriate power. Since the determinant of ϱ is χ,
this leaves only the following three possibilities for the image of ϱ:(1) The subgroup of GL2 (Fp ) of matrices with determinant ±1.
(2) The previous subgroup together with the the scalar element I with I 2 = −1.
(3) The group SL2 (Fp ) together with I.

26

F. CALEGARIThe third group does not have a non-scalar element of order 2 correponding tocomplex conjugation, and the first has traces which do not generate Fp2 . Hence theimage must be the second, which has projective image PSL2 (Fp ).
To conclude the argument, it suffices to show that there exists such a π. Consulting William Stein’s tables, one may takef = q + 4i · q 3 + 2 · q 5 − 8i · q 7 + . . . ∈ S3 (Γ1 (32), χ),2
for a quadratic
χ where√ i = −1. Since a3 , a5 , a7 ̸= 0, this form does not have√
CM by Q( −1) or Q( −2), so PSL2 (Fp ) is a Galois group for sufficiently large p,
which one could compute exactly if one wanted. My impression from√the notationin William Stein’s tables is that the fixed field of the kernel of χ is Q( −1), so thisis presumably the same family of examples that arises in Zywina. Other examples(in the range of William’s tables) are as follows:g = q + 2i · q 2 − 4 · q 4 + (3 − 4i) · q 5 + . . . ∈ S3 (Γ1 (20)),h = q + 3i · q 2 − 5 · q 4 − 3i · q 5 + . . . ∈ S3 (Γ1 (27))
Note that this argument requires slightly more than √pure thought; it was key that
there existed a non-CM form with coefficient field Q( −1), and there is no a priori
reason why there should exist any such form. For example, suppose one wanted
to generalize this argument to to PSp4 (Fp ). Then one would want to look for a
non-endoscopic Siegel cusp
√ form of weight (a, b) where (edit) 2a + b is odd with
Hecke eigenvalues in Q( −1) and quadratic Nebentypus character. Possibly suchthings exist but perhaps they don’t!
Comment 17.1 (David Zywina). Yes, your cusp form of weight 3 and level 32 gives
rise to exactly the same representations as in my paper! (I found this post when
doing a literature search for a note I am finishing up.) Amusingly, your cusp formof weight 3 and level 27 shows up at the end of Serre’s 1987 Duke paper [Ser03]. Heshows that the mod 7 representation attached to it produces PSL2 (F7 ) as a Galois
group over Q (unsurprisingly, the key is that the image contains a scalar matrix
with determinant −1). Serre was actually giving an example of his conjecture (hestarted with the PSL2 (F7 )-extension and then found the form), so he overlookedthat this cusp form also produces PSL2 (Fp )-extensions for all p ≥ 5!
Comment 17.2 (Persiflage). Concerning (from some anonymous comment): could
you please elaborate on the line “If p ≡ −1 mod 4, then, from the isomorphism
π c ≃ π ∨ ⊗ ∥ · ∥2 χ, there is an isomorphism ϱc ≃ ϱ ⊗ χ, where ϱc is the Galois
conjugate induced by complex conjugation. It follows that the projective image ofϱ lands in PGL2 (Fp )” Could you explain how the existence of an “inner twist” by c
implies that the projective image lands in PGL2 (Fp )? Where does the congruenceclass of p mod 4 play a role?
The assumption that p ≡ −1 mod 4 means that the residue field of the coefficient ring is Fp (i) = Fp2 (the case when p ≡ 1 mod 4 is easier and was dealt
with √
previously). Moreover, if p ≡ 1 mod 4, then there are two primes above p
in Z[ −1], and so there is no Galois action on the coefficient field. When p ≡ −1
mod 4, complex conjugation on the coefficients induces the automorphism c of Fp2 .
Two representations V and W correspond to the same projective representation if
and only if V ≃ W ⊗ χ for some character χ. The representation ϱ a priori lands
in GL2 (Fp2 ), and the projective representation lands in PGL2 (Fp2 ). The condition
that V actually arises from a GL2 (Fp ) representation is that V ≃ V c . The conditionPERSIFLAGE: MATH BLOG POSTS

27

that the projective representation lands in PGL2 (Fp ) is that V ≃ V c ⊗ χ for somecharacter χ.
Notes 17.3. See § 67 for some updates.

18. Galois representations for non-self dual forms, Part III
Sat, 27 Apr 2013
Here are some complements to the previous remarks on [HLTT16], following on
from §§ 15–16.
First, in order to deal with non-zero weights, one has to replace the Shimuravarieties Y , X, W by Kuga-Satake varieties over these spaces. This “only” adds
technical difficulties.
Second, in order to work over the most general bases F , one seems to require
good minimal models and compactifications XU , WU in characteristic p, for a prime
p which may be very ramified in F . This is a genuine problem. The way to avoid
this problem is amusing. It turns out that one only needs a good model of X ord
and W ord . In other words, one only has to understand integral models and toroidal
compactifications at the ordinary cusps. However, the ordinariness is exactly whatallows one to give appropriate models at these cusps, without having to deal withthe more complicated cusps except in some fairly superficial way (say by taking normalizations over an integral model of a universal moduli space of abelian varieties).
This seems quite clever.
Third, I was going to talk in more detail about n = 2, but having written
down the argument it seems a little pointless now, since it is not going to simplify
things very much. The only thing that is (perhaps) easier is to understand why thehigher direct images of the pushforward of the subcanonical bundle to the minimalcompactification vanishes; yet the example of A2 in the previous post gives theidea, I think. I was also going to talk about the combinatorics of the boundary andtheir relationship to the cohomology of GL(n), but on second thoughts I’m not.
ord∗∗
Fourth, how close is Hc,∂
(X ) to Hc,Betti(X), the compactly supported Betti
cohomology of the Shimura variety? It’s not so clear.
Fifth, the argument really only uses the ordinary locus in a fairly loose way,
namely, it is (in the minimal compactification) affinoid, and it is compatible with
Hecke correspondences. On the other hand, at finite level, this is pretty much the
only possible such choice. However, perhaps at infinite level there may be otherpossible choices (in a perfect[-oid] world, as it were. . . ).

19. Catalan’s constant and periods
Sat, 04 May 2013
There is a 60th birthday conference in honour of Frits Beukers in Utrech in July;
I’m hoping to swing by there on the way to Oberwolfach. Thinking about matters
Beukers made me reconsider an question that I’ve had for while.
There is a fairly well known explanation of why ζ(3) should be irrational (andlinearly independent of π 2 ) in terms of Motives. There is also a fairly good proof
that ζ(3) ̸= 0 in terms of the non-vanishinjg of Borel’s regulator map on K5 (Z). (I28

F. CALEGARI

guess there are also more elementary proofs of this fact.) A problem I would love tosolve, however, is to show that, for all primes p, the Kubota-Leopoldt p-adic zetafunction ζp (3) is non-zero. Indeed, this is equivalent to the injectivity of Soule’s
regulator map
K5 (Z) ⊗ Zp → K5 (Zp ).
(Both these groups have rank one, and the cokernel is (at least for p ≥ 5) equal toZp /ζp (3)Zp by the main conjecture of Iwasawa theory.) It is somewhat of a scandal
that we can’t prove that ζp (3) is zero or not; it rather makes a mockery out of theidea that the “main conjecture” allows us to “compute” eigenspaces of class groups,
since one can’t even determine if there exists an unramified non-split extension
0 → Qp (3) → V → Qp → 0
or not. Well, this post is about something related to this but a little different.
Namely, it is about the vaguely formed following problem:
Problem 19.1. What is the relationship between a real period and its p-adic
analogue?
Since one number is (presumably) in R ∖ Q and the other in Qp ∖ Q, it’s notentirely clear what is meant by this. So let me give an example of what I would liketo understand. One could probably do this example with ζ(3), but I would prefer
to consider the “simpler” example of Catalan’s constant. Here1111
G = − 2 + 2 − 2 . . . = L(χ4 , 2) ∈ R,
1 3
5
7
is the real Catalan’s constant, andG2 = L2 (χ4 , 2) ∈ Q2
is the 2-adic analogue. (There actual definition of the Kubota-Leopoldt zeta function involves an unnatural twist so that one could conceivably say that L2 (χ4 , 2) = 0
and that the non-zero number is ζ2 (2), but this is morally wrong, as the examples
below will hopefully demonstrate. Morally, of course, they both relate to the motiveQ(2)(χ4 ).)So what do I mean is the “relation” between G and G2 . Let me give two relations.
The first is as follows. Consider the recurrence relation (think Apéry/Beukers):
n2 un = (4 − 32(n − 1)2 )un−1 − 256(n − 2)2 un−2 .
It has two linearly independent solutions with a1 = 1 and a2 = −3, and b1 =−2 and b2 = 14. One fact concerning these solutions is that bn ∈ Z, and an ·gcd(1, 2, 3, . . . , n)2 ∈ Z. Moreover one has that:anlim= G2 ∈ Q2 .
n→∞ bnThe convergence is very fast, indeed fast enough to show that G2 ∈/ Q (see [Cal05]).
What about convergence in R, does it converge to the real Catalan constant? Well,a numerical test is not very promising; for example, when n = 40000 one gets
0.625269 . . ., which isn’t anything like G = 0.915966 . . .; for contrast, for this valueof n one has an /bn − G2 = O(2319965 ), which is pretty small. There are, however,
two linearly independent solutions over R given analytically by


5 1
903 1
136565 1665221271 1(−16)n1+−+−+ ... ,
256 n2262144 n467108864 n6274877906944 n8
n3/2

PERSIFLAGE: MATH BLOG POSTS

29


32261 1136565 1665221271 1(−16)n · log n
5 1
−
+
−
+
.
.
.
1
+
256 n27864320 n467108864 n6274877906944 n8
n3/2
,

(−16)n32261 130056525 1
1778169492137 1
1 1
+ 3/2
+
−
+
+
.
.
.
−
768 n27864320 n48455716864 n6346346162749440 n8nfrom which one can see that an /bn must converge very slowly, and indeed, one
has (caveat: I have some idea on how to prove this but I’m not sure if it works or
not):
an
1
=G−
bn
(0.2580122754655 . . .) · log n + 0.7059470639 . . .
So one has a naturally occurring sequence which converges to G in R and G2 in
Q2 . So that is some sort of “relationship” alluded to in the original question. Here’s
another connection. Wadim Zudilin pointed out to me the following equality of
Ramanujan:∞
4k
1X
  ∈ R.
G=2k2
k=0 (2k + 1)2kThis sum also converges 2-adically. So, one can naturally ask whether
∞

G2 =?

1X
2

4k

k=0 (2k + 1)2

  ∈ Q2 .
2kk

(It seems to be so to very high precision.) These are not random sums at all. Indeed,
they are equal to
1
1, 1, 1/2
·F;z
3/2, 3/2
2
at z = 1. Presumably, both of these connections between G and G2 must be thesame, and must be related to the Picard–Fuchs equation/Gauss-Manin connection
for X0 (4). This reminds me of another result of Beukers in which one compares
values of hypergeometric functions related to Gauss-Manin connections and elliptic
curves, and finds that they converge in R and Qp for various p to algebraic (althoughsometimes different!) values. Of course, things are a little different here, since thevalues are (presumably) both transcendental. Yet it would be nice to understandthis better, and see to what extent there is a geometric interpretation of (say)the non-vanishing of Lp (χ, 2) for some odd quadratic character χ. Of course, one
always has to be careful not to accidentally prove Leopoldt’s conjecture in thesecircumstances.
Notes 19.2. The claims here are provable but the general question remains vague
and mysterious. This post is also related to forthcoming work with Vesselin Dimitrov
and Yunqing Tang and will be updated later.

20. Exercise concerning quaternion algebras
Sat, 11 May 2013
Here’s a fun problem that came up in a talk by Jacob Tsimerman on Monday
concerning some joint work with Andrew Snowden:

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F. CALEGARI

Problem: Let D/Q(t) be a quaternion algebra such that the specialization Dt
splits for almost all t. Then show that D itself is split.
As a comparison, if you replace Q by Q, then although the condition that Dt
splits becomes empty, the conclusion is still true, by Tsen’s theorem.
This definitely feels like the type of question which should have a slick solution;
can you find one?

21. Equidistribution of Heegner points
Wed, 15 May 2013
I saw a nice talk by Matt Young recently (joint work with Sheng-Chi Liu and
Riad Masri, see [LMY13]) on the following problem. For a fundamental discriminant|D| of an imaginary quadratic field F , one has hD points in X0 (1)(C) with complex
multiplicationby the ring of integers of F . Choose a prime q which splits in F =p
Q( −|D|). One obtains a set of 2hD points in X0 (q)(C), given explicitly as follows:
C/a 7→ C/aq−1
for a in the class group and q one of the two primes above q in F . The complex
points X0 (q)(C) can be thought of as being tiled by q + 1 copies of the fundamental
domain Ω in the upper half plane.
Problem 21.1. How large does D have to be to guarantee that every one of theq + 1 copies of Ω contains one of the 2hK CM points by OF ?
This is the question that Young and his collaborators answer. Namely, one getsan upper bound of the shape |D| ≤ O(q m+ϵ ) (with some explicit m, possibly 20),
the point being that this is a polynomial bound. Note that this proof is not effective,
since it trivially gives a lower bound on the order of the class group which is a powerbound in the discriminant, and no such effective bounds are known.
I idly wondered during the talk about the following mod-p version of this problem. To be concrete, suppose that p = 2 (the general case will be similar). We
now suppose that D is chosen so that 2 is inert in F . Then all the hK points in
X0 (1)(F2 ) are supersingular, which means that they all reduce to the same curve
E0 with j-invariant 1728. Now, as above, choose a prime q which splits in F . The
pre-image of j = 1728 in X0 (q)(F2 ) consists of exactly q + 1 points.
Problem 21.2. How large does |D| have to be to ensure that these points all comefrom the reduction of one of the 2hK CM points by OF as above?
Since E0 is supersingular, we know that Hom(E0 , E0 ) is an order in the quaternion algebra ramified at 2 and ∞. In fact, it is equal to the integral Hamilton
quaternions H. If E and E ′ are lifts of E0 , then there is naturally a degree preserving injection:
Hom(E, E ′ ) → Hom(E0 , E0 ) = H.
The degree on the LHS is the degree of an isogeny, and it is the canonical norm on
the RHS. In particular, if E = C/a and E ′ = C/aq−1 , then one obtains a naturalmap:ψa : q−1 ≃ Hom(E, E ′ ) → H
preserving norms. The norm map on q−1 is N (x)/N (q−1 ). The image of the naturalq isogeny is simply ψa (1), whose image has norm q. Hence the problem becomes:

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31

Problem 21.3. If one considers all the 2hK -maps:
ψa : q−1 → H,

ψa : q−1 → H,

do the images of 1 cover the q + 1 elements of H of norm q?
Given a field F in which 2 is inert, it wasn’t obvious how to explicitly write down
the maps ψa , but this problem does start to look similar in flavour to the originalone. Moreover, to make things even more similar, in the original formulation overR one can replace modular curves by definite quaternion algebras ramified at (say)2 and q, and then the Archimidean problem now also becomes a question of a classgroup surjecting onto a finite set of supersingular points. In fact, this Archimedean
analogue may well be equivalent to the mod 2 version I just described! Young told
me that his collaborators had mentioned working with various quotients comingfrom quaternion algebras as considered by Gross, which I took to mean the finite
quotients coming from definite quaternion algebras as above. Hence, with any luck,they will provide an answer this problem.
Notes 21.4. I wasn’t really away of this paper, for example [Mic04, Thm 3]).

22. Finiteness of the global deformation ring over local
deformation rings
Sat, 18 May 2013(This post is the result of a conversation I had with Matt). Suppose that
ρ : GF → GLn (F)is a continuous mod-p absolutely irreducible Galois representation. For now, let’s
assume that F/F + is a CM field, and ρ is essentially self-dual and odd. Associated to this representation is a global deformation ring R (of essentially self-dual
representations) consisting of representations with no local restriction at primes
dividing p and the condition of being unramified at primes away from p. One also
has a (collection of) local (unrestricted) deformation rings for the set of primes v|p,combining to give a ring Rloc . Let us also assume that ρ has suitably big image (forexample, its restriction to F (ζp ) is adequate). Then we have:
Proposition 22.1. The map Rloc → R is finite.
(Matt and Vytas prove this in the modular (odd) case when n = 2 and F = Q,
although I’m not sure whether the paper exists yet [actually, I’m pretty sure it
doesn’t]. Possibly if I was listening closer to Matt’s talk at Fields I might haveremembered the argument, since I vaguely think it came up there, although possibly
only briefly.)
Here one has to be a little careful defining deformation rings in the local case,of course (for those worried by such issues, simply choose suitable framings). To
prove this, it suffices to prove the result after base change, so we may assume that
ρ is unramified at all primes, and completely trivial at all primes dividing p. By
Nakayama’s lemma, the problem above reduces to the following:

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Proposition 22.2. Let F ur be the maximal extension of F unramified everywhere.
Let Γ be the Galois group of F ur over F . Then Γ does not admit a continuous
essentially self-dual representation:
Γ → GLn (A)such that A is a complete local Notherian F-algebra of positive dimension.
This is a special case of the generalization of the unramified Fontaine–Mazur conjecture due to Boston. Recall that the group Γ may be infinite (Golod-Shafarevich),but that Fontaine–Mazur predicts that the image of any such representation into
any characteristic zero p-adic analytic group has finite image. Boston conjectured
that the same finiteness would hold for homomorphisms of Γ into GLn (A) for rings
like A = F[[T ]]. It turns out that even though the Fontaine–Mazur conjecture ishard, when A has characteristic p the conjecture is amenable to modularity lifting
theorems by comparison to a new deformation ring in regular weight.
The proof is as follows:
Step 1: Using lifting theorems (Theorem 4.3.1 from [BLGGT14]), we may assume, after a finite base change, that ρ is potentially ordinarily modular of level
one for some regular weight w.
Step 2: Using minimal modularity theorems in the ordinary case (Section 10
from Thorne’s Jussieu paper [Tho12], or Theorem 2.2.2 of [BLGGT14], both using
work of Geraghty), deduce that the minimal weight w ordinary deformation ring S
is finite over W (F), and hence that S/p is finite over F. Strictly speaking, theoremsof this kind are required to prove the previous result.
Step 3: Note that the minimal everywhere unramified deformations of ρ (i.e.,the deformations coming from Γ) of characteristic p are all ordinary of weight w,
because everything unramified is ordinary, and in characteristic p any two weights
are the same. Hence R/p is a quotient of S/p, from which it follows from thefiniteness of S that R is also finite.
While I am using the latest modularity lifting theorems here, weaker versions
for n = 2 with some local assumptions on ρ follow from 90’s era technology (sayTaylor’s Remarks on a conjecture of Fontaine and Mazur paper from 2000, or even
earlier if one assumes residual modularity).
Via the usual argument, this result also applies to even Galois representations ρ :GQ → GL2 (F) with large image. In particular, the unramified deformation rings inthese cases will be finite over W (F), and there will be at most finitely many counterexamples to the unramified Fontaine–Mazur conjecture in characteristic zero fora fixed residual representation. One can also apply it to many classes of higher
dimensional non-self dual representations by taking irreducible summands of ρ⊗ρ∨ .
For example, one can take any representation of Q whose image contains SLn (Fp )if n is even, since then the associated (n2 − 1)-dimensional representation Ad0 (ρ)
restricted to an auxiliary CM field is irreducible, odd, self-dual, and adequate forlarge enough p. Similar remarks apply to representations over an arbitrary field Fwith generic enough image by taking the tensor induction down to Q.
If one starts allowing ramification at auxiliary primes, things become a little
harder. One fix is to build the auxiliary primes into the local deformation ring Rloc ,although this might be considered cheating. The problem is that one cannot deducePERSIFLAGE: MATH BLOG POSTS

33(in general) that more general ordinary deformation rings S are finite in the nonminimal situation. Although perhaps one can get by with the Taylor trick in somecontexts. One should be OK with GL2 by Ihara’s Lemma.
Notes 22.3. One can handle the non-minimal case as well, fairly robust arguments
are given in my paper with Patrick Allen [AC14].

23. Scholze on torsion 0
Thu, 13 Jun 2013This will be the zeroth of a series of posts talking about Scholze’s recent preprint,
available here (now published [Sch15b]). This is mathematics which will, no question, have more impact in number theory than any recent paper I can think of. The
basic intent of this post is to commit to future posts in which I will discuss thedetails. I should remark that Scholze’s writing is pretty clear, so these posts will
mainly be for my own benefit rather than yours.
Here are some of the specific points that I might cover:
Basics: The Hodge–Tate Period map, Perfectoid spaces, etc. To be honest, I
will probably skip the details here to begin with, and discuss them only at points
where they become fundamental for understanding.
Theorem IV.3.1: The action of Hecke on the completed cohomology groups
e i (Z/pn Z) for Shimura varieties is detected by the action of Hecke on classicalH
cuspidal automorphic forms. Although it may end up being no easier to consider,this result is already intersting in some quite degenerate cases. For example, this is
new even for X = U (2, 1)/Q and i = 1 (Gee and Emerton’s results, for example,
are contingent on the relevant Galois representations being three dimensional —now one knows that they are!). A very similar example is the case of a compact
inner form of U (2, 1) (so called Rogawski lattices) or, more generally, the simple
Shimura variety of Kottwitz-Harris-Taylor type. Can one show in those cases thate i vanishes outside degree zero and outside the middle dimension? A weaker quesH
tion: can one compute the completed cohomology in degree one? Compare with thework of Pascal Boyer.
Local Global Compatibility: Suppose one is in the ordinary case. Then the
HLTT approach [HLTT16] (via congruences, discussed previously in § 15, § 16,or § 18, should allow one to establish some cases of local-global compatibility. At
ramified primes ℓ ̸= p, the HLTT approach should also work, especially if one is also
willing to assume that the residual representation is absolutely irreducible (usingbase change arguments). What can one do in the torsion case?
The Nilpotent Ideal: Scholze ultimately constructs Galois representations overT/I for an ideal I such that I m = 0. The necessity of this ideal arises from a spectral
sequence argument. (The parameter m only depends on the degree of the field andn.) The Calegari–Geraghty modularity lifting argument (in the minimal case) canstill be made to apply even with the presence of this ideal if one is in the minimal
case, but not in the non-minimal case which will require m = 1 (the Taylor Ihara’s
avoidance trick requires more precise control than the minimal case). Are there any
circumstances (extra assumptions, etc.) in which one can prove that m = 1?

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Notes 23.1. Of course, § 91 and [ACC+ 23] is relevant for some questions raisedhere.

24. Scholze on torsion, Part I
Sun, 16 Jun 2013This is a sequel to § 23, although as it turns out we still won’t actually get toanything substantial — or indeed anything beyond an introduction — in this post.
Let me begin with some overview. Suppose that X = Γ\G/K is a locally symmetric space, where G is a semi-simple group which does not admit discrete series.
To be concrete, suppose that G = SL2 (C), and that Γ is an congruence subgroupof level N in the Bianchi group SL(OF ) — recall here that F is an imaginary quadratic field. Since days of yore (Langlands, Clozel, Fontaine–Mazur, etc.), everyone
has expected that there should exist a bijection between the following objects:(1) Regular algebraic cuspidal automorphic representations π for GL2 (AF ) oflevel N and weight zero (= the same infinitesimal character as the trivial
representation).
(2) Cuspidal (= non-boundary) cohomology classes H 1 (Γ, R) which are eigenforms for the ring of Hecke operators T.
(3) Weakly compatible families of two-dimensional Galois representations of F
which are irreducible of level N and Hodge–Tate weight [0, 1].
(4) Irreducible semi-stable p-adic Galois representations of weight [0, 1] andlevel (determined in the usual way) dividing N .
(5) Abelian Varieties of GL(2)-type over F which don’t have CM by F .
The equivalence between (1) and (2) follows from Matsushima/Franke. The (conjectural) relationship between (1) and (3) is the problem of reciprocity in the Langlands programme. It consists of two directions; existence (i.e. constructing Galois
representations from automorphic forms) and modularity (i.e. showing nice Galois
representations are automorphic). Both of these directions are difficult. The passage
from (5) ⇒ (3) ⇒ (4) is easy, whereas (4) ⇒ (5) is basically the Fontaine–Mazur
conjecture (not so easy). Actually even that last statement is not quite correct:
if one knows that V is pure of weight one with non-negative Hodge–Tate weights
and arises up to twist in the cohomology of some smooth proper algebraic variety,then it should actually arise in cohomology without having to twist and hence comefrom an Abelian variety; proving this, however, is probably hard (as in Standard
conjectures hard). Note that (1) ⇒ (5) follows, in the case of classical modular
forms, from a geometric construction of Shimura, but that idea doesn’t work here,
and in fact this arrow is completely open and we shall say no more about it. In theparticular case of imaginary quadratic fields, Harris, Soudry, and Taylor showedin 1992 that (1) ⇒ (3) under certain favourable conditions. This case is slightly
exceptional in this regard, since there exist functorial transfers of such π to GSp(4)which do contribute to the (q, K)-cohomology of Shimura varieties and hence can bedirectly related to the coherent cohomology of Shimura varieties (although not directly to Betti cohomology, because the corresponding weights are not regular.) As
readers of this blog know, only very recently, Harris–Lan–Taylor–Thorne [HLTT16]
established the same result for GLn over a CM field. (Small lie: not all the desired

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35

properties of the corresponding Galois representations, — i.e. local-global compatibility — have been established. I think Ila Varma is working this out for her thesis.)
It was observed early on that the cohomology groups of X are not, in general, torsion free. So what then does a torsion class represent? Computations by Grunewald,
Helling, and Mennicke in an 1978 paper suggested that torsion classes (specifically,
two-torsion with Hecke eigenvalues in F2 ) in these groups should be associated toGL2 (F2 ) = S3 Galois representations of the field F . Apparently there are evensome unpublished notes from Grunewald in 1972 doing similar things, although I
have only ever heard rumors of their existence (to be fair, I heard those rumoursfrom Grunewald, so they’re probably pretty reliable). So very early on there werehints that a further story was going on between torsion classes and Galois representations that wasn’t immediately related to the (conjectural) story coming fromautomorphic forms. The first general and precise conjecture along these lines were
formulated by Ash in his 1992 Duke paper, with further refinements by Ash–Sinnott,
and Ash–Doud–Pollack. Unlike the previous speculations of Grunewald, these conjectures were precisely formulated and falsifiable, and in the spirit of Serre’s original
conjecture. Moreover, Herzig did actually come along and falsify them, by finding amore natural prediction for the set of possible Serre weights which turned out to bedifferent from the formulation of Ash et. al., and Herzig’s formulation subsequently
proved (numerically!) to give the right answer in these cases (Edit: see comments,
this is not quite correct). At any rate, for quite some time, we have expected thatmod-p torsion classes give rise to Galois representations, and following the conjectures of Serre, Ash, and others, one can be quite precise about exactly the local
properties the corresponding Galois representations should have at primes of bad
reduction. What is perhaps more recent is the idea that, especially for groups G
with no discrete series, that torsion is not merely a technical nuisance, but rather
is the source of “most” of the interesting Galois representations. In particular,• The phenomenon whereby Galois representations coming from the countably many classical automorphic forms are dense in a suitable universal
deformation ring (Böckle, Gouvêa–Mazur, Chenevier) will be totally false
when G does not have discrete series. On the other hand, the representations coming from torsion should cut out all of the universal deformationring.
• That in order to answer the most pressing questions concerning reciprocity(even in characteristic zero!) one needs to understand torsion classes.
For one take on this, I might suggest reading Section 1.1, Speculations on p-adic
functoriality of [CM09]. For another interesting perspective, you should also readthis, as well as the accompanying review.
So let us assume then that studying torsion representations and associating themto Galois representations is an Important Goal. How do we construct them? An
observation also going back a long way (I believe to Harder??) is the following.
Even though G may not admit discrete series, there may exist a group H containing
a parabolic P with G as a Levi. If H does admit discrete series, then there will
exist a Shimura variety XH whose Borel-Serre compactification will have at least
one boundary component which is a torus bundle over XG , and as a result one
obtains a map (with some mixing of degrees) H ∗ (XG , Z) → H ∗ (X H , Z). Now one
is theoretically in better shape, because this map should be compatible (in somesense) with Hecke operators, and the latter group has a chance to admit comparisons

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F. CALEGARI

to étale cohomology groups which do come with Galois representations. There arethree immediate problems:(1) The compactification X H will be singular, except in the case of modular
curves.
(2) Given a long exact sequence for (co)homology relative to a boundary divisor, it’s not clear whether the cohomology in the boundary ends up in
H i−1 or H i .
(3) Just because a class [c] has an interesting Hecke eigensystem doesn’t meanthat that étale cohomology sees an interesting Galois representation.
The third issue is a genuine problem. If one has a Hecke eigenclass [c] ∈ H ∗ (X)in the etale cohomology of a Shimura variety, even in characteristic zero, then all
one can deduce is that the corresponding Galois representation is annihilated by thecorresponding characteristic polynomial of Frobenius. But this is not always enoughto get the correct Galois representation! It’s probably worthwhile to consider two
examples.
First, a somewhat degenerate example. Let G = GL(1)/Q, G = R× , H =SL(2)/Q, and H = SL2 (R). Now XG is (for some level structure) a finite set ofpoints, and XH is a modular curve with cusps. The boundary map realizes XG asa set of cusps in XH . The Hecke operators act on cusps by the degree map, i.e.
Tp [c] = (1+p)[c]. This coincides with the action of Hecke operators on H 0 (XH ). So,
in the etale cohomology group H 0 (XH ) we have a Hecke eigenclass which we imagine(looking at the Hecke operators) to be associated to the Galois representation 1 ⊕ χ
where χ is the cyclotomic character. Yet H 0 (XH ) is one dimensional, and so weonly see half of the Galois representation, namely, the trivial character. Now as
it turns out, the other piece of the Galois representation can be seen in H 1 (XH ),which is now mixed because XH is not projective (so the cuspidal part has motivic
weight one, and this other piece of the Eisenstein series has weight two). So evenin this trivial case, we see that a Hecke eigenclass in etale cohomology may havea less interesting Galois representation than the Hecke eigenvalues might suggest.
From the Eicher–Shimura relation, we do get that the (trivial) Galois representation
which does occur is annihilated by the characteristic polynomial of Frobenius (σ −1)(σ − χ(σ)), indeed it is annihilated by the first factor.
Second, let G = GL(2)/F , G = GL2 (C), H = U (2, 2)/Q, and H = U (2, 2).
Here H is taken to split over F . The cohomology of (a torus bundle over) the
Bianchi group maps into the cohomology of U (2, 2). The characteristic polynomialsof the Hecke operators are, morally, the following. If ρ is the (conjectural) Galois
representation associated to an eigenclass on the Bianchi group, then r = ∧2 (ρ⊕ρc )is a six dimensional (reducible) representation which is a direct sum r = s ⊕ ψ ⊕ψ c for a four dimensional representation s = ρ ⊗ ρc and a Grossencharacter ψand its conjugate (which are related to the central character of the original form
and its conjugate). Now the characteristic polynomials of Frobenius on this Galois
representation are, by Eichler–Shimura, the characteristic polynomials of Hecke on
the image of this cohomology class in the cohomology H ∗ (XH ). Without assuming
one has ρ, one can phrase the above purely in terms of Satake parameters, but thisway of saying it makes clearer what is going on, even though we don’t know yet that
ρ actually exists. If one could find the Galois representation r (and in particular s)
inside the etale cohomology of XH one would (almost) be done, but instead, theclasses which actually turn up in etale cohomology in these degrees are the reduciblePERSIFLAGE: MATH BLOG POSTS

37

terms in r corresponding to the Grossencharacters rather than to the interesting
representation s we are looking for. So as above, even in characteristic zero, one
has the interesting Hecke eigenclass, but not the Galois representation.
These examples suggest that to understand what is going on we first need to geta better understanding of Shimura varieties. Most of the recent history of understanding Shimura varieties (and the Galois representations associated to automorphic forms) has concentrated on the cohomology arising from cuspidal automorphic
representations. In this classical setting, the automorphic representations have aclassical avatar as global sections of certain coherent bundles on XH . (For example,
classical modular forms of weight ≥ 1 are global sections of the line bundle ω ⊗k .)If we want to restrict to cusp forms, we can also take the corresponding extensionof these sheaves to minimal (or toroidal, doesn’t matter) compactifications which
vanish appropriately at the boundary. If we denote these automorphic bundles by
Esub , then another way of saying this is that the action of Hecke operators T onM
H 0 (X H , Esub )Eis now understood if XH is, for example, a Shimura variety of unitary type overa totally real field. Even getting this far is a somewhat monumental task that
required, amongst other things, Ngo’s work on the Fundamental Lemma, work of
Kottwitz, Clozel, some large fraction of Jussieu, the work of Shin, and many more.
In fact, as far as local-global compatibility goes, the ink is barely dry on the mostrecent work. Now we can at least state, in vague terms, the following:
Theorem 24.1. [Sch15b, IV.3.1]: For (many) Shimura varieties XH , the actionof T on torsion classes in Betti cohomology factors through the action on coherent
cusp forms in characteristic zero.
Two examples: If XH is the modular curve, then this says that the action of
Hecke operators on H 1 (XH , Z/pn Z) can be realized by the action on classical modular cuspidal eigenforms modulo powers of p. Given how we think about modular
forms, this is almost tautological, because, by Eichler–Shimura, we can pass between cohomology classes and classical modular forms (in this case, we can evendo this via the Hodge decomposition of H 1 ). However, there is a little wrinkle: we
do see Eisenstein classes in Betti cohomology, and this theorem says that we canrealize these as coming from cusp forms, so this result also implies that there exist
cusp forms which are congruent to Eisenstein classes modulo pn . Since we are ultimately interested in classes coming from the boundary of some compactification,
we don’t want to ignore this case. Still, it’s not so difficult to prove.
If XH comes from U (2, 1)/Q (so it is a arithmetic complex hyperbolic manifoldof real dimension 4, also known as a Picard modular surface), then we can look at
the group H1 (XH , Zp ). The characteristic zero classes here are known to correspondto endoscopic automorphic representations (and thus to not exist in the co-compactcase) and are understood. However, unlike in the modular curve case, we no longer
know that this group is torsion free, and in general, it may not be. So, a priori, all
we know about the torsion classes and their Hecke operators is that there exists a
Galois representation which is annihilated by the characteristic polynomial of Tp ,using Eichler–Shimura. These polynomials are all of fixed degree (three in this case),but that doesn’t give any lower bound on the dimension of this representation. Thisis an even more stark example of the well known phenomenon that Eichler–Shimura

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F. CALEGARI

is pretty much useless for constructing Galois representations outside the case ofdimension two where knowing both the trace and determinant tells you a lot. For
example, suppose you have an irreducible representation V of a finite group G
in characteristic zero such that all the elements of g have a minimal polynomial of
degree at most d: then you can’t a priori bound the dimension of V ! As an example,
the extra-special 2-group of order 21+2n has a representation of dimension 2n all of
whose elements have images satisfying the degree two polynomials x2 − 1 = 0 or
x2 + 1 = 0. So, before Scholze, we could not say anything about the dimensions of
mod-p Galois representations arising from torsion in the first homology of U (2, 1).
However, using Scholze, we can now deduce that any such representation comesfrom a classical cusp form, and hence must (in this case) have dimension three!
Comment 24.2 (Molesworth). In fairness, I’m not sure that Ash etc claimed tohave written down a complete list of weights, so it’s a little strong to say that FH[Florian Herzig] falsified them.
Comment 24.3 (Persiflage). You are correct: I just checked Ash–Doud–Pollack [ADP02].
They say “Note that the conjecture makes no claim of predicting all possible weightsthat yield an eigenclass with ρ attached.”

25. Scholze on torsion, Part II
Wed, 19 Jun 2013This post follows on from § § 23–24.
Section V.1: Today we will talk about Chapter V. We will start with Theorem
V.1.4. This is basically a summary of the construction of Galois representations in
the RACSDC case, which follows, for example, from work of Shin. We know a little
bit more than this theorem states (namely, local-global compatibility).
Corollary V.1.7 is just the statement that the cohomology groups H 0 (X, Esub )for the sub-canonical extensions of automorphic sheaves E are computed by forms
π whose transfer to ResF/Q GL(n) are RACSDC. The sub-canonical extension corresponds to imposing a vanishing condition at the cusps. For example, the subcanonical extension of ω k on the modular curve is ω k (−∞). There is a nice actionof the Hecke algebra T on this space, which is compatible with the associatedGalois representations in all the expected ways (Satake parameters to Frobenius
eigenvalues) at the unramified primes. So far, this is all classical (as of 2011).
Determinants: (See [Che14]) We will be using congruences to obtain Galois
representations, but the information that gets glued is really the Hecke eigenvalues. So one wants a convenient way to pass from one to the other. The classicalapproach with modular forms is to remember the “standard” Hecke operators Tx
which correspond to the traces of Frobenius. Knowing the trace is enough to determine a two dimensional representation away from characteristic 2, if one has residual irreducibility. This is the theory of pseudo-representations. Naturally enough,for larger dimensional Galois representations, it helps to remember more than the
trace, namely, the entire characteristic polynomial. The corresponding theory wasworked out by Chenevier. Namely, given an n-dimensional representation ρ of thegroup G over a commutative ring A, there is a map:
D : A[G] → Mn (A) → A

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39

given by formally extending ρ in the obvious way and then composing with thedeterminant. For example, if n = 2, thenT (g) := D([g] + [1]) − D([g]) − D([1]) = Tr(ρ(g)).
Now the map D has to satisfy a bunch of formal properties due to the constraintsof coming from an n-dimensional representation. Writing all these down gives thecorrect notion of Chenevier’s generalized “determinant.” (Original paper here.) For
those who like pseudo-representations, note that when n = 2, one can define D
using the formula:T (g)2 − T (g 2 )D(g) =,
2
where T is the trace. So for n = 2 in characteristic greater than two, the notions areequivalent. And indeed Chenevier’s notion of determinants is the same as a pseudorepresentation whenever n! is invertible, but is better behaved in small characteristics. Determinants satisfy the nice properties that pseudo-representations do, andthat Galois representations sometimes don’t (but do in the residually absolutely
irreducible case), namely:(1) You can glue determinants: D → A/I and D → A/J which agree on
A/(I + J) to get a determinant D → A/(I ∩ J).
(2) Given a formal variable X, there is a natural determinant map D : A[X][G] →A[X] such that D(1 − Xg) is the characteristic polynomial of g if the determinant comes from an actual representation.
Here I follow Scholze in using Det(I − X · M ) rather than Det(M − I · X) as
the definition of a characteristic polynomial — this is just a bookkeeping issue (thedreaded arithmetic versus geometric Frobenius). Returning to automorphic forms
from coherent cohomology, since H 0 is torsion free, the module T is flat over Z.
Since the characteristic zero forms give rise to Galois representations coming fromRACDSC forms, we naturally obtain a determinant map:
D : Zp [GF ] → Tsuch that D(1−X ·Frobx ) is exactly as one would expect. (This is Corollary V.1.11).
Note that the ring Tc , which arises at this point, is just the inverse limit of thecorresponding classical T over all p-power levels; this is defined in Chapter IV which
we shall talk about later.
Segue on Completed Cohomology: I have to recall here a few basics aboutcompleted cohomology (one reference is [CE12].) I already know about completed
cohomology (and so do many of my loyal readers) so I don’t really feel obliged to say
too much about it, but since most of you have been sent here from Quomodocumque,I will cough up a few pointers. The basic definition (for any congruence arithmetic
manifold corresponding to a group G) is as follows:e i (X, Z/pn Z) := lim H i (X(K), Z/pn Z).
H
K→

Here the limit is over shrinking compact open subgroups K of G(Zp ). The tamelevel is fixed and can be included in the notation somewhere. One can also adorn thecohomology groups in the usual way, namely, by considering compactly supported
cohomology. So what’s the point of completed cohomology? Apart from having anatural action of G(Qp ), which is always the type of group one wants to act on a
candidate space for automorphic representations of any kind, a matter of experienceand intuition suggested (to Matt and me) that it should be the “correct” space of40

F. CALEGARI

automorphic forms modulo pn when G(R) does not have discrete series (and evenwhen it does). One way to justify this is via the following four properties, the final
one conjectural:e i (X, Z/pZ) are co-finitely generated(1) The completed cohomology groups H
over Λ = Fp [[G(Zp )]]. This latter ring has nice properties, e.g. after shrinking the group G(Zp ) slightly to get a powerful torsion free pro-p group, Λ
is a local Noetherian ring which is Auslander regular (see Lazard [Laz65]
and also Venjakob, which is [Ven02]).
e i (X, Qp /Zp )∨ , which are finitely generated(2) The Pontryagin dual groups H(by part one and Nakayama’s Lemma and the usual long exact sequences)
are not torsion Λ = Zp [[G(Zp )]]-modules if and only if one is in middle
degree and the corresponding real group admits discrete series (see [CE09]).
(3) The completed homology groups satisfy a Poincaré duality spectral sequence. The completed cohomology groups are compatible with the Hochschild–
Serre sequence from which one can recover classical cohomology groups.
(4) Given a torus bundle, or more generally a nilmanifold, the completed cohomology disappears outside degree zero.
(5) Conjecturally: for any reductive algebraic group there will be a domie i (X) in degree i = q0 which will have co-dimension l0 as
nating term Ha Λ-module, where 2q0 + l0 is the real dimension of XG , and the degrees[q0 , q0 + 1, . . . , q0 + l0 ] are exactly the degrees in which tempered automorphic representations contribute to cuspidal cohomology. More directly, l0
for a semi-simple group is the rank of G(R) minus the rank of the maximal
compact. For example, l0 is equal to zero if and only if the real group admits discrete series. Hence this bullet point is a conjectural generalizationof point (2). As an example, in the case of GL2 over an imaginary quadratice 1 (Fp ) should have codimension exactly
field, the completed cohomology H
one.
(For the last three points I’ll refer you once again to [CE12].)
Section V.2: The key starting point, as mentioned last time, is that one canrelate the cohomology of the group we are interested in — ResF/Q (GLn ) — to thecohomology of Shimura varieties by realizing the first group as the Levi M inside
a maximal parabolic P inside a group G corresponding to a Shimura variety. The
first step is to compare the cohomology of what we are interested in (coming fromthe Levi M ) to the cohomology of the boundary piece coming from the parabolic P
inside G containing M . This is pretty standard: what happens is that the resulting
space XP which actually occurs in the boundary of the Borel–Serre compactificationBSXG
of XG is a torus bundle over XM . Well, not literally always a torus bundle,
but rather a nilmanifold N coming from the unipotent part of P . The nilmanifold
fibres spread the cohomology around by a Künneth type formula like a Frenchman
expectorating over-oaked California Chardonnay into a spittoon. (Usually this fibration arises as a quotient from a fibration with a contractible fibre, which meansthat the cohomology really is just the derived product of the cohomology of thebase and the cohomology of N , so it’s not really so bad.) One way to avoid thismess is by passing to completed cohomology. On the boundary this has the effect ofcollapsing all the torus like directions in the nilmanold, and obtaining a map from

PERSIFLAGE: MATH BLOG POSTS

41

the completed cohomology of the arithmetic manifold corresponding to the Levi
into the completed cohomology of the total space. Compare with equation (1.4) of
this survey again.
Hecke Operators from M to G. One thing we have to understand is how tocompute the Hecke operators at unramified primes on the completed cohomologyof the boundary of XG in terms of the action of the Hecke operators on the original
object of interest XM . Let us fix an unramified prime x which is prime to everything.
To orient you, we are at the top of page 82 of Scholze. I’m going to be more prosaic
in my notational choices and write TG , TP , and TM for the local Hecke algebras at
the prime x (Scholze does all the unramified primes at once). Yes, I know this is an
abuse of notation, because here the groups G, the parabolic P and the Levi M arereally the local versions at the prime x. (You will cope.) There are natural maps:
TG → TP → TM .
Let’s actually consider what these are in the case when M comes from GL(2)over an imaginary quadratic field F in which x splits, and G comes from U (2, 2)
which also splits over F . So locally at x, the group G is just GL(4), and M is theLevi GL(2) × GL(2), and P is what it obviously has to be. In this case, we haveisomorphisms:
TG ≃ Zp [X1± , . . . , X4± ]S4 ,√TP ≃ Zp [Y1± , Y2± ]S2 × Zp [Z1± , Z2± ]S2 .

Perhaps we are required to adjoin x to both sides in order to normalize this appropriately. Consider it done. Now the map TG → TM is the one sending (X1 , X2 , X3 , X4 ) 7→
(xY1 , xY2 , x−1 Z1 , x−1 Z2 ). The choice here must be coming from the choice of M(for a fixed torus) corresponding to a choice of subgroup S2 × S2 of the Weyl group.
One can write down analogous formulas for the inert and ramified primes. The corresponding maps of Satake parameters indicates that the if our original eigenclass
has a Galois representation ρ, then the Hecke eigenvalues of the class which has
been pulled back is associated to ρ∨ ⊕ ρc . (Edit: In the previous version I omitted
the dual. Note that ρ∨ det(ρ) = ρ for n = 2. End Edit) Now this statement seems
to be somewhat in conflict with my previous post, where I claimed that the actionof the Hecke algebra on the cohomology of U (2, 2) corresponded to the Galois representation ρ⊗ρc . This is because of a subtlety which I think I can explain. Suppose
you start from a classical modular form f and base change it to a Hilbert modular
form fE over a real quadratic extension. Then the corresponding map of Satakeparameters is just the obvious one corresponding to the restriction of the Galois
representation. In particular, if α, β are the Satake parameters of a local unramified
component πx of f , and if x splits in the quadratic field and y is a prime above x,
then πy of fE will have the same Satake parameters, and fE will have the same
Hecke eigenvalue for Ty that f has for Tx . However, the actual Galois representation
occurring inside the etale cohomology of the Hilbert modular surface is not the restriction of the Galois representation to E, but rather the (four dimensional) tensor
induction. This also reflects an important point: we will not be finding the desired
Galois representation inside etale cohomology (which, apparently by an argumentof Clozel and Harris, is impossible), but rather we will simply be “following the
Hecke eigenclasses.” In this context, for example, cuspidal automorphic representations for U (2, 2) contain all the information for the associated four-dimensional
representations, but the ones occurring in cohomology are (tensor inductions!) of∧2 . That is why in this post we see the Hecke eigenvalues as looking like the direct

42

F. CALEGARI

sum ρ∨ ⊕ ρc , whereas the action on cohomology via Eichler–Shimura looked like∧2 (ρ∨ ⊕ ρc ), which contains ρ ⊗ ρc up to twist.
The arguments on the lower half of page 82 are just related to the fact thatthe boundary of the compactification on XG can have a number of components,
and these components can have their own boundary, and so on. If one takes thecase where XG corresponds to U (2, 2) over an imaginary quadratic field, then the
only boundary components are (torus bundles over) Bianchi manifolds XM , andthe only boundaries that they have are hyperbolic cusps. In particular, in this case,using remark (4) on completed cohomology above, the completed cohomology ofe k (∂XG ) (denoting X BS ∖ XG by ∂XG ), is given by
the boundary H
G


k
n
e (∂XG , Z/p Z) = IndG(Zp ) H
e k (XM , Z/pn Z) .
H
P(Zp )
So we are interested in the Hecke action on the right hand side, which we have
now transferred to the left hand side. (Of course, the local Hecke algebras combine
by taking tensor powers to get the Hecke algebra at all unramified primes, whichsurjects onto the corresponding global Hecke algebras TG and TM .) There is anatural long exact sequence of completed cohomology associated to a manifoldwith corners as follows:e k−1 (XG , Z/pn Z) → H
e k−1 (∂XG , Z/pn Z) → H
e ck (XG , Z/pn Z) → H
e k (XG , Z/pn Z) → . . .
... → He k−1 (∂XG ),So to get a Galois representation (or, to begin with, a determinant) on H
we can start by finding determinants for the two surrounding terms.
Special Case: Let’s continue discussion the special case where XM is a Bianchi
manifold, and XG comes from U (2, 2) which splits over the corresponding imaginary quadratic field. The key term of interest will be (for the Bianchi manifold)e 1 (XM ) or, equivalently, H
e 1 (∂XG ). In fact, by Hochschild–Serre, the completedH
1e (XM ) captures all the interesting Hecke actions coming from torsion
cohomology H
in Bianchi groups as long as one localizes away from the Eisenstein primes comingfrom the cusps. The cusps in the Borel–Serre compactification of the Bianchi groupare elliptic curves with CM by the underlying imaginary quadratic field. The differe 1 proved themselves to be a real pain
ence between the classical classes in H 1 and H
in my book with Akshay, because when one wants a numerical correspondence, one
can’t ignore Eisenstein terms. Yet blessedly, in this context, we can localize awayfrom them. Hence the key terms are those in the following boundary exact sequence:e 1 (XG ) → H
e 1 (∂XG ) → H
e c2 (XG )H
Let’s consider the first term. The group U (2, 2) has real rank two. In particular,
by super rigidity, any non co-compact lattice in U (2, 2) will have the congruencee 1 (XG ) is trivial! The point is that if all thesubgroup property. It follows that H
finite quotients of a lattice in U (2, 2) come from congruence quotients, then pullingback over all such quotients kills everything. Actually, this is not strictly correct,
because completed cohomology only pulls back over p-power quotients, and there
may be cohomology coming from the tame level. However, it is easy to see (byHochschild–Serre) that any such cohomology will be Eisenstein. In particular, after
localizing at a non-Eisenstein (in the appropriate sense) ideal, we get an injectione 1 (∂XG ) to He c2 (XG ), and thus from Theorem IV.3.1, we obtain a determinantfrom He 1 (XM ) (localized away from Eisenstein ideals) without
to the Hecke algebra of H
any need to quotient out by an ideal with fixed zero power as in Corollary V.2.6.

PERSIFLAGE: MATH BLOG POSTS

43

I don’t think this trick will really work in any other examples, however, since it’svery hard to say anything in general about H 2 . (There is recent work on on stablecompleted co/homology here, but that will never be enough to give something usefulin this context.)
General Case: The general case is now quite similar, except know to understande k (∂XG ) one needs to understand both boundary terms. There is also going to beHsome loss of information coming from the corresponding extension class. If one hade ∗ (XG ) and He c∗ (XG ), then one would immediately get Corollary
determinants on H
V.2.6 with an ideal I with I 2 = 0. However, Theorem IV.3.1 (which is being invokede c∗ (XG ). Now He ∗ (XG ) is related to its compact cousin by ahere) only applies to H
Poincaré duality spectral sequence, but this will once again spread out some terms
and necessitate replacing I 2 = 0 by some power involving the dimension. At any
rate, while there is room for improvement in general, there is still the fundamental
problem (mentioned in part zero!) of controlling whether this boundary cohomology
is going forwards or backwards in the long exact sequence above (or worse, being
mixed). I’m going to give some heuristics next time on what one expects should
happen (short answer: after localizing at a nice maximal ideal, it should work out
as well as the Bianchi case, but that will be hard to prove.)
Note that Scholze actually works with classical cohomology here, and then relates
it back to completed cohomology using Hochschild–Serre on p.86. The point ineither argument is that all the terms in the spectral sequence (on every page)are, by Theorem IV.3.1, modules for the Hecke ring Tc which acts on coherent
cohomology. Hence the limit terms have filtrations by a fixed bounded number ofsuch objects.
Next time, I’ll say a little more about how one might expect the “simplification”
in the Bianchi case above to apply more generally, and I’ll talk about the final
section V.3 of chapter V, in which we extract the n-dimensional representations
from our 2n-dimensional determinants.

26. Scholze on torsion, Part III
Sat, 22 Jun 2013This post follows on from § § 23–25. Before I continue along to section V.3, I want
to discuss an approach to the problem of constructing Galois representations fromthe pre-Scholze days. Let’s continue with the same notation from last time, where
XM is the symmetric space whose cohomology is of interest, and X = XG is the
Shimura variety with Borel–Serre compactification X BS whose boundary contains(simplified assumption: is) a generalized torus bundle XP over XM . If we localize ate n (XM ) shoulda “non-Eisenstein” ideal, then the completed cohomology groups H
vanish outside a single degree q0 . For this discussion, let us define non-Eisensteinclasses to be those which do not occur in degrees ≤ q0 in H ∗ (XM ). By Hochschild–
Serre, any cohomology class in lowest degree (after localization) always survives
in the completed limit, so even if one doesn’t assume the expected vanishing in
e q0 (XM ) will contain all the information about thehigher degrees, the module H
q0classes in H at classical level after localization. Hence, to obtain the desired
Galois representations for these classes, one wants to prove:e q0 −1 (X BS ) after localization.
(1) The vanishing of H

44

F. CALEGARI(2) There are Galois representations (of the correct form) associated to classes
e cq0 (X BS ).
in H
The hope was that one could try to prove this via the following idealized argument. There is a spectral sequence:e jBM , Λ) ⇒ H
e d−i−j ,Exti (Hwhere d = 2 · dim(X) = 2n is the real dimension of the Shimura variety X. There is
an identical sequence with the roles of completed homology and completed BorelMoore homology reversed. Note that the completed homology groups are (Pontryagin dual) to the cohomology groups, which relates compactly supported cohomology
to homology and cohomology to Borel-Moore homology. The non-commutative Ext
groups in the spectral sequence vanish for any value of i that is less than the coe ∗ is torsion
dimension of the corresponding module. Recall from last time that Hjexcept for the middle degree j = n. Now suppose that one can show that thee ∗ have sufficiently large co-dimension outside thecompleted homology groups Hjmiddle degree. Then from these bounds (and from trivial bounds on the cohomology of the boundary) the spectral sequence should degenerate, and one should haveisomorphisms of the following form (after localization):
e n−i = Exti (He nBM , Λ), i ≤ n,H

e n+i = 0, i ≥ 0,H

e nBM = Hom(He n , Λ).
H

(To recall, even though we are localizing at an ideal whose avatar on H ∗ (XM ) is
maximally non-Eisenstein, the corresponding ideal on H ∗ (XG ) will be Eisenstein.)
From these equalities, we see that to understand the action of the Hecke operators on completed cohomology, we are reduced to understanding the action on thecompleted cohomology in middle degree, which we know to be a module of positive
rank and hence (even after localization) contain many cusp forms which are known
to have interesting Galois representations. At the very least, this would prove theexistence of the residual Galois representations associated to such a non-Eisenstein
ideal m. The approach I am outlining here is the one in the (currently non-existent)
paper that Matt and I had planned to write. Let’s suppose that one attempts toapply this approach in the Bianchi case. There’s no issue in defining Eisensteinclasses here, since the classes that occur in H 0 (XM ) are easy to understand, ande 1 (X BS ) vanishes,
q0 = 1. So the first step in the above program is to show that H
at least if we pass to finite tame level. As we noted last time, this follows from thecongruence subgroup property which is known because U (2, 2) has real rank two
and the corresponding lattice in this group is (obviously) not co-compact. Here thee j is
Shimura variety has complex dimension four. So one only has to show that H
small for j = 2 and j = 3. In particular, one wants, explicitly, that:e 2 ) ≥ 4, codim(He3) ≥ 3
codim(HThe dimension of Λ = Zp [[G]] is, for reference, 1 + dim SL4 (Zp ) = 16. As noted previously, we know that these cohomology groups are torsion and so have co-dimension
at least one. The proof of this result ultimately relied on facts concerning the growthof spaces of automorphic forms. However, it is impossible to determine anything
further about the codimension by naïve automorphic considerations, because already Λ/p has co-dimension one but no characteristic zero points. So, to prove thisconjecture, one really needs to understand the torsion in the cohomology of Shimuravarieties. This was where, basically, we were stuck. Note that even understandingPERSIFLAGE: MATH BLOG POSTS

45

e 1 in this case took a powerful result. Understanding He 2 is already much harder.
H
As the real rank increases, it won’t be the case that such completed cohomology
groups completely disappear, since there will exist not only trivial stable classes incharacteristic zero, but also exotic torsion classes which will be related to K-theory
and regulators (as can be seen [Cal15]). One implication of our conjectures (as noted
above) is that the completed cohomology groups vanish for Shimura varieties above
the middle dimension. Scholze proves this! (IV.2.3). However, he doesn’t prove ite j are small for small j, and instead deduces a (weaker form)by showing that the Hof such an estimate in reverse. I think it’s an interesting problem to understandH 2 (Γ, Fp ) for groups where the only characteristic zero classes are invariant underG, in both the stable and non-stable range. The first case I mentioned previously,
and there is something in this direction (in the second case) in [CV19, §4.5].
Section V.3 OK, continuing on from last time, we now have a determinantof dimension 2n with image in A0 /I = T/I for some ideal I with I m = 0 for
an integer m which only depends on dim(X). The goal is now to extract an ndimensional determinant, i.e., to recover ρ from ρ∨ ⊕ ρc . Of course, the idea is not
to do this from simply one class, but rather allowing twisting, so that we also knowrψ = ρ∨ det(ρ)ψ −1 ⊕ ρc ψ c for some Hecke character ψ. We may as well take ψ tobe a collection of characters of Q, so that ψ c = ψ.
Let’s first make some simplifying assumptions, namely, that the ideal I = 0, that
we are in characteristic zero, and that the image of r is through a finite group G,
and the image of all the twists factors through the group Γ := G × Z where ψ is a
finite order character of the second factor, and ψ 2 ̸= 1. We would like to imagine
that there are equalities:
r = W =? U ⊕ V,rψ = Wψ =? U ψ ⊕ V ψ −1 .

Because the two factors of Γ commute, it follows that [ψ ⊗ Wψ ] − [W ] is a virtual
character of Γ. Evaluating this character on the pairs G ∼ (g, 1) ⊂ Γ defines aclass function on G. Normalizing by ψ 2 (1) − 1 ̸= 0, this class function applied to
Frobx is the sum of the Satake parameters at x corresponding to U , and we deducethat [U ], and hence also [V ], are virtual characters (with rational coefficients) of
G. It now suffices to promote [U ] to an actual character. The virtual characters [U ]and [V ] tautologically promote to virtual characters of Γ which decompose under
the second factor into trivial representations. It follows that [U ψ] and [V ψ −1 ] are(rational) sums of irreducible representations which decompose under the second
factor as direct sums of the representation ψ or ψ −1 . Assuming that ψ ̸= ψ −1 , there
can be no cancellation in [U ψ] + [V ψ −1 ], from which it follows that [U ] is alreadyan actual character.
In general one has to modify this argument to work more integrally as well as
to be compatible with the ideal J. As I told Toby Gee, “without having looking at
this yet, it must essentially be trivial.” So, if you are like me, you can just ignore
the following which took me a non-trivial number of hours to work out:
(1) We take the characters ψ to be characters of Gal(Q(ζℓ∞ )/Q) of ℓ-power order, where ℓ is prime to two and p and anything else inconvenient including
the ramified primes. This auxiliary prime may vary.
(2) Since we are going to allow ψ to have order some arbitrarily large power ofvarious primes, it is convenient to extend scalars to A = A0 ⊗ W (Fp ). HereA0 is the Hecke algebra acting with coefficients modulo some fixed power

46

F. CALEGARI

of p. It’s useful to work with both rings, however, because whilst A accepts
characters of all orders, A0 is literally a finite ring, which is convenient for
finiteness arguments. We would like to show that the twisted determinants
corresponding to rψ have values in A/Iψ for the same A. This amounts toshowing that, at the level of our original locally symmetric quotient XM ,we can twist by a sufficiently nice character ψ and not change the Hecke
algebra, except for extending scalars. This is straightforward, and is what
is going on at the top of page 88.
(3) If we have two determinants with a pair of corresponding ideals I and Iψ
with I m = Iψm = 0, then clearly Ie = I + Iψ satisfies Ie2m−1 = 0. So,
at the cost of increasing the nilpotentcy, for any character ψ, we get two
e Note that if I and Iψ are both trivial,
determinants with values in A/I.
ethen so is I.
(4) We would also like the ideal Ie to be independent of ψ. Actually, we don’t
need this, it will suffice to note that we can take Ie∩A0 to be independent of
ψ. Because A0 is finite, there are only finitely many such ideals, and so we
can take one that occurs for infinitely many primes ℓ and infinitely manyof the corresponding characters ψ.
(5) For any fixed character ψ, our determinant (which has twice the required
dimension) will be defined on a finite quotient ofΓ := Gal(L∞ /F ) = Gal(L/F ) × Gal(F∞ /F ),where L/F is finite and L∞ , F∞ are the pro-ℓ cyclotomic covers of L, F
respectively. This should hopefully look similar to our simplified problem
in characteristic zero. We have two determinants D and Dψ with the property that the characteristic polynomial of Frobenius Frobx (which exists fordeterminants) is:(a) Of the form Px∨ (X)Pxc (X) mod I for D.
(b) Of the form Px∨ (X/χ(g))Pxc (Xχ(g)) mod Iψ for Dψ .
These polynomials Px∨ (X) and Pxc (X) are what they obviously shouldbe, namely, the polynomials with inverse roots given by the appropriate
Satake parameters. (Or more accurately, with coefficients given by the appropriate Hecke operators.) Because these are determinants, these products
are locally constant on the group Γ because they are coming from honest
Galois representations of rank 2n. We would like to decompose these into
products of two determinants of rank n. In the characteristic zero case, wetook a character ψ such that ψ 2 (1) − 1 ̸= 0 and used this as a fulcrum
on which to tease out the representation U . Here we do something similar.
A first step is to show that each of the four polynomials above is locallyconstant. We choose an element 1 ∈ Gal(F∞ /F ) and a deep enough character χ so that χ2m (1) − 1 ̸= 0 for all m = 1, . . . , n. We now find an open
neighbourhood of (G, 1) where D and Dχ are constant. Let a(x) be thelinear term of Px∨ (X), and let b(x) be the linear term of Pxc (X). Then we
deduce that the following two terms are locally constant:a(x) + b(x),a(x)ψ(x) + b(x)ψ −1 (x).

So, because ψ 2 (x)−1 ̸= 0, we deduce that a(x) and b(x) are locally constant,and so a(x) mod Ie∩A0 is also locally constant. Given this, one proves thatPERSIFLAGE: MATH BLOG POSTS

47

the quadratic terms are also locally constant in the same manner, and by
induction one has the result for the entire polynomial. Thickening the open
neighbourhood up, one proves the same result for the entire group Γ minus
the piece coming from Z ∼ Gal(F∞ /F ), which gives us Lemma V.3.4.
Then by choosing a different auxiliary prime ℓ′ , one patches to get a welldefined class function on G in Lemma V.3.5.

(6) So now we have a class function D on the Galois group Gal(L/F ) withvalues in characteristic polynomials (now of the right dimension!) in A0 /I(dropping the tildes), and we want to promote it to a genuine determinant.
Of course, over finite rings we can’t use the language of virtual characters.
What Scholze does next is use the fact that we have such a decomposition
for infinitely many different characters in order to glue enough of them
together to obtain a determinant map
D : A[G × Z] → A[t]/I,D(1 − Xγ k g) = Pg∨ (X/tk )Pgc (Xtk ),where γ is a generator of Z and I has the expected properties of nilpotence.
This consists of Lemmas V.3.6 and V.3.7.

(7) Now we are at Lemma V.3.8. Bugger it, this is taking a long time, andquite possibly nobody is interested in these specific details. Let me cut somecorners and replace determinants by pseudo-representations. We deducefrom the above that we are in the following situation: we have a degree 2n
pseudo-representation:
T : G × Z → A[t],T (g, m) = a(g)tm + b(g)t−m .

We want to deduce that a(g) and b(g) are both pseudo-representations ofdegree n. We are allowed to use the fact (which is obvious) that a(g) andb(g) are not pseudo-representations of degree strictly less than n. (Actually,is it obvious? It’s certainly obvious for n = 2 that a(g) and b(g) are not acharacter. So let’s assume n = 2. Ah, I see by passing to the trivial element
we can compute that a(1) = b(1) = n, so it is obvious.) Now, if we abstractslightly and drop any knowledge about a(g) and b(g) other than they areclass functions, the best we can hope to prove is that a(g) and b(g) are both
pseudo-representations of degrees A and B respectively, where A + B = 2n.
This is what we do. Since T is a pseudo-representation of degree 2n, we
have the following identity:
X(−1)d Tσ (gi , mi ) = 0.
S2n+1

In fact, this identity on class functions characterizes pseudo-representationsof degree at most 2n, the only other information coming from evaluating on
the identity. Suppose
P we take the mi to be sufficiently generic integers sothat all the sums
±mi are distinct. Now let us partition the mi into two
sets MA , MB of cardinality A + 1 and B respectively, where A + B = 2n.

48

F. CALEGARI

Consider the coefficient of tC in the sum above, where we take
X
X
mi
mi −
C :=
MA

MB

The corresponding coefficient must vanish. Moreover, because of the waythat the mi were chosen, we know exactly what terms can arise with thiscoefficient: explicitly, the terms in MA must come from a(g), and the terms
in MB must come from b(g). Hence we recognize the coefficient to be (upto sign)


!
X
X

(−1)d aσ (gi )
(−1)d bσ (gi ) .
SA+1 ↷MA

SB ↷MBWe deduce that, for any decomposition A + B = n, either a(g) is a pseudorepresentation of degree at most A, or b(g) is a pseudo-representation ofdegree at most B −1. Taking B to be the smallest integer for which b(g) is a
pseudo-representation, we deduce the result (such an integer exists becauseb(g) is at least a degree ≤ 2n pseudo-deformation). We are, mercifully, done.
Looking at Scholze, I think this lemma (and even roughly the argument) is
quite similar to the proof of Lemmas V.3.8-V.3.15 but this is much easier,at the cost of assuming that p ≥ n.
It looks as though one can probably skip step 6 simply by choosing thevalue of t ∼ ψ(1) to generate a sufficiently generic extension of A0 inside
A, although I guess that’s how one does step 6 anyway.
Section V.4 is just a matter of putting things together. Next time: onto Chapter
IV!

27. Scholze on torsion, Part IV
Sat, 29 Jun 2013This is a continuation of § § 23–26.
I was planning to start talking about Chapter IV, instead, this will be a very
soft introduction to a few lines on page 72. At this point, we have reduced the
problem of constructing Galois representations for torsion classes on a wide class of
locally symmetric spaces to the equivalent problem for Shimura varieties. Naturally
enough, the Shimura varieties which arise in this context will not be projective. Howe c∗ (X)ever, the problem of attaching Galois representations to Hecke actions on H
is still a very interesting one in the compact case. The difficulties that arise in the
non-compact case are somewhat orthogonal to the issue of constructing Galois representations, so I don’t think much is lost (at this point) in considering the compact
case. (MH tells me that one of the main ingredients for dealing with issues concerning the boundary may well be the Hebbarkeitssatz, II.3.) A good case to keep inmind are the simple Shimura varieties of Kottwitz-Harris-Taylor type, and even thesimple case of ball quotients coming from U (2, 1) will be of interest. Honestly, eventhe case of modular curves will be of interest. Modular curves are not compact, ofcourse, but this is the one non-projective case in which the minimal and toroidal
compactifications coincide and are smooth, so the boundary causes (relatively) little difficulty. A related problem is to understand the action of Hecke operators on

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49

torsion in coherent cohomology. In some sense, Scholze reduces the problem to thiscase, so we shall begin by considering this problem. Note that already in this casethe problem is no longer trivial even for classical modular curves, where one may
have torsion in H 1 (X, ω).
Coherent Cohomology: Let E be an automorphic vector bundle on X. Supposethat X is smooth over Zp , so that it makes sense to impose some nice integral
structure on E, and hence to consider the coherent cohomology groups:H ∗ (X, E/p)
If X is non-compact, then denote (also by E) the sub-canonical extension to asmooth toroidal compactification. This cohomology group has a natural Hecke action. How does one construct Galois representations associated to the Hecke action
this object? Let’s consider the first non-trivial case, where X is a modular curveand E = ω. There’s no problem understanding H 0 , because (via the Hasse invariant) this will be related to classical spaces of modular forms, so the problem is tounderstand H 1 . The first step is to understand what H 1 is as a vector space. To
compute the cohomology of a projective curve, we can take a covering by (two)affines and compute Cech cohomology. To do this, we first need to find two affines.
In anticipation of having something sufficiently natural in order to understand theaction of Hecke, we let S denote the supersingular locus and U = X ∖ S. For now
let’s let the other affine be V . Then the Cech complex is the following:
H 0 (U, ω) ⊕ H 0 (V, ω) → H 0 (U ∩ V, ω)
Here U is the ordinary locus. The space H 0 (U, ω) is the space of ordinary modular
forms, and we may relate the Hecke action on this (infinite dimensional) space tothe Hecke action on classical modular forms by noting that:(1) For any section c ∈ H 0 (U, ω), there exists a power sn of the Hasse invariants such that sn · c extends to H 0 (X, ω m ) for some integer m.
(2) The ordinary locus U is preserved by Hecke operators, and moreover multiplication by the Hasse invariant s is Hecke equivariant.
The problem is that it’s hard to find a second open affine V which is preserved
by Hecke, let alone admits an analogue of the Hasse invariant. In this case, we caninstead do the following. Take V to be an infinitesimal neighbourhood of S, (that is,the completion of X along S). Then V is stable by Hecke. Imagine for conveniencethat there is only one supersingular point. The cohomology H 0 (V, ω) of V has afiltration by the order of vanishing at (each) supersingular point, the first piece
consisting of simply functions H 0 (S, ω) on the supersingular point. There exists a
section B p−1 (see [Edi92, Prop 7.2]) which is Hecke equivariant. This approach isused Emerton/Reduzzi/Xiao to construct Galois representations for torsion classes
in the coherent cohomology of Hilbert modular varieties (Note that one would also
want these representations to satisfy certain local properties at the prime p, which
is more subtle in general, but has been done at least for modular curves at leastin the residually irreducible case in my paper [CG18a] with David.) If one thinks
about applying this method in the general case, there are two obvious issues. The
first, which is perhaps not impossible to overcome, is that one needs to constructa suitable stratification of the Shimura variety by pieces which one understands
and for which one can construct suitable Hasse-invariant type sections which allow
one to pass to very ample sheaves whose cohomology vanishes, and hence reduce
the problem to degree zero. The second is that, at least in the context of Scholze,

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one is working at a level which is very ramified at p. Certainly all of the discussion
above was predicated on X having good integral models at the prime p. It’s easier
to find good integral models when the corresponding Shimura variety is smooth!
At level X(pn ), there do exist integral models (obviously no longer smooth). It’s
convenient to assume that the open modular curves X(pn ) are projective, because
the issues at the cusps are orthogonal to what is happening here. So what do theylook like? Well, they are proper and flat, which is nice. The general problem to theconstruction is that the torsion subgroup E[pn ] of an elliptic curve E is no longer
etale (and so certainly not locally isomorphic in the etale topology to (Z/pn Z)2 ),but it is at least finite flat of rank p2 . So all one needs to do is to impose enoughextra structure on the finite flat group scheme in order to recover the correct object
on the generic fibre and yet have enough points in the special fibre. Katz–Mazur
do this by considering a so-called “Drinfeld basis”
ϕ : (Z/pZ)2 → E[pn ]where there is a corresponding equality of Cartier divisors (see 3.1.2 of KM). In
particular, given a point xn one gets a level structure Pn , Qn ∈ E[pn ] given by theimage of the two generators.
So how does one understand the tower of varieties X(1) ← X(p) ← X(p2 ) . . .,either integrally or even just on the generic fibre? The ordinary locus up the tower iseasy to understand. Let’s first consider the rigid analytic varieties corresponding to
the generic fibre. There are sections X ord (1) → X ord (pn )∞ from the ordinary locus
to the component of the ordinary locus containing infinity, because, for ordinary elliptic curves, we still have etale locally a canonical isomorphism E[pn ] = Z/pn ⊕µpn ,giving an appropriate trivialization. Moreover, the the action of GL2 (Zp ) is transitive on the cusps, and so one sees all of the ordinary locus in this way. Thinking more
integrally, we can see more directly from Serre–Tate theory that (for all points) at
level one the completed local rings will be smooth. However, because Z/pn ⊕ µpn
does not admit any deformations, the covering maps will be smooth at ordinary
points and so the complete local rings at any ordinary point will remain smooth.
It follows that the interesting geometry will be taking place over the supersingulardiscs. One can try to understand what is happening by looking at the corresponding
completed local rings at supersingular points. Suppose one takes a compatible sequence of supersingular points (in the special fibre) in such a tower. The base point
corresponds to a supersingular elliptic curve E0 over Fp which has a corresponding
formal p-divisible group G0 , now of height two. What Weinstein teaches us is thatwhilst the completed local rings An of xn on X(pn ) will be hard to understand,there is still hope to understand the completion\
A = (limAn )→over the ring OK , which is the completion of W (ζp∞ ). By universality, the Drinfeld level structure gives rise to two parameters Xn , Yn in An which lie inside themaximal ideal. The Weil pairing (we’ve added a consistent sequence of roots of
p-power roots of unity) gives a relation of the form ∆n (Xn , Yn ) = ζpn . Jared showsthat these are essentially all the relations in the limit ring A, which thus has avery nice description. We will come back to this example, because I suspect that
understanding this result will be important.

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The Lubin–Tate tower There’s also a local analogue of this picture, namely
the Lubin–Tate tower. Recall that the Lubin–Tate space M0 is the universal deformation ring of a commutative height h formal group G0 over k = Fp , where h = 2.
It turns out that M0 is smooth of relative dimension h − 1 over the Witt vectors
W (k). The smoothness is the “same” as the smoothness of the modular curve oflevel one at a supersingular point. It makes sense to consider level structures in the
Lubin–Tate context also, where now the nth layer Mn of the Lubin–Tate tower consists of triples (G, ι, α) with Drinfeld level structure, as in the Katz–Mazur model.
Quite explicitly, the K-points are given as follows:(1) G is a formal group over OK ,(2) G is a deformation of the height h formal group G0 over k, and ι : G0 →G × k is an isomorphism,(3) αn (Z/pn Z)h → G[pn ] is an isomorphism.
If we go up the entire tower, there is a natural action of GLh (Zp ) in the limit.
×
If D is the corresponding division algebra, then there is an action of OD
on (each)piece of the tower, given by replacing G by a prime-to-p isogeny. In order to havericher actions of GLh (Qp ) and D× on this tower (not only on the cohomology) it
makes sense to modify it slightly (while enlarging the component group in a waythat doesn’t change the intrinsic geometry) by considering a trivialization of therational Tate module α : (Qp )h → V (G). Here we now consider deformations up
to isogeny, although we remember a quasi-isogeny on a nilpotent divided power×
thickening of k as well so as not to lose the action of OD. The combined actionof these groups on the compactly supported cohomology of the tower realizes the
local Langlands correspondence. The proof (for h = 2) is to realize this towergeometrically (or at least the cohomology) as the “supersingular part” of the towerof modular curves, and then use global facts concerning automorphic forms. In
fact, this is how Harris-Taylor prove local Langlands in general. The corresponding“space” is not literally a rigid space (but more on perfectoid spaces later), but one
can ask for a description of the Cp -points of M . To this end, one may construct socalled period maps. I plan to come back to this in some detail, but for now let me
simply say that these maps (constructed in this context in differing contexts andlevel of generality by Fargues, Weinstein, and Scholze) have their roots in Tate’s
p-divisible groups paper, where by taking OCp -points one may split the p-divisiblegroup into a p-adic Hodge filtration, and the corresponding period map records theslope of the corresponding line as an element of P1 (more generally, one obtains
a point in a Grassmannian). Let me mention at this point that I have studiously
avoided thinking about this whole chapter in the world of Shimura varieties formany years, and it always had the reputation to me as something done by Very
Smart People like Mantovan and Fargues, and I have been rewarded in my lazinesssimply by waiting for the moment where the correct way to view these objects has
started to emerge, and there’s someone around like Jared Weinstein who (apart frombringing new ideas) writes and lectures so beautifully well. I certainly recommend
reading his papers and lecture notes to understand what is going on (instead of
having to sort through the partially digested version I have produced for you here.)
Scholze also writes well, thank god.
Page 72: Very roughly, one does the following:(1) Understand the tower (either the Lubin–Tate tower or the corresponding
tower of modular curves) as an actual geometric object X (perfectoid space).

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F. CALEGARI(2) Construct a period map π : X → P1 (or F ) using p-adic Hodge theory.
(3) Use the first two steps to construct a formal model X, which will havesections arising via pull-back from some ample line bundle on P1 .
(4) Note that the construction of these sections only depends on the p-tower,
and so are Hecke equivariant with respect to all the other Hecke operatorsand can thus serve as a replacement for the Hasse invariant, and multiplication by these sections allows one to pass back to characteristic zero forms
in H 0 , which, by virtue of the control one has over the geometric context,
one may identify with classical modular forms.
As Matt explained to me, one can understand the image of the ordinary locus
under π to be P1 (Qp ), which should correspond to the fact that ordinary Galois
representations have splittings already before having to pass to Cp . This also fits
into the Lubin–Tate story and the period map to the Drinfeld upper half plane(which has P1 (Qp ) excised), as occurs in the paper of Fargues linked to above.
We also see here that the ordinary locus under the period map factors through thecomponent group π0 , with the natural action of GL2 (Qp ) permuting the cusps. In
particular, all the ordinary points are mapping in the special fibre to P1 (Fp ), which
doesn’t look at all like the usual story at all. This is related to footnote 4 on page
72.
Question for the the audience: is it obvious how one can extract the classicalcoherent cohomology groups H ∗ (X, E) at level one from H ∗ (X ∗ , E)?

28. Effective motives
Wed, 03 Jul 2013This is a brief follow up concerning a question asked by Felipe. Suppose we assume the standard conjectures. Let M be a pure motive, and consider the following
problems:(1) Problem A: (“effectivity”) Suppose that M has non-negative Hodge–Tateweights. Then is M effective?
(2) Problem B: (“ordinary primes”) Does the Hodge polygon = Newton polygon for infinitely primes p?
(3) Problem C: (“Katz”) Suppose the characteristic polynomials of Frobenius
have coefficients in Z. Then is M effective?
An affirmative answer to Problem C implies an affirmative answer to Problem A.
Conversely, a positive answer to Problems A & B implies a positive one for Problem
C.
The relevance of Problem A was for deducing that a weight zero regular algebraic
cuspidal automorphic form for GL(2)/F could be associated to an abelian varietyof GL2 -type over F . I claimed that this was probably “Standard Conjectures hard.”
It seems that this is partly right and partly wrong. As mentioned previously, if M
has weight zero, then Problem A already follows from Kisin–Wortmann (always
assuming the standard conjectures), because then M will be an Artin motive.
As was pointed out to me, the case of weight one follows from the Hodge conjecture. Namely, the Hodge realization gives a polarized Hodge structure of weight
one which gives a polarized complex torus. By Riemann, such a torus is actuallyan abelian variety A, which (using the standard conjectures) one can descend to F .

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This argument doesn’t obviously extend to the general case, because the image ofthe period map from (say) pure Motives with Hodge–Tate weights [0, k] to polarized
Hodge structures will not be surjective for Griffiths transversality reasons. As an
aside, it was also pointed out that the Hodge conjecture is not one of the standard
conjectures.
When I asked Deligne about Problem A, he politely told me(1) There’s no evidence for Problem A beyond the fact that it would be nice,(2) The Hodge conjecture is false, and(3) Grothendieck already mentioned that his (Grothendieck’s) modification of
the generalized Hodge conjecture implies that the answer the Problem A is
positive.
Here the generalized Hodge conjecture says (roughly) that a sub-Hodge structureof H k with weights in the range [k − q, q] to [q, k − q] arises via the Gysin map froman algebraic cohomology class on an ≥ q-codimensional subvariety. In particular, if
M has non-negative Hodge–Tate weights and is of weight w, and M (n) is effective
inside some smooth proper variety X, then M gives rise to a sub-Hodge structureof H w+2n (X) with weights from [n, n + w] to [n + w, n], and hence come from somealgebraic subvariety Y of codimension at least n. However, the Gysin map on etale
cohomology involves a Tate twist by Qp (n), and so (using the standard conjectures)one recovers M effectively in Y . Grothendieck also points out that, in the case when
M has weight one, the generalized Hodge conjecture follows from the usual Hodge
conjecture after replacing X by X × C for proper smooth curves C, essentially by
the same argument of the previous paragraph. (I guess one also has to use the easyfact that any abelian variety is a quotient of a Jacobian.)
Talking of Deligne and Grothendieck, Benson Farb sent me the following link to
an interview of Deligne by MacPherson:
Deligne interview
which contains the following slightly terrifying exchange about Grothendieck:
MacPherson: I’ve heard people say that he [Grothendieck] was always verykind to students when they didn’t understand, but if someone was older and had
pretensions he could be less . . .
Deligne: That’s quite possible, and I think he was completely willing to explain
something once, I don’t think he would have be willing to explain it three times,
even to students.
(In my original memory of this passage, “three times” was replaced by “twice.”)29. Life on the modular curve
Tue, 24 Sep 2013
Alice and Bob live on the modular curve X0 (1) = H/PSL2 (Z). What does theworld look like to them, assuming that they view the world in hyperbolic perspective?
To those who are not used to hyperbolic geometry, there may be a few mild
surprises. Suppose that Alice is at the point x = i and Bob is at y = 10i. Let us
also imagine that Alice is looking in the direction of the cusp along the projectionof the geodesic given by the y-axis. What does she see? Take a moment to think
about it if you like; we will give the answer in the next paragraph.

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Lifting Bob to the universal cover, there are infinitely many Bobs spaced equallyalong the horosphere (10i + t). A naive guess is that all of these Bob’s would fillout Alice’s field of vision. But this can’t be true; since geodesics in H are given bysemi-circles perpendicular to the x-axis, most geodesics through x = i don’t crossBob’s horosphere. In fact, Bob only takes up about 10◦ of Alice’s vision, and those
Bobs who are at (10i + n) for large integers n appear almost to be directly in frontof Alice (although a long way away). Of course, Alice also sees copies of herself
receding similarly into the distance directly in front of her.
All this and more can be seen in the 80’s inspired video game of my undergraduate summer students Jasmine Powell and Justin Ahn (funded by the NSF!). The
basic setup is as follows: you are a cube wondering around on X0 (1) and you needto shoot the monsters, which are in the shape of a pill. Occasionally, some bonus
feature will appear (extra shields, freeze, extra life, etc.) which you can collect.
Some mathematics that is hiding in the background but is only partially relevantfor game play: the monsters travel along closed geodesics, and the goodies appear
at CM points. The game was also partly inspired by the video not knot. Here’s alink to a video capture from the game:
Link to the video(The transition to video has made it look a little wonky.) If you notice carefully,
you will see that at one point in the video you crash into yourself by passing throughthe cone point i, losing a life.
The α-release of the game itself can also be downloaded here (sorry, macintosh
only). Please play around with it and offer suggestions and improvements! Various
possibilities include upgrading to a 3-manifold (probably a Bianchi manifold), andalso the ability to pass to congruence covers X0 (p) of X0 (1).

30. Virtual congruence betti numbers
Fri, 27 Sep 2013
Suppose that G is a real semisimple group and that X = Γ\G/K is a compact
arithmetic locally symmetric space. Let us call a cohomology class tautological if it isinvariant under the group G. For example, if X is a 3-manifold, then the tautologicalclasses are all multiples of either the trivial class in H 0 or the fundamental classin H 3 . We say that X has positive Betti number if there exist any non-tautologicalclasses in the cohomology of X. One can pose the following question:e → X such that
Problem 30.1. Show that there exists a finite congruence cover Xe has positive Betti number.
X
An automorphic way of phrasing this question is as follows: do there exist any
automorphic forms besides the trivial representation for the Q-group G associated to Γ. If G admits discrete series, then the result is obvious for automorphic
reasons (from the trace formula, by de George-Wallach). If X has non-zero Euler
characteristic, then the result is obvious for topological reasons. In fact, as I leantfrom Gross one day at tea, these two situations coincide (this certainly follows from
Borel-Wallach, even in the stronger form that the contribution from each π via
Matsushima’s formula has zero Euler characteristic if it is not a discrete series; I’m
not sure if there’s a slicker argument).

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The problem is obviously related to the virtual positive Betti number theorem ofAgol, but there are a few important subtle differences. The first is that we insist thate is congruence. Hence, the problem remains open for a general arithmetic
the cover X
3-manifold. Second, we also allow (as we must) cohomology in any degree. Anotherexample to consider is G = U (2, 1). In this case, X is a compact complex hyperbolic
manifold. It is an open problem whether such manifolds have virtual positive first
Betti number. In contrast, by a theorem of Rogawski, they certainly don’t havevirtual positive first Betti number in congruence covers, although they clearly dohave virtual positive Betti number in congruence covers for the two equivalentreasons given above.
What I want to do in this post is discuss a related problem, namely, can one finde which all fail to have positive Betti number?
arbitrarily large congruence covers X
Specific examples of this kind (for a compact arithmetic 3-manifold X) were given
in my paper with Dunfield (conditional on local-global compatibility of certain
Galois representations, now known), and Boston–Ellenberg shortly thereafter founda different (unconditional) argument using group theory (which applied to the same
example). I want to explain how to generalize these results to higher dimension,
contingent on computations which might be hard to carry out explicitly.
Choose:• An imaginary quadratic field F .
• A prime p which splits as pp in F .
• A central simple algebra D/F with local invariants 1/N and −1/N at theprimes dividing p.
Associated to D is a maximal lattice Γ in G/K = SLN (C)/SUN (C) whose
quotient is a compact finite volume orbifold of real dimension N 2 −1. For sufficiently
large n, the congruence covers X(pn ) are√manifolds which are K(π, 1) spaces with
fundamental group Γ(pn ). When F = Q( −2), N = 2, and p = 3, one recovers the
manifolds considered in my paper with Nathan.
Let me now make another definition. Let FS be the maximal pro-p extension of
F (ζp ) unramified outside the primes dividing p.
Definition 30.2. The prime p is very regular in F if the map:
Gal(Qp /Qp ) → Dv ⊂ Gal(FS /F )is surjective for either v|p.
The notion of very regular primes arose in my latest paper on K-theory [Cal15]
and completed cohomology in the stable range, but more on that later. One lastdefinition: say that an ideal m of a Hecke algebra T is Eisenstein if the image ofany Hecke operator T in T/m coincides with multiplication by the degree deg(T ).
This is how T acts on the trivial representation. We then have the following:
Conditional Theorem 30.3. Suppose that p is very regular, and that m is an
Eisenstein maximal ideal. Then for all n there is an equality:H ∗ (X(pn ), Zp )m ⊗ Qp = H ∗ (SU(N ), Qp )In particular, if the only maximal ideals of T on H ∗ (X(p), Zp ) are Eisenstein, then
all the X(pn ) are rational SU(N )-homology spaces.

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F. CALEGARI√
Example 30.4. The prime p = 3 is strongly regular for F = Q( −2), and – by
a computation – the only maximal ideals of T on H ∗ (X(p), Zp ) for N = 2 are
Eisenstein. Of course, a rational SU(2)-homology space is a homology 3-sphere.
Proof. Suppose that there is exists a non-trivial class in the cohomology of X(pn ).
It will give rise to an automorphic representation π which is tempered, because
X are Shimura manifolds for which we can show (reference?) have no endoscopic
forms. Hence, by HLTT or Scholze [HLTT16, Sch15b], there exists a corresponding
Galois representationr(π) : GF → GLn (Qp )that is unramified away from p. We now assume (this may be proved soon, but thisis the reason for the “conditional” in the statement) that we know enough about
local-global compatibility to deduce that this representation is also ordinary at theprime p′ . Note that the reason it should be ordinary is that the level is prime top′ , and since the quaternion algebra is ramified at this prime we know that πp′
is Steinberg. We deduce that r(π) is completely reducible after restriction to Dv
for v = p′ . The Eisenstein assumption and the ramification assumption imply thatr(π) and hence r(π) factor through Gal(FS /F ). Hence, using the fact that p is very
regular, we immediately deduce that r(π) itself is reducible and ordinary. It follows
that, after semisimplification, r(π) is a direct sum of characters, which leads to aneasy contradiction.
□
Experts will recognize this argument as a generalized and more streamlined version of what appears in my paper with Nathan. One may naturally ask whether
there is a generalization of the Boston–Ellenberg argument as well. Emerton and I
already explained that the correct way to view that argument was as follows. Whatone really wants to prove is that the partially completed cohomology groups:e ∗ (p) = lim H ∗ (X(pn ), Fp )H→

all vanish identically outside degree zero. For 3-manifolds, it suffices to prove thise 1 . For what X might one be able to prove such vanishing? As Matt and I
for H
explained in our paper on 3-manifolds, for all these groups to vanish there has tobe a delicate balancing act between the dimension of the group acting on completed
cohomology and the dimension of the manifold. For example, it is crucial that there
is an equalityYdim(G/K) = dim( G(Fv ))S

where one partially completes at primes S above p. (Otherwise one obtains animmediate contradiction by Hochschild–Serre.) In the case at hand, this inequality
is satisfied, since:
dim(G/K) = dim(SLN (C)) − dim(SUN (C)) = N 2 − 1 = dim(SLN (Zp ))Hence, it is really possible that all the completed cohomology groups may vanishin this case. In fact, if one instead considers the split group GL(N )/F , then thepartially completed cohomology groups do vanish in the stable range exactly forvery regular primes. (This is where the definition of very regular primes comesfrom.) By Nakayama’s Lemma, one can explicitly compute at some finite level toPERSIFLAGE: MATH BLOG POSTS

57

e ∗ (p) vanish or not. In fact, it suffices to compute that thedetermine whether the Hmaps:
H ∗ (G(p), Fp ) → H ∗ (X(p), Fp )are isomorphisms, where G(p) is the congruence subgroup of SLn (Zp ). If one wanted
to find an explicit example where these theorems
applied for N ≥ 3, the first place√
to look would probably be to take F = Q( −2), p = 3, and N = 3. One would thenhave to compute the cohomology of a certain 8-dimensional manifold! (The resulting
manifolds would potentially all be rational SU(3)-homology space = rational S 5 ×
S 3 -homology space). This computation is within the realms of plausibility. To ruleout characteristic zero representations, we can pass by functoriality to the split
side. So, if there is a characteristic zero class which is not Eisenstein mod-p, that
residual representation
also has to occur at low(ish) level inside the cohomology√
of GL3 (Z[ −2]). This is the sort of cohomology that people like Gunnells might
almost be able to compute!
Notes 30.5. Certainly the required local–global compatibility results are now
known, e.g [ACC+ 23], so Conditional Theorem 30.3 is unconditional.

31. Abelian varieties
Wed, 30 Oct 2013
Jerry Wang gave a nice talk this week on his generalization of Manjul’s work on
pointless hyperelliptic curves to hyperelliptic curves with no points over any field of
odd degree (equivalently, Pic1 is pointless). This work (link here, also [BGW17]) is
joint with Manjul and Dick, so the exposition is predictably of high quality. But I
wanted to mention a result that arose during the talk which I found quite intriguing.
Namely, given the intersection X of two quadrics P and Q in projective (2n + 1)space, the variety of projective n-spaces passing through X turns out (over thecomplex numbers) to be an abelian variety. For n = 1 this is pretty familiar, but,for general n, I hadn’t seen any construction like this before. It gives, for example,
explicit constructions of equations for abelian varieties in surprisingly low degree.
It brought me back to a lecture I once went to by Beauville as a graduate student
when he talked about intermediate Jacobians (wait — perhaps this construction
also has to be isomorphic to an intermediate Jacobian . . . ). Is it possible (in someweak sense) to classify all varieties whose variety of maximal linear subspaces isan abelian variety of suitably high dimension? Are there varieties in which thisconstruction gives rise to abelian varieties which are not isogenous to Jacobians?
The geometric result is due (independently) to several authors, but, in a solo paper
here, Jerry showed that the result is true arithmetically, and, even better, theconstruction can more precisely be described as giving an explicit torsor for thecorresponding Jacobian. This very nicely generalizes the classical picture between
pairs of quadrics and 2- and 4-descent.
Comment 31.1 (Jack Thorne). Dear Persiflage, I cannot resist mentioning myfavorite example of this kind of construction, which relates to smooth hyperplanesections H of G(4, 8). H has dimension 15; it has primitive cohomology only in the
middle degree, which gives a Hodge structure of dimension 6 and level 1. Thus theintermediate Jacobian is a PPAV.

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F. CALEGARI

Over the complex numbers, the PPAV which arise this way are exactly the
Jacobians of the non-hyperelliptic curves X of genus 3. What about over a generalfield K of characteristic zero? Then the Jacobian of X arises from a K-rational
hyperplane section exactly when the curve X has a K-rational flex in the canonicalembedding.
Comment 31.2 (Wholesome Breakfast). The variety of Pn−1 s in X is indeed anintermediate Jacobian, see [Don80].

32. Local representations occurring in cohomology
Tue, 05 Nov 2013
Michael Harris was in town for a few days, and we chatted about the relationshipbetween my conjectures on completed cohomology groups with Emerton and therecent work of Scholze. The brief summary is that Scholze’s results are not naively
strong enough to prove our conjectures in full, even for PEL Shimura varieties. Motivated by this discussion, I want to give two quite explicit challenges concerning the
mod-p cohomology of arithmetic locally symmetric spaces. The first I imagine will
be very hard — it should already imply a certain vanishing conjecture of Geraghty
and myself which has strong consequences. However, the formulation is somewhat
different and so might be helpful.
Fix an arithmetic locally symmetric space X corresponding to a reductive group
G over Q. Let ℓ and p be distinct prime numbers. Consider the completed cohomology groupsb d (Fℓ ) = limH d (X(K), Fℓ ),b d (C) = limH d (X(K), C),
H
H
→

→

where we take the completion over all compact open subgroups. The limit has anaction of G(A) for the finite adeles A, and so, in particular, has an action of G(Qp ).
b d (Fℓ )? Here is a guess:
What irreducible G(Qp ) representations can occur in H
Conjecture 32.1. If the smooth admissible representation π of G(Qp ) occurs as
b i (Fℓ ), then there exists an irreducible reprean irreducible sub-representation of H
sentation Π of G(Qp ) in characteristic zero such that:(1) The Gelfand–Kirillov dimension of Π is at least that of π. Equivalently,nndim ΠK(p ) ≫ dim π K(p ) .
(2) Let rec(Π) and rec(π) be the Weil-Deligne representations associated to Πand π respectively by the classical local Langlands conjecture and the mod-ℓlocal Langlands conjecture of Vigneras. Then(rec(Π))ss ≃ (rec(π))ss .
b j (C) for some j ≤ i.
(3) The representation Π occurs in H
Roughly speaking, this conjecture says that the irreducible representations occurring in characteristic p are no more complicated than those which occur incharacteristic zero. One naive way to try prove this conjecture would be to showthat any torsion class lifts to characteristic zero, at least virtually. This conjecture
is too strong, however, as can be seen by considering K-theoretic torsion classes instable cohomology — the mod 3 torsion class in H 3 (GLN (Z), F3 ) can never lift toPERSIFLAGE: MATH BLOG POSTS

59

characteristic zero for sufficiently large N because the cohomology over Q is zerofor all congruence subgroups by a theorem of Borel. The conjecture as stated seems
very hard.
In a different direction, here is the following challenge to those trying to understand completed cohomology through perfectoid spaces. (I expect one can prove
this by other means, but I would like to see a proof using algebraic geometry.)
Problem 32.2. Fix an integer d, and let Xg be the Shimura variety corresponding
to the moduli space of polarized abelian varieties of genus g. Prove that, for ge d (Xg , Fp ) is finite over Fp .
sufficiently large, the completed cohomology group H
An equivalent formulation of this problem is to show that the only smooth ade d (Xg , Fp ) are one dimissible GSp2g (Qp )-representations π which occur inside H
mensional.
Notes 32.3. This is still completely open and really interesting. It is of course
related to the congruence subgroup problem.

33. DaleksThu, 05 Dec 2013
I’ve wanted to write a post about the new Doctor Who series for a while, but thisis not that post. Instead, this post is about a Macintosh game called Daleks, which
I first played on a Mac 512 (running OS 3) in the mid-’80s. Research indicates that
this game was based on a Unix game called robots, and that some wag came up
with the idea of rebranding it under the name of the classic Doctor Who monster.
The first version I played had a very peculiar high score table: all the high scores
were attributed to a fellow named “fingers,” and the high scores were not in any
sort of numerical order. Moreover, no matter what one scored, it was impossible topermanently make it onto the high score list. My second encounter with the gamewas during a summer research program with Alf van der Poorten in 94/95, where I
was impressed to find that he had broken 10000. Later, I had a copy on an ancientlaptop given to me by my brother, and still later, I played classic Daleks in classic
mode under OS X. I am not ashamed to say that I am proud of my high score,
15670, a feat which is probably meaningless to almost everyone. Anyway, today’s
post is about some mathematical problems related to this game. If you have a mac
computer, I recommend playing around with some current incarnations of the game,for example super daleks (presumably robot is available on Gnome games as well):Consider the following game: the Doctor is positioned on the lattice Z2 at theorigin (0, 0), and daleks are distributed on the rest of the lattice with uniform
density ρ ∈ (0, 1). It turns out that it is more convenient to work with the parameterq = 1 − ρ, although all the graphs below are drawn with respect to ρ. On each
move, the dalek at point P moves to the unique neighbouring square (out of 8)
which is closest to the origin in the taxicab metric. In particular, daleks always
move diagonally towards the origin unless they lie on one of the axes. If two or
more daleks occupy the same square, then they crash and are destroyed, leaving apile of debris which remains at that square forever. Moreover, any other dalek which
later moves on to the same square now occupied by the debris is also destroyed. If
a dalek reaches the origin unscathed, the Doctor is exterminated. However, if the

60

F. CALEGARI

debris resulting from dalek collisions prevents all other daleks from reaching theorigin, then the Doctor survives. What is the probability that the Doctor survives?
(In the computer game the Doctor can also move about, but not in our simplified
version.)If a dalek starts on either the diagonal or the anti-diagonal then it will nevercrash with another dalek (in general, daleks can only crash on the axes). Hence, we
modify the game by forbidding daleks from either of these diagonals. This effectively
separates the playing area into four quadrants which do not interact, and so we may
as well confine ourselves to a single quadrant, and assume that all daleks lie in thequadrant (1, 0) + Q where Q = (x, y) with x ≥ |y|. A sample game is as follows,with the positions at time t = 0, 1, and 2. The Doctor will win this game, because
the debris at (1, 0) will prevent all other daleks in the quadrant from reaching theorigin (they will crash into the debris and be destroyed):

Figure 1. Three consecutive frames: the daleks are all destroyed
Definition 33.1. Let Qd be the truncated quadrant consisting of (x, y) with |y| ≤x < d. Let wd denote the probability of surviving the game where daleks only exist
with density ρ in the quadrant (1, 0) + Qd .
For example w0 = 1, and w1 = q = 1 − ρ. It is clear thatX
1 − wd =E(P ),P ∈(1,0)+Qd

where E(P ) is the expectation of being exterminated by a dalek which originatesat point P . (If there is a dalek which kills the Doctor, it is unique.) Note that E(P )is independent of d, providing that P ∈ (1, 0) + Qd .
Definition 33.2. The occludation O(P ) of P consists of the squares R different
from P where a dalek at square R will reach the origin before or at the same timeas P , and which reach the x-axis at least as near to the origin as P reaches thex-axis.
For example, here is a point P = (1, 0) + (4, 3) together with its occludation
shaded in blue (the green region to be described later):
The daleks R ∈ O(P ) are those for which P is in the shadow of R, namely, thoseR which eventually occlude P from the origin (thus the name). It is not a great
name, but I couldn’t think of anything better. Explicitly, if P = (x, y), thenO(P ) ∪ P = {(a, b) ∈ (1, 0) + Q such that a < x, and a − |b| ≤ x − |y|}

PERSIFLAGE: MATH BLOG POSTS

61

P

D

Figure 2. The occludation of P = (5, 3)
The Doctor can only be killed by dalek at point P if the occludation O(P ) is emptyof daleks. The reason is that any dalek R in the occludation can only crash at points
on the x-axis where P must eventually travel, and R will reach this point either ator before P does. We have
|O(P )| = |O((x, y))| = x2 − y 2 − 1.
On the other hand, conditional on the assumption that the occludation contains
no daleks, then the probability that P exterminates the Doctor only depends on y;
namely, it is equal to the probability of surviving the game with Qd and d = |y|.
This the quadrant not in the occludation of P . For example, with P = (5, 3) as in
Figure 2 above, the dalek at P (if the occludation is empty) will reach the doctor
only if the daleks in the green region, corresponding to Q3 , do not survive. It follows
that2
2
E(P ) = q |O(P )| (1 − q)w|y| = q x −y −1 (1 − q)w|y| ,and hence
1 − wd =dX
X

E((n, m)) =n=1 |m|≤n

d
X
X

2

2

q n −m −1 (1 − q)w|m| .

n=1 |m|≤n

We may simplify this slightly by writing
wd−1 − wd = (1 − wd ) − (1 − wd−1 ) =

X

2

2

q d −m −1 (1 − q)w|m| ,|m|≤dThis simplifies even further to(wd − wd+1 ) − q 2d+1 (wd−1 − wd )
X
X 2 2
2
2
=
q (d+1) −m −1 (1 − q)w|m| − q 2d+1
q d −m −1 (1 − q)w|m||m|≤d+1|m|≤d

2d= 2q (1 − q)wd ,and hence, subject to w0 = 1 and w1 = q,
wd+1 = (3q 2d+1 − 2q 2d + 1)wd − q 2d+1 wd−1 .

62

F. CALEGARI

The resulting recurrence relation for wd gives a decreasing convergent sequence (foreach fixed q and also in Z[[q]]) with limit
w∞ = q − 3q 3 + 3q 4 − 2q 5 + 2q 6 + 4q 7 − 13q 8 + 13q 9 + . . .
Here is a graph of this function in Figure 3 (with respect to ρ, remember thatρ = 1 − q):Figure 3. The chance of survival w∞ for dalek density ρ
Although it appears from the graph that the maximum occurs at ρ = q = 1/2,
closer inspection reveals that the optimal density is ρ = 0.517208 . . . The maximum
value is approximately ∼ 0.28116 . . ., which means that, on an entire plane with all
four quadrants, the largest possible chance of winning (with daleks on the diagonal
and anti-diagonal removed) is approximately 1 in 160. Note that as ρ → 1, we
certainly have w∞ → 0. As ρ → 0, it is also clear that one should expect the first
dalek on a line to survive, which means that w∞ should tend to 0 as q → 1 (as see
in Figure 3), that is not yet apparent from the formula above.
33.3. Reverse The Polarity. Here is a different way to estimate w∞ , this timefrom below. In order for the Doctor to survive, two or three daleks must eventuallycoincide at (1, 0). Call such daleks savior daleks. All savior daleks must be in thesame row, and at least one such dalek must lie on the edge of the quadrant. Let us
now consider the probability sn that one will be “saved” by a dalek in the nth row.
If P = (n, n − 1) is a savior dalek, then the dalek P creates the first crash at the
point (0, 1), and no dalek exterminates the Doctor before this point. It follows thatno daleks may occlude P , and hence O(P ) must be free of daleks, with the possible
exception of −P . Note that |O(P )| = n2 − (n − 1)2 − 1 = 2n. Suppose that −P isnot occupied. Then (assuming that O(P ) is empty) P will be a savior dalek if andonly if the remaining restricted quadrant of size n − 1 would otherwise result on
the Doctor being exterminated at the final term, equivalently, the probability that,from a quadrant of size n − 1, the Doctor would be exterminated by a dalek in thelast row. Yet the probability of this is(1 − wn−1 ) − (1 − wn−2 ) = wn−2 − wn−1 ,and hence the contribution to sn is
2q 2n−2 (1 − q)(wn−2 − wn−1 ).

PERSIFLAGE: MATH BLOG POSTS

63

On the other hand, if both P and −P are to be savior daleks, then one simply
requires that, in addition to the rest of occlusion O(P ) being empty, that in theremaining quadrant of size n − 2 (removing the final row and the occlusion) no
Doctor is exterminated, and this has probability wn−2 . Hencesn = 2q 2n−2 (1 − q)(wn−2 − wn−1 ) + q 2n−3 (1 − q)2 wn−2 .
Let tn be the probability that there exists a savior dalek at a row at most n. Thenclearly
tn =

n
X

sm .

m=1

Moreover, we naturally have inequalities wn ≥ tn , andw∞ = lim wn = lim tn ,
n→∞

n→∞

where the limit is pointwise and q ̸= 1. However, the behavior of wn and tn is quite
different in the regime ρ → 0 or q → 1, as (Figure 4) of w5 ≥ t5 shows. This is
not so surprising, as ρ → 0 one expects that w∞ = t∞ = 0, but the dalek which
destroys the doctor will be expected to become further and further away.

Figure 4. Upper and lower bounds: t5 ≥ t∞ = w∞ ≥ w5

33.4. Behavior as ρ → 0. In order to estimate the behavior of w∞ as q → 1,
we consider the following problem: What is the probability that the first row withany dalek contains exactly two daleks, and that at least one of these daleks liesat the edge of the quadrant? In such a situation, the daleks necessarily annihilate
one another at (1, 0), and the Doctor is saved. Call the resulting function A(q), so
w∞ ≥ A(q). Since there are (4n − 1) pairs of elements in 1, . . . , 2n + 1 which contain64

F. CALEGARI

at least one of the end points, we have
∞
∞
X
X
2
2
A(q) =(4n − 1)q n (q 2n−1 (1 − q)2 ) = (q − 1)2 q −2
(4n − 5)q n
n=1

n=2

= (q − 1)2 q −2

5+q+

∞
X

2

4nq n − 5

n=0

∞
X

!
qn

2

n=0

!
∞
5 X n25n2+q+2q|n|q −= (q − 1) q2
2 n=−∞
n=−∞
!
∞
∞
X
X
2
2
5
q n (1 + O(1)) + O((q − 1)3 ).
= (q − 1)2 2|n|q n −2 n=−∞
n=−∞
∞
X

2 −2

Let q = e−τ . As q → 1, we have τ → 0, and so 1 − q ∼ τ . On the other hand, by
Poisson summation, we have
r
∞
X
2
π
e−n τ ∼+ O(τ N ),
τ
n=−∞
∞
X

2

|n|e−n /τ ∼

n=0

τ
τ2
1 1
− −
−
+ O(τ 3 ),
τ
6 60 252

from which it follows easily that A(q) ∼ 2τ ∼ 2(1 − q), and thus
lim sup w∞ ≥ 2(1 − q).
q→1

In fact, we can actually prove that12
w∞ (e ) = + O √ .
ττIn other words, the simple model above is very accurate in the limit q → 1. However, the combinatorics required to prove this are actually somewhat involved andannoying, and this is a blog, so I will omit it here. (The arguments are somewhat
timey–wimey.)−τ

33.5. A conditional game. Consider the game which is pre-conditioned on thefirst square (1, 0) being empty. Since that square containing a dalek is not consistent
with survival, the new game results in a win with probability:
w∞
w∞
c∞ =
=
.
1−ρ
q
Apropos of nothing, here’s Davros enjoying a cuppa in Figure 5.
33.6. The TARDIS. Suppose that the Doctor has a TARDIS. This allows him,
at any point, to dematerialize and the materialize somewhere else. In the context
of the classic daleks game, the player appears at a random point in the plane withuniform distribution. Although this doesn’t quite make sense on an infinite plane,
we can take it to mean that we have moved sufficiently far away from the axes
that it is as if the game has started again. Hence this will be the context in which
we shall consider rematerialization, namely, as if the game has started again. The
catch with using the TARDIS is that the Doctor may materialize next to a dalek,
in which case he is immediately exterminated. The optimal strategy is to continue
to continue rematerializing until one has a winning game. The chance of surviving

PERSIFLAGE: MATH BLOG POSTS

65

Figure 5. Davros enjoying a cuppa
a rematerialization is (1 − ρ); the resulting game is the same, but now conditional
on not being annihilated by the initial dalek, hence is equivalent to the conditional
game described above. It follows that the chances of survival are:(2 − q)w∞.
d∞ := w∞ + (1 − w∞ )(1 − ρ)(c∞ + (1 − c∞ )(1 − ρ)(c∞ + . . . =1 − q + w∞
The asymptotic behavior of this function as ρ → 1 (or q → 1) requires the correct
asymptotic w∞ ≃ 2(1 − q), and from this we can deduce thatd∞ → 2/3 as ρ → 0.
In this case, we see that the optimal probability is that the density ρ tends to zero.
A graph of d∞ is in Figure 6.

Figure 6. The chance d∞ of surviving with a TARDIS
33.7. The Sonic Screwdriver. Like John Nathan-Turner, I find the sonic screwdriver to be somewhat ridiculous. Although it does exist in some versions of thegame, I will only mention a minor modification here. The “sonic” in the game allows
the Doctor to survive for one round when he would otherwise be exterminated; it
has only one use. We shall additionally assume that the sonic can only be used on
the very first round. This essentially changes the game (at the beginning) into theconditional game described above. If one is allowed to use the TARDIS as above,
the resulting probability of winning isw∞.
(c∞ + (1 − c∞ )(1 − ρ)(c∞ + (1 − c∞ )(1 − ρ)(c∞ + . . .) =q(1 − q + w∞ )

66

F. CALEGARIAs ρ → 1, this function tends to 1, and as ρ → 0, it tends to 2/3. The behavior ofthis function in a neighbourhood of 0 appears to be of the form
2/3 − Aρ1/2 + . . .
for some constant A, possibly around 0.3. Note that this function is not monotone(see Figure 7); the most dangerous density of daleks is approximately ρ = 0.127,
where the resulting probability of surviving dips below 3/5.

Figure 7. The change of winning with the sonic screwdriver

Figure 8. The vanilla game at the optimal value ρ ∼ 0.517 —the Doctor lives!
Notes 33.8. I’m not sure if anyone ever read this post. I made some small clarifications and added another diagram.

PERSIFLAGE: MATH BLOG POSTS

67

34. The mystery of the primes
Sat, 04 Jan 2014
No, this is not the sequel to Marcus du Sautoy’s book, but rather a curious observation regarding George Schaeffer’s tables of “ethereal” weight one Katz modular
eigenforms (which you can find starting on p.64 of his thesis, ultimately downloadable from Proquest but available more directly from the unstable link here). Let
N be a positive integer, let χ be an odd quadratic character of conductor dividing
N , and let p be an a prime not dividing N . Recall that the reduction map between
spaces of Katz modular forms:
M1 (Γ1 (N ), χ, Zp ) → M1 (Γ1 (N ), χ, Fp )is not surjective in general, although it will be surjective for all but finitely many p.
For what pairs (N, p) is the map not surjective? As originally observed by Mestre(and predicted by Serre), such pairs do exist. One way to think of the primes whicharise in this way are as the primes dividing the torsion subgroup of H 1 (XH (N ), ω),
where H ⊂ (Z/N Z)× is the subgroup of squares, and XH (N ) is the corresponding
modular curve (as a stack, if necessary) over Z[1/N ]. The reduction mod-p map isHecke equivariant; let m denote a maximal ideal of T in the support of the cokernel.
Associated to m is a Galois representation:ρ = ρm : GQ → GL2 (Fp )which is unramified at primes not dividing N (including p). It is not necessarily thecase that ρ does not lift to characteristic zero, although this is typically the case forthe examples arising in the tables (and is always the case if the image of ρ contains
SL2 (Fq ) for some q ≥ 5). Not surprisingly, it turns out there are no such forms forsmall N . The reason is that the fixed field K of the kernel of ρ would be a highdegree field with a root discriminant which (for very small N ) would violate theGRH discriminant bounds of Odlyzko, and for smallish N would still give fields ofunusually low root discriminant.
Of course, as N increases, there do exist many such forms, sometimes in quite
large characteristic. However, something peculiar happens in the range of the tables, namely, there is not a single example with N prime. This leads to the (incredibly) vague question: can this be predicted in advance? If there is going to exist a
PGL2 (F199 ) representation unramified outside N for small N , is it more likely thatN = 82 (see here) rather than N = 83? One can try to use heuristics predicting
the number of fields with certain ramification behavior, but these heuristics aremuch better behaved for fixed Galois groups G = PGL2 (Fp ) or G = PSL2 (Fp ) andincreasing discriminant, not in the regime of fixed root discriminant and Galois
group G as above for varying p. Is there any conspiracy ruling out certain kinds ofnumber fields with small root discriminant ramified at a single prime? For example,
if you fix some arbitrary constant, say M = 1000, do there exist infinitely many
primes p such that there is a number field K different from Q which is unramified
away from p and has root discriminant less that M ?
These questions are hard to pin down, because they are really questions concerning the law of small numbers. Namely, they ask about the behavior/distribution ofvarious quantities in the range before asymptotic behavior begins. Since the asymptotic behavior is (in these contexts) already mostly conjectural, it’s probably hardto say anything intelligent about these even more delicate questions. (Idle question:

68

F. CALEGARI

are there similar problems for which one does understand what happens before theasymptotic regime begins, even heuristically?)
Here’s one reason to consider these questions. Suppose one wants to compute“ethereal” Siegel modular forms. At what level does one first expect to find suchforms? The numerics above suggest that it might be easier to find such forms at
small composite levels rather than prime levels. Is that a reasonable inference?
Comment 34.1 (Dick Gross). I thought about this question when writing mypaper on companion forms [Gro90], and basically gave up. It’s predicting when aline bundle of small degree on a curve has a larger space of sections (mod p). That’s
why Serre’s criterion is so subtle — this jump occurs precisely when you have an
odd 2-dimensional modular representation which is unramified at p which does notcome from a 2-dimensional representation over C. Serre’s conjecture in weight 1
was what attracted me to the subject of companion forms in the first place, and I
found it amusing that it was precisely the weight 1 situation that I couldn’t resolve
completely. Fortunately, Robert Coleman understood what I was doing much better
than I did, and finished it off [CV92]. I should say that Mestre’s computations for
p = 2 were more convincing than any of the proofs!
Comment 34.2 (Persiflage). In response, I noted: even classical (odd) Artin representations can give rise to torsion classes, exactly when their mod-p reductions
admit “extra” unramified p-adic deformations. For example, consider a modular
representation:
ρ : GQ → S3 ,→ GL2 (C).
Let K/Q be (any of the) corresponding imaginary cubic fields inside the fixed field of
the kernel of ρ. If p ≥ 3 is prime, then ρ admits a non-trivial unramified deformation
to Fp [ϵ]/ϵ2 exactly when p divides the class number of K. This deformation willbe (by [CG18a]) Katz modular but does not come from characteristic zero, so it
will give rise to torsion in H 1 (X, ω), or equivalently mod-p classes which don’t lift
to characteristic zero. The smallest example (of the exact flavour above) occurs forthe field
K = Q(θ)/(θ3 − θ2 + 7θ − 6)of discriminant −3 · 521, with class number hK = 5. As Dick then noted: the cubic
field K has a unit group of rank 1, so its class number will rarely be divisible by p
— by the Cohen–Lenstra heuristic — and the existence of unramified deformations
is still a sporadic phenomenon.
Notes 34.3. See the remark of George Boxer in §112

35. Gross Fugue
Sat, 11 Jan 2014
Here are some variations on the theme of the last post § 34, which is also related
to a problem of Dick Gross.
In this post, I want to discuss weight one modular forms where the level varies
in the “vertical” aspect (that is, N is a growing power of a fixed prime, rather thansimply an increasing integer). First of all, consider the spaces
S1 (Γ(M · ℓn ), C)PERSIFLAGE: MATH BLOG POSTS

69

for fixed M and growing n. For example, if M = 1, the corresponding Galois
representations are associated to number fields unramified outside a single prime ℓ.
Given a cusp form f , the twists f ⊗ χ by a finite order character of ℓ-power order
will also be modular (possibly with larger n), so all the finiteness statements below
should be interpreted “up to twist.”
The first observation is that there exist only finitely many exceptional cusp
forms (with projective image A4 , S4 , A5 ) because, by a theorem of Hermite, there
are only finitely many fields with a fixed Galois group unramified outside a fixed
set of primes. (This echos a very general conjecture which says [very loosely] that if
one fixes an infinitesimal character and varies the level in a ℓ-adic tower, one should
only see finitely many automorphic forms which do not arise via functoriality fromconstructions using discrete series.)
The second observation is that all the other cusp forms are easy to describe:they are induced from finite order characters of a fixed number of easily determined
quadratic fields K.
So far so good. But what happens if one replaces C by Fp , or more generally
Qp /Zp ? Here is the following optimistic guess:
Question 35.1. For a fixed prime p ̸= ℓ, are there only a finite number of nonliftable forms in the p-power tower?
Here we have to take the usual caveats — not only do we have to take into
account twisting, but also the GL2 (Qℓ )-action (old forms).
This question is supposed to be a GL(2)-analogue of Washington’s famous theorem on the p-part of the class group in the ℓ-adic cyclotomic tower. We shall seethat it is more than an analogy. What will the source of torsion classes be?
(1) One source are Galois representations: ρ : GQ → GL2 (Fq ) with big image
that are unramified outside ℓ (with q a power of p). Of course there are
only finitely many such representations for any fixed q, but some heuristics
I learnt from Akshay convince me that there should only be finitely manyeven if one varies q over all powers of p (taking into account twisting, ofcourse).
(2) Another source of torsion comes from deformations of big image Galois
representations ρ as above, or from representations with projective image
one of the exceptional groups. Since each unramified deformation ring will
be finite, each ρ should only give rise to finitely many extra torsion classes.
(3) A third source of torsion classes comes from reducible indecomposable
representations. The residual representations ρ which arise in this way
occur when L(0, χ) is divisible by p for an odd character χ of finite order.
In particular, there are only finitely many such representations which occur
exactly if all but finitely many L-values L(0, χ) are prime to p, where χis an odd character of conductor M times a power of ℓ. But this exactly
the content from Washington’s Theorem (the oddness assumption is not,however, necessary).
(4) The final class come from deformations of dihedral representations. If
ρ is the induction of a character ψ of K/Q, then the tangent space to theunramified deformation ring of ρ gives rise to torsion classes when there
are no everywhere unramified classes in H 1 (Q, Ind(ψ/ψ c )) — the unramified dihedral representations in H 1 (Q, ηK ) are seen globally. By inflationrestriction, this is equal to a certain invariant part of the class groups of the

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anti-cyclotomic tower. There are non-vanishing results concerning L-valuesof Hida that are relevant here, although I haven’t checked to see if they
imply the finiteness statement or not.
The only way to start thinking about answering this question is to think in termsof the torsion in the cohomology of modular curves. But, I confess, I do not reallyhave any ideas on how to prove it. (To be honest, I still find Washington’s proof
very mysterious.)
On related matters, it would be nice if one could prove — say by analytic means— that H 1 (XH (N ), ω) has torsion (prime to N ) for all sufficiently large N . Taking
N to be a power of a prime, this would give a different construction of non-solvable
Galois representations unramified outside a single prime (for all ℓ) from the one
suggested by Dick and carried out in for ℓ ∈ {2, 3, 5, 7} by Dembélé and others.
Moreover, although (as in those examples) it would involve the group PSL2 (F) as asimple factor, the residue characteristic would be different from ℓ rather than equal
to ℓ in the previous constructions. (George Schaeffer told me he tried computing
torsion coming from XH (343) but didn’t find any.)√ It might also (for suitable H)
give a lower bound for π1 (OK ) where K = Q( −D) which is better (at least forsome primes) than one gets from class numbers.

36. Local crystalline deformation rings
Sat, 08 Feb 2014
I just returned from a very pleasant conference in Puerto Rico courtesy of theSimons Foundation (general advice: if you live in Chicago, always accept invitations
to conferences in January). One thing I learnt from Toby Gee was the following niceobservation. Suppose that
ρ : GQ → GL2 (Fp )is a modular Galois representation, which for convenience we shall assume is unramified outside p. Consider deformations of this representation which are crystalline
with fixed Hodge–Tate weights [0, k − 1] where k is even. According to Kisin, theglobal minimal crystalline deformation ring contains a point on every component of
the corresponding local crystalline deformation ring. (All discussions of components
refer to the generic fibres.) One natural question is how many components the local
deformation rings actually have (when the weight is very small, it’s usually the casethat there is only one such component and it is smooth — this was crucial in theoriginal Taylor–Wiles method before Kisin). For higher weight, one can distinguishbetween components which are “ordinary” and “not ordinary”, but it is not clear
what else there is. (Indeed, Kisin seemed to think some years ago that this would
be it, using the meta-argument that amongst any finite set one should be able todistinguish different points by some naturally available property.)
Now suppose we also now assume that ρ is locally reducible. According to Buzzard’s conjectures, all the slopes of the global crystalline lifts of ρ will be integral.
Suppose one wants to prove this by local methods. Then one is ultimately led toconjecturing that each component of the local crystalline deformation ring has afixed integral slope (recall we are in the locally reducible case, this is certainly false
for locally irreducible representations in general). As a first consequence, one seesthat in very high weights there will be many different components. Moreover, if one

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takes a different global representation ϱ which is the same locally as ρ, then theset of slopes arising from lifts of ϱ will be the same as for ρ. These ideas do notquite give a complete conjectural explanation of why Buzzard’s slope conjectures
are true, but it is a good start.
Something that is a little disturbing in this picture, however, is the case when ρis reducible. It becomes clear that, in high weight, there will be many crystalline
representations with reducible residual representations, but the set of components
of local crystalline deformation space which have a global point will be a proper
subset of the set of components (assuming that components can be distinguishedby slope). For example, all the slopes at level one when p = 2 are (besides theEisenstein series) ≥ 3, but there certainly exist modular forms of higher tame levelwith the same local residual representation of slope one. So is there any way to
predict when a reducible representation will have a global lift on any component oflocal deformation space?
In fact, the failure of lifts in the reducible case is an old problem. In the mostnaive sense, one can find reducible representations at levels where there are no cusp
forms, but to play the game honestly we should also allow (globally) reducible lifts.
Perhaps the first genuine example corresponds to extensions:
1 → Z/pZ → V → µp → 1
where the extension is completely split at p but ramified at an auxiliary prime N .
These representations are locally split and so certainly admit local lifts (namely,
Zp ⊕ Zp (1)). If p ≥ 3, then such extensions exist whenever N ≡ ±1 mod p, but(by Mazur) one knows that there exist weight two level Γ0 (N ) lifts only whenN ≡ +1 mod p (in fact, one can prove the analogous claim that there only exist
global crystalline lifts with the appropriate conductors under the same congruence
condition). This is related to the general problem of understanding when certain
reducible representations can be lifted to cusp forms, which seems to be a tricky
problem (Ken Ribet’s student Hwajong Yoo has thought about this, see [Yoo19]).
This also reminds me of a fact I learnt from Kevin Buzzard. Take the representation
ρ : GQ → GL2 (Q2 )associated to the cusp form ∆. Then there exist lattices for this representation suchthat the corresponding residual representation is any one of the four (three nonsplit) extensions of Z/2Z by itself which are unramified outside 2. (Question: does
this immediately imply the same is true for all 2-adic representations coming fromlevel one modular forms?)

37. The thick diagonal
Fri, 14 Mar 2014
Suppose that F is an imaginary quadratic field. Suppose that π is a cuspidal
automorphic form for GL(2)/F of cohomological type, and let us suppose that it
contributes to the cohomology group H 1 (Γ, C) for some congruence subgroup Γ ofGL2 (OF ). Choose a prime p which splits in F so that π is ordinary at v|p. Hida
proves that the corresponding cohomology class lives in a Hida family H over theappropriate weight space, which in this case is (up to connected components) just
Λ = Zp [[X, Y ]]. However, unlike the classical situation, this Hida family will not

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be flat, because the specialization to any local system which is not invariant undercomplex conjugation is necessarily finite. Thus the support D of H has co-dimension
at least one over Λ. Hida proves that it does indeed have co-dimension one.
What does the support D of H look like? Let us suppose that we are normalizing
Λ so that the point X = Y = 0 corresponds to π. One can imagine two possibilities:(1) D contains the diagonal ∆ : X = Y .
(2) the components of D passing through X = Y = 0 only contains finitely
many classical points.
It seems as though these are the only possibilities. Certainly, by a Zariski closure argument, D either contains the diagonal ∆ or intersects it in finitely many
points. Hence, it is true that if the first condition does not hold, then the components passing through [0, 0] contain only finitely many crystalline automorphic
forms. However, there could be more classical points on D, namely, those of parallel weight but non-parallel finite order nebentypus character. To be concrete, thepossible points of Spec(Λ) which may give rise to automorphic forms have (withsome normalization) the following shape:
1 + X 7→ (1 + p)k ζ,

1 + Y 7→ (1 + p)k ξ,

where ζ and ξ are p-power roots of unity, and k is a non-negative integer. So one isreally considering not simply the intersection of D with the diagonal ∆, but withthe thick diagonal ∆∆, which is the union of the infinitely many translates of ∆ byp-power roots of unity. In particular, the Zariski closure of ∆∆ is all of weight space.
I wrote a paper with Barry Mazur [CM09] where, as an illustrative example, we
found an explicit Hida family which did not satisfy the first condition and claimed
that it therefore satisfied the second, whereas we should only have made the weakerclaim that D (which was irreducible in this particular case) contains only finitely
many crystalline points. (The main point of the paper was, by studying infinitesimal
deformations of Artin representations, to give evidence that D should only evercontain the diagonal when π is either a base change form or CM.) The error was
pointed out to me by David Loeffler [Loe11, §5.1].
I am pleased to say, however, that my student Vlad Serban has overcome thiserror [Ser22] (see also [Ser18])! Namely, suppose one has a non-trivial power seriesΦ(X, Y ) ∈ Zp [[X, Y ]], and suppose that
Φ((1 + p)k ζ − 1, (1 + p)k ξ − 1) = 0
for infinitely many triples (k, ζ, ξ) with k a non-negative integer, and ζ, ξ, p-powerroots of unity. Let D be a component of the zero set Φ(X, Y ) = 0 passing through(0, 0). Then, after possibly replacing the roles of X and Y , Vlad proves the following.
Either:(1) D contains the diagonal ∆,(2) Φ(ζ − 1, ζ N − 1) = 0 for all p-power roots of unity ζ, for a fixed N ∈ Zp .
Certainly the latter is possible, because one could have Φ(X, Y ) = (1 + X)N −(1 + Y ). In fact, he proves a more general theorem than this for all the components(not necessarily passing through (0, 0). After translation, this amounts to working
over ramified extensions of Zp .
This theorem allows one to prove (with finite computation) that any particular
D only contains finitely many points (when that is true). It also shows, without
any computation at all, that D either contains ∆, or it only contains finitely manyPERSIFLAGE: MATH BLOG POSTS

73

classical points of weight different from π. A nice way to think about this theorem
is that it is of the flavour as the multiplicative Manin–Mumford conjecture. That is,
one is intersecting a sub-variety with a particular arithmetically defined discrete set(inside ∆∆), and one wants to deduce that this can only happen for a well definedgeometric reason. In fact, if one replaced Φ(X, Y ) by a polynomial with coefficientsover C and specialized to the case when k is always zero, then this would exactly
be the Multiplicative Manin-Mumford conjecture in two dimensions.
As a special case, letting k = 0, one ends up with the following pretty result.
Suppose that Φ(X, Y ) ∈ Zp [[X, Y ]] is a power series, and suppose thatΦ(ζ1 − 1, ζ2 − 1) = 0
for infinitely many pairs of p-power roots of unity. Then the zero set of Φ contains atranslate of Gm . This exactly answers the puzzle asked by Jordan here. Explicitly,
it says that the only quotients of Zp [[Z2p ]] of co-dimension one which have lots of“arithmetic” points really do come from a one-dimensional subgroup!
I think that this special case (with k = 0) is probably easier than the generalcase, because one has other methods available. The argument was, however, inspired by a result of Hida which came up during his last number theory seminar
at Northwestern. Translated into the language of this post, Hida’s rigidity lemmacorresponds to the puzzle of Jordan above in the case when Φ(X, Y ) = Y − F (X)for some function F (X) ∈ Zp [[X]].

38. The congruence subgroup property for thin groups.
Sun, 09 Mar 2014
I finally had a chance to visit Yale, which (by various orderings) is the fanciest
US university at which I had never given a talk (nor even visited). The town itself
struck me, at first, as a cross between Oxford and New Jersey. That aside, mycoffee research led me to Blue State Coffee, which was more than up to the task of
preparing a decent 8 ounce latte. (As a comparison, it is significantly better than
Small World Coffee in Princeton. Small World has all the correct hipster attitude
without enough of the corresponding aptitude.) Mathematically, I had a great chatwith Hee Oh and Gregg Zuckerman over several hours. At one point, I raised thefollowing idle question about thin groups.
Problem 38.1. Let G = SLN (R) where N ≥ 2. Let Γ be an arithmetic lattice in
G. Suppose that Φ ⊂ Γ is a subgroup such that the following two conditions are
satisfied:(1) The Zariski closure of Φ in G is G.
b →Γ
b is injective.
(2) The induced map of profinite completions: ΦThen is Φ necessarily of finite index in Γ?
If Γ = SLN (Z), then the first condition implies that the image of the induced
map of profinite completions has finite index; I presume this is true more generally.
Hence the question asked can be phrased as follows: “can congruence subgroups be
determined by their pro-finite completions?” Alternatively, in the opposite direction,
one can ask: “are there thin groups which satisfy the congruence subgroup property?” I have no particular reason to believe that the answer to the question above

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is positive, and I might even guess that one could write down a counter-example,
but I don’t know how to write one down myself.
On the other hand, suppose that the answer to the question is positive. Thenit might prove useful for determining whether, given a finitely presented group
H := ⟨G | R⟩ and an explicit homomorphism:
ϕ:H→Γ
whether its image has finite index or (even more strongly) whether ϕ is an isomorphism onto a finite index subgroup. Namely, if the image of ϕ does not have finite
index, then a positive answer to the question above would imply that H must havea finite quotient which does not come from Γ, and (since finite quotients of H may
be enumerated) this leads to an algorithm which terminates if ϕ has infinite index.
On the other hand, if H does have such a quotient, then certainly ϕ will not be an
isomorphism onto a finite index subgroup.
This problem explicitly came up in some work of Curt McMullen (see question
5.6 of this paper), who produced explicit maps of various finitely presented groups
into lattices (not quite in SLN (R), but one can of course ask the more generalquestion for lattices in semi-simple groups of rank at least two) and asked whether
these maps were isomorphisms onto finite index subgroups. So the hope is that (inthe contexts in which one expected the answer to be negative) this could always be
answered by considering the pro-finite completion of the finitely presented groupin question. Alas, I believe that I explicitly tried to find non-congruence quotientsof the associated explicitly presented groups (in contexts where one expected ϕ tohave infinite index) and didn’t find any (not that I carried out this computation inanything approaching a sophisticated manner, of course).
Comment 38.2 (Anon). For some related (mainly negative) results see Bridson–
Grunewald’s paper [BG04b].

39. Robert Coleman
Tue, 25 Mar 2014
I was very sad to learn that, after a long illness with multiple sclerosis, Robert
Coleman has just died.
Robert’s influence on mathematics is certainly obvious to all of us in the field.
Most of my personal interaction with him was during my last two years as a graduate
student at Berkeley. We would chat in his office, and sometimes have lunch at Nefeli
caffe. Kevin and I had recently made some modest progress on Kevin’s crazy slope
conjectures, and much of that time with Robert was spent with me presenting crazyideas and predictions on the white board in Evans Hall while Robert looked on withhis classic look of amused skepticism. There would also be the occasional wine andcheese in his office, especially if an old visitor was in town.
I certainly didn’t know him as well as many others did, but I felt very honored
that he asked me to accompany him (as a grad student assistant) to China for his
ICM address. As it happened, the relevant hotels in China would not allow him tobring Bishop (his guide dog) along with him, so he didn’t end up going.
Mathematically, Robert was very original. I have no plans to attempt to summarize his research, but I just want to discuss one problem which he had thought

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75

about in recent years, namely, what the eigencurve looked like at the boundaryof weight space — especially in light of the description given by Kevin and Lloyd
Kilford when N = 1 and p = 2. Suppose one is given a Fredholm determinant∞X
det(1 − U T ) = P (T ) = 1 +an T nn=1where an ∈ Λ = Zp [[X]], and one wants to understand the spectrum of U at
the “boundary” of weight space, that is, when the valuation of X goes to zero.
For example, an interesting collection of points near the boundary are the classicalpoints with highly ramified nebentypus character. If an is not divisible by p, then thevaluation of an at a specialization of X close to one will coincide with the valuationof the reduction mod-p of an as an element of the discrete valuation ring Fp [[T ]],that is, it will be determined by the smallest non-zero coefficient of an modulo p.
Robert’s idea was to study the “halo” of the eigencurve, which intuitively speaking,should be an object cut out by a compact operator Uχ in characteristic p withcharacteristic power series P (X) mod p. If the valuations of the elements an (X)mod p define a Newton Polygon N , then the Newton Polygon at some point on the
eigencurve which is sufficiently close to the boundary should be a simple multipleof N . This is one of my favourite problems! I know Robert has some ideas on how
to approach this problem, but unfortunately I don’t know exactly what they wereor how much progress he had made. One natural question is whether this structure
will ultimately be purely explainable in terms of p-adic local Langlands. One evenmore basic question is what happens numerically on components of the eigencurve
corresponding to a representation ρ which is absolutely irreducible after restriction
to a decomposition group at p; I presume one sees the same behavior, but has
anyone checked this? Perhaps the easiest example to check would be to computethe slopes of forms on S2 (Γ1 (11 · 2n ), χ), where χ has conductor 2n .
Matt Baker has some further recollections of Robert here, and he also invites hisreaders to share there memories there.
Comment 39.1 (Toby Gee). It won’t surprise you to hear that I wanted to attack
this problem with p-adic local Langlands and R = T, but we never got anywhere.
I think Kevin Buzzard and I did think about this a little in 2006 — of course these
representations are still trianguline, so you can look in Colmez and see a concrete
description of p-adic Local–Langlands, and then try to compute reductions mod p.
Given that the expected answer is so simple, you might hope that there was some
nice structure that you’d see that would explain it, but we didn’t spot anything.
Then again, I think we were sufficiently disillusioned with the whole approach tothese kinds of questions that we didn’t even explicitly bash out a single example,
which is presumably possible.

40. Are Galois deformation rings Cohen–Macaulay?
Wed, 02 Apr 2014
Hyman Bass once wrote a paper on the ubiquity of Gorenstein rings [Bas63].
The first time they arose in the context of Hecke algebras, however, was Barry’s
Eisenstein ideal paper, where he proves (at prime level) that the completions Tm
are Gorenstein for all non-Eisenstein maximal ideals m of T except possibly those

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which are ordinary of residual characteristic two. He also shows that the completions at Eisenstein primes are also Gorenstein, although this is trickier and makes
fundamental use of the assumption that the level is prime. The Gorenstein propertyof various Hecke at non-Eisenstein maximal ideals was crucially used by Wiles todeduce non-minimal modularity lifting theorems. In the late 90’s, including aroundthe time I started graduate school, it seemed as though all Hecke algebras in weight
two were going to be Gorenstein (localized at non-Eisenstein ideals). One case remained, however, namely when char(k) = 2, andρ : GQ → GL2 (k)has the property that ρ is unramified at 2 and, moreover, the image of Frobenius at
2 is a scalar. (The other cases having been dealt with by results of Mazur, Wiles,
Ribet, and Buzzard.) But then it turned out, amazingly, that T was not always
Gorenstein. Lloyd Kilford found a counter-example at level N = 431. The naturalplace to look, of course, is at GL2 (F2 ) = S3 -representations. They have to comefrom a quadratic field K with class number divisible by three and such that 2 splits
completely in the corresponding unramified degree three extension of K. It also
makes sense to work at prime level, because this will make computing the integral
Hecke ring easier. The condition that 2 splits in K forces ∆K to be congruent to 1
mod 8, which certainly means the class number is odd. The condition that 2 split
in the corresponding cubic field is more subtle; if the class number of the field was
3, then this would be equivalent to the primes in K above 2 splitting principally inK, but this can’t happen for norm reasons. So one has to start with a quadratic
field K with ∆K ≡ 1 mod 8 and class number h = 3h′ for some h′ ≥ 1, and suchthat the class given by [p] for the prime above 2 does not generate the 3-Sylow
subgroup. The smallest prime number with this property is . . . N = 431. So it fails
at the first opportunity!
Nowadays we know, at least in the analogous context when p is odd and weare in weight p, that the appropriate Hecke algebras are Cohen–Macaulay. But weunderstand that the reason that these global Hecke algebras have these properties
is because the local Hecke algebras have nice properties. The idea of deducing facts
about the global Hecke algebra in the process of proving modularity lifting theorems started with Diamond, who found the first improvement to the Taylor–Wiles
method. Essentially, given an R = T theorem, one has a presentation of T as aquotient of a (power series over a) local deformation ring by a sequence of parameters. If the local deformation rings are nice (Complete Intersections, Gorenstein,
Cohen–Macaulay, etc.) then so is the global Hecke ring. Now this is only true inthe contexts where ℓ0 = 0; otherwise one is taking the quotient by “too many” relations (that is, not a sequence of parameters), and so there’s no longer any reason
to expect that T has those nice properties unless ℓ0 = 1 and T is finite.
So now we come to the question: are all local deformation rings Cohen–Macaulay?
Well, perhaps there is not really any reason to suppose that they are. Perhaps evenworse, there is a paper [San14] by Fabian Sander, a student of Vytas, proving that acertain deformation ring is not Cohen–Macaulay. But I am not deterred. My issue
is that one has to take the correct deformation ring. And the correct deformation
ring is the one that should include the extra data corresponding to the local Hecke
operators which may not come (at an integral level) from the Galois representation.
To take a well known example, consider ordinary p-adic representations of weightp. From a characteristic zero ordinary representation, one can always recover the

PERSIFLAGE: MATH BLOG POSTS

77(unique) eigenvalue of Frobenius on the unramified quotient. But this is not possibleat the integral level, because (for example) ρ could be locally trivial. This exactly
corresponds to the fact that in weight p, the Hecke operator Tp does not have to lie
in the algebra generated by the other Hecke operators (the “anemic” Hecke algebra— was that term coined by Ken Ribet?). In order to prove modularity theorems, it
usually suffices to work with the anemic Hecke algebra, but when one does include
data which captures Tp (or Up ) the local deformation ring is (in this case) Cohen–
Macaulay, as was shown by Snowden. So, for example, I would conjecture thatthe ordinary deformation ring (in any dimension) which includes the local Galois
information corresponding to all the Hecke operators is Cohen–Macaulay.
Is there any real evidence for this guess besides the fact that it would be useful? Well, not really. But it would provide a systematic local Galois explanation
for why deformation rings are torsion free, which is consistent with the guess that,appropriately defined, one should latex R = T theorems on the nose, not just after
looking at (say) MaxSpec. Of course, all of this is in the residually globally irreducible setting. Note that one reason to care about integral modularity statements
is that most of the time, one would expect both R and T to be torsion anyway.
Notes 40.1. Some conjectural progress on these questions has been made by mystudent Chengyang Bao, see § 152.

41. A Preview of Barbados/BellairsMon, 21 Apr 2014
This post is probably not so interesting unless you plan to travel to the Caribbean
in a few weeks. The website for the conference is offline, so I thought I might update
attendees on what might be happening, at least those who read my blog.
There are two hours of talks in the morning by me and two hours of talks inthe evening. Warning: the paragraphs below are not necessarily in one-to-one
correspondence with talks.
Part I: I will give an overview of the Taylor–Wiles method in something approaching its original formulation (so without Kisin’s modifications). I may givethe circular proof of modularity for GL(1) as an example. I will then start talking
about modular forms of weight one. I will give the details of local-global compatibility as proved in my paper with David, first in the irreducible case, and then via
a modification of this method in the general case (using results which will be in
Joel Specter’s thesis).
Background I: Jared and Peter will give a background talk on the geometry of
Shimura varieties, with an emphasis on the case of modular curves, and (possibly)
also that of Siegel 3-folds.
Part II: I will introduce the general strategy developed by myself and Davidin [CG18a] to prove modularity lifting in the ℓ0 = 1 and ℓ0 = ℓ0 situations, inparticular, the details of our patching lemma. I will outline how the method naturally breaks up into several different constituent problems (constructing Galois

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representations, proving local-global compatibility, proving vanishing of cohomology outside certain ranges, representation theoretic problems arising from Taylor–Wiles primes). I will then apply these strategies to prove minimal modularity lifting
theorems for weight one modular forms in the residually irreducible setting.
Background II: David will talk about Kisin’s modification of the Taylor–Wiles
method. Toby Gee will talk about how to prove local-global compatibility and whatthat means in a (somewhat) general setting.
Part III: I will discuss the geometry of local deformation rings for GL(2). Topics
to be covered here include classical questions of multiplicity one and two, as well
as non-minimal modularity lifting theorems in weight one.
Background III: Sug Woo will discuss the relation between cohomology andautomorphic forms and how the Eichler–Shimura isomorphism generalizes to higher
dimensions. Jack Thorne will discuss Taylor–Wiles primes for GL(n).
Part IV: I will talk about my work with David concerning minimal modularity
lifting theorems for low weight Siegel modular forms. This will consist of generalizing
some of the ingredients from GL(2), such as local-global compatibility results, andvanishing results of Lan-Suh. I also discuss an approach to Taylor–Wiles primes inthe torsion setting for GL(n).
Part V: I will talk about completed cohomology in low degree. I shall explain
my results with Emerton on the stability of completed cohomology, and the computation of these groups using K-theory.
Related Research: I have asked a number of people to talk about their recent work on topics related to this conference. This includes George Boxer who hasagreed to talk about coherent cohomology and generalized Hasse invariants, andIla Varma who will talk about local-global compatibility for non-self-dual representations at ℓ ̸= p.

42. A postview of Bellairs/Barbados
Wed, 14 May 2014
I am just recovering from my trip to Barbados for the McGill sponsored conference at the Bellairs institute (§ 41). I thought it was a wonderfully enjoyable
conference, for many reasons. The first is that I got to give 14 hours or so of talks,
and I like the sound of my own voice. What was unique, however, was the reallyhigh level of the audience, not just in terms of technical strength, but in terms of
their knowledge of the particular topics which were being discussed. Usually when
you have a chance to talk to a specialized audience, you only have 50 minutes to
speak, and for at least for the first 20 minutes or so you should not assume that youraudience is au fait with all the latest technical developments in the subject. On theother hand, the contexts in which one has multiple hours to give details (such as amini-course or graduate class) it’s often the case that the target audience is graduate students first encountering the material. At this conference, practically half
the audience had written papers proving modularity lifting theorems! I surveyedsome participants beforehand on how long I should spend reviewing the basic theory of Galois deformations, and the answers typically ranged from 1 to 5 minutes.
In reality, I gave a 150 minute “background” talk on the first morning, althoughPERSIFLAGE: MATH BLOG POSTS

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by background here I really mean Wiles’ proof of minimal modularity lifting for
irreducible modular Galois representations of GQ .
I broke the mold of previous Bellairs conferences by scheduling an additional
talk in the afternoon, so typically we had some 6-7 hours of lectures per day. Thissounds a lot, but when it is divided up into only three speakers and spread out fromearly morning to late evening, it didn’t seem so much at all. (We still had plentyof time every day to snorkel at the reef, and even one free afternoon to go on aboat tour and swim with the turtles. Even Sug Woo’s 200+ minute talk just flew
by, although it was accompanied by rum drinks.) In addition to the backgroundtalks I mentioned previously, , there were also research talks by Peter Scholze, Jack
Thorne, George Boxer, Ila Varma, and David Geraghty (I may blog about some ofthese talks later). I think this was the first conference in which I learned somethingfrom every single talk. Of course, I did get to suggest many of the participants, so
in a way this conference was designed for me.
One outcome of the conference is that I feel confident that we will have unconditional modularity lifting theorems for GL(n)/Q in the next five years. Of course,
it’s always dangerous to make predictions.
Finally, apropos of nothing, I hope to have more posts in the future whose
keywords include both “Richard Taylor” and “Turtles.”Notes 42.1. Since [ACC+ 23] was written within five years of this conference, theprediction was correct.

43. Thurston, Selberg, and random polynomials, Part I.
Wed, 21 May 2014
Apart from everything else, you could always count on Bill Thurston to ask interesting questions. This is the first of a small number of posts which were motivatedin part by figure two from this paper, and this accompanying MO question. I liked
this problem enough to give it as a thesis problem to my student Zili Huang, andmuch of what I discuss below arose from this project (see [CH17]).
Say that an algebraic integer α is Perron if |α| ≥ |σα| for every conjugate σα of
α. One immediately observes that α must be real. Say that a monic polynomial isPerron if it is irreducible and has a Perron integer as a root. Thurston’s question is(roughly) to describe the distribution of Perron algebraic integers, especially thosechosen in some (small) fixed interval in R. This question has several interpretations,but one experiment Thurston does is to take 20,000 monic polynomials of degree 21
with integer coefficients in [−5, 5], and plots the quantities σα/α ∈ B(1) for all the
conjugates of the 5,932 resulting Perron polynomials such that the corresponding
Perron integer was in the interval [1, 2]. The result is this:The first observation is that this graph has (apart from some noise coming fromreal roots) rotational symmetry. The next observation is that the roots tend to
be concentrated in a ring of some radius, which (from experiment) becomes moreconcentrated the more one restricts the range in R of the Perron integers one isconsidering. The first question is: can one explain this graph, and does it reflectreality (that is, the actual distribution of Perron integers)?
The answers to these questions turn out to be: yes, and no. The first problem isthat it is hard (a priori) to “randomly” generate Perron algebraic integers of large

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Figure 9. σα/α ∈ B(1) for Perron α ∈ [1, 5] which are roots ofrandom monic polynomials with coefficients in [−5, 5] ∩ Zdegree in [1, 2]. Knowing a bound on the roots places a bound on the coefficients,
but a randomly chosen polynomial with coefficients satisfying the required bounds
will almost always have a root larger than 2. Thus Thurston “cheats” with hisalgorithm, making the coefficients of his polynomials very small in order to increase
the probability that the largest root will also be small. (Full disclosure, Thurston
makes no claims that his algorithm reflects reality, and explicitly asks whether itdoes so or not.) The issue is then whether this will skew the distribution of theroots. It turns out that it does! To explain why this might not be surprising, let’stalk about the size of the spaces over which Thurston is sampling. Let ΩP
21 be the setof monic polynomials of degree 21 with real coefficients and with a unique largest
real root λ ≤ 2. Thurston is sampling over a space with 1120 lattice points andvolume 1020 . On the other hand, it turns out that the volume of ΩP
21 is equal to2399∼ 2.249 × 1060 .
324 512 710 1111 139 175 193
So Thurston was only really sampling a 10−40 th of the entire space! Thurston’s
picture can be explained as follows: polynomials with (suitably) small coefficients(contingent on the initial and final coefficients not being too small) tend to have all
their roots clustering uniformly around the disc of radius one. This follows in theradial direction by a famous theorem of Erdös and Turán, and for the absolute values
it follows (in a related way using Jensen’s formula) from a paper of Hughes andNikeghbali here. So the apparent “radius” in Thurston’s picture is just representing
1/R, where R is the approximate size of the Perron integers being considered.
It turns out that, in reality, most of the conjugates of Perron integers have size
comparable to the Perron integer itself. That is, the correct version of Thurston’s
picture should show the roots clustering (roughly) uniformly around the boundary.
OK, now a pause when I look at Thurston’s graph and see that the radius is not
something like a half as I claimed above, but something much smaller. So I just

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81

repeated Thurston’s experiment, and out of 20,000 monic polynomials with coefficients randomly chosen in [-5,5], only 1011 were Perron polynomials with largest
root less than 2, and the resulting picture came out like this:Figure 10. σα/α ∈ B(1) for Perron α ∈ [1, 2] which are roots ofrandom monic polynomials with coefficients in [−5, 5] ∩ Z
Here one really sees the (misleading) accumulation around the radius 1/2. I’m
guessing that Thurston actually kept all polynomials whose largest root was in [1, 5],which would account for the larger success rate for choosing Perron polynomials aswell as the smaller radius. This is also consistent with how Thurston describes thecorresponding graphs in the MO question rather than in Figure 2 of his preprint.
So how does one study Perron integers? Let us re-wind slightly and discuss a moreelementary problem. How does one count algebraic integers? The most natural way
to count algebraic integers is to order them by height. However, Thurston’s problem
clearly suggests a different measure, namely, to count by the size of the largest
conjugate. This has a profound effect on some of the statistical properties underconsideration. Roughly, algebraic integers ordered by height are much more likely
to have a small number of “outliers” with large absolute value, whereas when one
orders by the size of the largest conjugate, most of the other conjugates accumulate
around the circle with radius the size of the largest root as the degree goes toinfinity.
The problem of understanding algebraic integers of bounded size (where bybounded we mean a bound on the largest conjugate) amounts to understanding
the lattice points in a certain region of RN . Now as long as one fixes the degree
and increases the bound, such counting problems (including this one) typically reduce to a volume problem. (One also uses the fact that almost all polynomials are
irreducible, and that the regions are “nice” in some explicit way, i.e. not Cantor
sets.) Moreover, the corresponding regions are essentially (up to a simple stretching) independent of the bound. Hence the key region to understand is the regionΩN ⊂ RN of monic degree N polynomials all of whose roots have absolute value atmost one, and the region ΩPN ⊂ ΩN consisting of such polynomials whose largest

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F. CALEGARI

root is real. Of course, one is not only interested in the volumes of these regions,but also the integrals of various quantities. As an example, one can consider the
integralZ
P (T )|aN |α−1 dV

CN (T, α) =ΩNwhere P ∈ ΩN represents the monic polynomial at any point, and aN is the constantterm. Evaluating this integral at α = 1 and taking the leading term (in T ) recovers
the volume. On the other hand, there are some other relations. A fairly simple
computation shows that4
CN −1 (1, 1),Vol(ΩPN) =N (N + 1)
which is how one can compute the left hand side exactly for any N . In order toevaluate these integrals, it makes more sense to integrate not over the “coefficientspace” of polynomials, but rather the “configuration space” of roots. The coefficientspace is naturally stratified by the number of real and complex roots. For thatreason, it makes sense to decompose ΩN asaΩR,S
R+2S=N

where ΩR,S corresponds to polynomials whose roots all have absolute value at mostone and have signature (R, S) (since we are interested only in integrals, we elide
issues concerning whether one wants these spaces to be open or closed or somewhere
in between). As a special case, let’s think about the integral CN,0 (T, α) where one
restricts the integrand to ΩN,0 . The configuration space is simply [−1, 1]N . On theother hand, the map from configuration space to coefficient space is just given interms of the symmetric polynomials, and the corresponding Jacobian matrix is the
Vandermonde determinant. Hence, taking into account the action of SN on thefibres, one finds thatZY
Y α−1 Y1
(T − xi )|xi − xj |dx1 . . . dxN .
CN,0 (T, α) =xi
N ! [−1,1]NThis is now very reminiscent of the classical Selberg Integral. There is somebeautiful mathematics related to the Selberg integral; let me direct you here for anice survey (see [FW08]). The integrals arising here are, however, not quite Selberg
integrals except for some very degenerate cases.
Once you start writing these integrals down, and computing some of them (byhook or crook), there are a number of problems which naturally come to mind. For
example,Question 43.1. What is the probability that a random polynomial all of whoseroots have absolute value at most one is Perron?
Answer 43.2. By explicitly computing the ratio of the volume of ΩP
N to ΩN , youfind that the answer is 1/N if N is odd and 1/(N − 1) if N is even (this checks out
for N = 1, 2).
Question 43.3. Given a polynomial all of whose roots have absolute value at mostone, what the expected number of real roots?
Question 43.4. What the probability is (at least in even degree) that the polynomial has no real roots at all?

PERSIFLAGE: MATH BLOG POSTS

83

Having asked these questions, it is then sensible to ask the same questions forother ways of choosing random polynomials. The classical way to choose a real
random polynomial is to write
f (x) = aN xN + . . . + a0
where the ai are independent normal variables with mean zero (this is the Kac
ensemble). To what extent do the statistics of random polynomials with this measure mirror the constrained problem consisting of polynomials all of whose roots
have absolute value at most one? Obviously, it depends on the type of problem
one considers. The most classical problem for real polynomials concerns counting
the expected number of real roots. A famous theorem of Kac says that, under theensemble above, the expected number of real roots is approximately 2/π · log(N ).
I recommend reading this paper for an introduction to the subject; I learnt thesethings from chatting with Peter Sarnak at the IAS.) The methods of Kac also showthat the real roots concentrate for large N around −1 and +1. In fact, the complex
roots also concentrate along the unit circle as well. How does this compare to ourconstrained model? First of all, the real roots in Kac model either lie in [−1, 1] orin [−∞, 1] ∪ [1, ∞]. Certainly our polynomials have no roots in the larger region.
If one restricts the Kac polynomials to [−1, 1], then the expected number of realroots decreases to 1/π · log(N ). This is in some sense easy to see from the previous
formula, because the map on coefficients ak → aN −k is measure preserving and inverts the roots. In fact, a stronger result follows from Kac. If one takes an interval[a, b] strictly contained inside [−1, 1], then the expected number of real roots in thepolynomial for sufficiently large N converges to
Z
1
1 b
.
π a 1 − T2
This gives another strong indication of how the roots are concentrating at the points+1 and −1. OK, so now let us return to our constrained model consisting of monicpolynomials all of whose roots have absolute value at most one. How many realroots does one expect such a polynomial to have? There’s a natural mapΩN −1 × [−1, 1] → ΩN
which sends P (x) to P (x)(x − T ). The Jacobian of this matrix turns out to beequal to |P (T )|. On the other hand, the map is not one to one, rather, the imageof ΩR,S has multiplicity R. Hence, if Z(P ) denotes the number of real roots of thepolynomial P , thenZ
Z 1Z
Z(P ) =|P (T )|dV
ΩN

0

ΩN −1

The left hand side (after dividing by the volume) gives the expected number of realroots. So one is again reduced to a Selberg type integral. In this case, one apparently
has (based on some Zagier-like integral guessing mojo, but unfortunately not yet
Zagier-like integral proving mojo) for N = 2m,
1
DN

Z|P (T )| =ΩNm
X
2m − 2k + 1 2m − 2k2k

1
22m 2mm
k=0

2m + 1

m−k

k

!
T

2k

m 
X
2m − 2k2k
k=0

m−k

k

!
T

2k

and there is a similar formula for N = 2m + 1. After some analysis to estimate theresulting integral of the RHS from T = −1 to 1, it turns out that, for large N , the,

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F. CALEGARI

expected number of real roots is approximately
1
log N ,πwhich is exactly in accordance with the Kac model! (See [CH17] for details). Indeed,
if one restricts to real roots in an interval [a, b] strictly in [−1, 1], then one also
obtains the same integral formula as in the Kac ensemble. So, somewhat surprisingly
to me, the number of real roots in [−1, 1] behaves in a very similar way whether one
considers Kac polynomials or monic polynomials all of whose roots have absolutevalue at most one.
What then of the other problems? Given a polynomial in the Kac model of evendegree 2N , what is the probability that is has no roots in the interval [−1, 1]?
This problem was explicitly addressed in [DPSZ02] by Dembo, Poonen, Shao, andZeitouni here, where they show (in a wide class of models) that this occurs withprobability O(N −b/2+o(1) ) for some universal constant b/2 which they do not determine, although they estimate based on numerical evidence that b/2 = 0.38 ± 0.015.
What happens in our constrained model? Once more it comes down to a Selberg-like
integral, this time computing the ratio of volumes:
Z
dV
Ω0,N
Z
dV
Ω2N

It turns out that one can compute this explicitly as a product of factorials. Moreover,
one can compute the exact asymptotic in this case as N → ∞, and the resulting
probability is′
2C√, where C = 2−1/24 e−3/2·ζ (−1) = 1.24512 . . .
3/8
2π(2N )
(It may be hard to read in the exponent, but that is the derivative of the Riemannzeta function ζ ′ (−1) at −1. That may seem strange, but in fact this is a fairly
typical constant that comes up in asymptotics of the Barnes-G function, whichis exactly the type of expression (a product of factorials) which turns up in theevaluation of the relevant integrals.) Now the result of [DPSZ02] does not apply in
our case (where the coefficients are a long way from being independent), but given
the similarity in the distribution of real roots between our polynomials and the Kacmodel, we naturally make the following conjecture:
Conjecture 43.5. The constant b/2 is equal to 3/8.
Optimistically, one might even try to prove this conjecture by showing that thestatistics of our collection of polynomials mirror those of the Kac polynomials for
sufficiently large N .
Next time: we discuss a more concrete relationship between random polynomials
and our models in terms of limits of gap probabilities. But let me also leave youwith the following teaser question: What is the probability that the largest root ofa polynomial of degree N is real?
Notes 43.6. Conjecture 43.5 was proved in this paper [PS18] by Mihail Poplavskyi
and Grégory Scheh, (not indexed by MathSciNet)PERSIFLAGE: MATH BLOG POSTS

85

44. Thurston, Selberg, and random polynomials, Part II.
Sat, 24 May 2014
Problem 44.1. What is the probability that the largest root of a polynomial isreal?
Naturally enough, this depends on how one models a random polynomial. If wetake polynomials of degree N which are constrained to have all of their roots to beof absolute value at most one (with respect to the normalized Lebesgue measure
on RN ), then, as mentioned last time, the probability that the largest root is realis either 1/N in odd degree N and 1/(N − 1) in even degree. A priori, this seems
surprisingly small. However, the roots of such polynomials are accumulating on theunit circle, and it’s easier for complex roots to be near the unit circle than real roots.
So let’s instead consider the Kac model of polynomials f (x), where the coefficientsare chosen to be independent normals with mean zero. If you ask for the probability
that the root whose absolute value is closest to one is real, then I suspect that theanswer will be approximately 1/N . However, what about the largest root? The firstobservation is that the expected number of real roots is 2/π log N , so a the mostnaïve guess is that the probability that the largest root is real is approximately(2/πN ) log N . If you like, you can pause here and guess whether you think this is
too high, too low, or about right.
A useful observation is that, instead of considering the largest root, we can consider the smallest root. This is because the map ak → aN −k is measure preserving
and inverts the roots. On the other hand, the behavior of random Kac polynomials
in large degree inside the unit circle starts to approximate the behavior of random
power seriesf (x) = a0 + a1 x + a2 x2 + . . .
where the ai are all normally distributed with mean zero and standard deviation
one. It’s easy to see that f (x) will have radius of convergence 1 with probability
one. So we might instead consider what the probability is that the smallest root ofa random power series is real. However, in this case, it is quite elementary to seethat this probability P∞ is strictly between zero and one. Quite explicitly, consider
the subspace of power series such that the following inequality holds:
111|a0 | + |a1 − 2| + 2 |a2 | + 3 |a3 | + . . . ≤ 1.
2
2
2
This region has positive measure (easy exercise). On the other hand, for all such
power series, one can apply Rouché’s theorem for the contour |2x| = 1 to see thatf (x) and 2x have the same number of zeroes inside this disc, and hence f (x) has
exactly one root of absolute value less than 1/2. By the reflection principle, thisroot is real. It follows that the probability that the smallest root of f (x) is real is
positive. Equally, one can consider the region:1
1
1|a0 − 1| + |a1 | + 2 |a2 − 8| + 3 |a3 | + . . . ≤ 1,2
2
2
and by applying Rouché along |2x| = 1 and comparing with 1 + 8x2 , the corresponding f (x) will have exactly two roots inside this ball, and from the inequalities
above it follows that neither of them will be real, and hence P∞ ≤ 1.
The same argument shows that if PN is the probability that the smallest (orlargest) root of a Kac polynomial is real, then there are uniform (independent of

86

F. CALEGARI

N ) estimates 0 ≤ a ≤ PN ≤ b ≤ 1 for all N . Naturally enough, one should expectthat PN converges to P∞ . This is true, and the rough idea is to show that, withprobability approaching one (as N → ∞), one can apply Rouché’s theorem todeduce that the smallest root of f (x) is real if and only if the smallest roots of its
truncation fN (x) is real. The key idea here is that, for the truncation fN (x), mostof the roots of fN (x) will be uniformly
Q distributed along the unit circle, and so thecontribution of the relevant factor |x − α| to fN (x) will not be too small. Henceone can usually apply Rouché along the contour |x| = β as long as there are noroots of fN (x) of absolute value too close to β.
The computations above also allow one to give effective gaps between P∞ andeither zero or one (by estimating the measure of the corresponding regions as translates of |ai | ≤ 1/2i+1 ), although these estimates are not so sharp. Namely, the probability that the smallest root of a random power series is real is at least 0.256%
and at most 99.999999999999917%. Some numerical data suggests, however, thatthe probability that the largest root of a random Kac polynomial (of large degree)
will be real is approximately 52%. I have some undergraduates working with me
this summer, and one of their projects will be to see if they can prove that theprobability is really strictly larger than 50%, or at least to find a good an estimate
as they can.
One may ask what happens for other ensembles of polynomials. One naturalclass to consider is the so-called binomial polynomials, where the ai are now normal
with mean zero and variance n!/i!(n − i)!. Here the previous argument doesn’t (apriori) work. On the other hand, as Boris Hanin (a Steve Zelditch student from
Northwestern who is leaving for a postdoc at MIT next year) pointed out to me, it
actually
does: to fix it, one should scale all the roots of the relevant polynomials by√
N , and then there really is a limit distribution as N → ∞, given by power seriesf (x) =a0
a1 x a2 x2+ √ + √ + ...
0!
1!
2!

where the normalized ai are normals with standard deviation one. Note that thesepower series have an infinite radius of convergence with probability one. The probability that the smallest root is real will once again be strictly between 0 and 1.
In order to prove convergence of PN (by applying Rouché’s theorem), one needs
to know that the relevant 2-point correlation functions behave reasonably enough;I’m hoping to get Boris to work out and write down the details here. Numerically,
the limit probability in this case is somewhere around 62%.
44.2. Gap Probabilities. I speculated last time on some conjectural relationshipbetween the space of real monic polynomials ΩN all of whose roots are at mostone, and the space of random Kac polynomials of degree N as N goes to infinity.
But now I wanted to point out a more direct an elementary relationship between
ensembles of random real polynomials and our space ΩN . A gap probability is theprobability that the eigenvalues/roots of some ensemble avoid some region of thecorresponding parameter space. Let’s compute this for a very large gap. That is,
let’s compute the probability that a random polynomial has all of its roots less thanT as T → 0.
Let’s consider the Kac model of random polynomialsf (x) = a0 xN + a1 xN −1 + . . . + aNPERSIFLAGE: MATH BLOG POSTS

87

where the ai are chosen independently from a normal distribution with mean zeroand standard deviation one. Hence we are asking: what is the probability that all
the roots of f (x) have absolute value at most T ? This is simply the integralN +1 Z2
1√e−|x| /2 dx
2π
P ΩN (T )where P ΩN (T ) is the space of polynomials (not necessarily monic) all of whoseroots are at most T . There is a mapΩN (T ) × [−∞, ∞] → P ΩN
given by (λ, P ) 7→ λP with Jacobian |λ|N +1 . Hence we can write our quantity as
an integral over ΩN (T ), which turns out to beN +1 Z ∞ Z

2
2 2
2
2
1
√
|λ|N +1 e−(λ −a1 λ −...−aN λ )/2 dx.
2π−∞ ΩN (T )We can now compute the integral over λ directly, and then scaling ΩN (T ) to ΩNin the usual way, we find that the probability that all the roots have absolute value
T is (N +1)/2
ZdV2
Γ(N/2 + 1)
T N (N +1)/2
2 a2 + T 4 a2 + . . . + T 2N a2 )N/2+1
π
(1
+
T
ΩN
1
2
N
Now suppose that T → 0. Then the integral over ΩN converges to the volume ofΩN , and we obtain an exact asymptotic that all the roots are (highly) concentrated
at zero. In fact, one can do this computation with any probability measure µ which
decays sufficiently at infinity.
Curiously enough, we can also ask (in the setting of random polynomials subject
to some reasonable measure µ for each i) what the probability is that a randompolynomial has R real roots contingent on all the roots of that polynomial being
less than T . It turns out that, as T → 0, the answer in this case is simply the ratioof the volume of ΩR,S to ΩN (with R + 2S = N ). This answer does not depend
at all on µ. The explanation for this is that, having subjected the polynomials to
the constraint that all the roots have absolute value at most T for small T , one isrestricting to some tiny region where the measure is constant, and so it is converging
to a scaled version of Lebesgue measure.
Notes 44.3. The summer students did not succeed, I would still like to see thisquestion answered. Problem 44.1 was recently asked here (this time in anothermodel where the ai are uniformly chosen from [−1, 1]). Unfortunately, the questionis raised without any of the accompanying insight or explanation given here, andadditionally comes with a speculative conjecture (and clearly intuitively wrong)which at least is debunked in one of the answers. A certain combination of lack of
insight with a speculative nonsense conjecture which takes some effort to disprove
is mother’s milk to that website.

45. I don’t know how to prove Serre’s conjecture.
Fri, 30 May 2014

88

F. CALEGARII find it slightly annoying that I don’t know how to prove Serre’s conjecture for
imaginary quadratic fields. In particular, I don’t even see any particularly goodstrategy for showing that a surjective Galois representation — say finite flat withcyclotomic determinant for v|p —ρ : GF → GL2 (F3 )is modular of the right level. The first problem is that the strategy used by Wiles
does not work. The results of Langlands-Tunnell imply the existence of an automorphic form π √for GL(2)/F which has an associated finite image Galois representation
into GL2 (Z[ −2]) with projective image A4 that is “congruent” to ρ modulo a prime
above 3, but there is no way to realize this congruence in cohomology. An analogous
example over Q would be that the (known) modularity of a surjective even Galois
representation:
ρ : GQ → SL2 (F4 ) = A5
has no implications for the modularity of the corresponding even complex representation with projective image A5 (which is “congruent” modulo 2), because there
is no way to relate them via Betti or coherent cohomology.
One context in which we have a fairly satisfactory answer to Serre’s conjecture
over imaginary quadratic fields is for representations ρ which are the restriction ofan odd representation of GQ . (I guess one also has modularity in some CM cases,that is, inductions from CM extensions H/F .) So, if we give ourselves modularity
lifting results (surely a requirement to get anywhere), one could imagine trying
to play some sort of game using the 3-5 switch to construct a chain between a
representation which comes from Q and the target representation. Or, perhaps,√
one can play the 3-3 game using abelian surfaces with real multiplication by Z[ 7].
However, there’s a big hole in this strategy: the 3-5 game presupposes that once
you know that ρ is modular of some level, you know it at minimal level. So now one
runs into the problem of level lowering. Alternatively, if you want to play the 3-5
game Khare–Wintenberger style, you really have to construct minimal lifts. But
such lifts will not (in general) exist over imaginary quadratic fields.
This seems to be a serious problem. The only general strategies I can imagine
involve being able to push the torsion classes around to different groups using some(as yet unknown) functoriality for torsion classes. (For example, find minimal lifts
over some large CM extension F ′ /F , prove modularity over F ′ , and then invoke
non-abelian base change for torsion classes to recover modularity of the original representation.) The other argument would be to examine the corresponding Eisensteinclasses for U (2, 2)/F . This seems a little fishy, however; one would really want to
see these representations inside (etale) cohomology in order to invoke some kindof Mazur principle, but as we have noted previously, the Galois representations ofinterest don’t actually live inside the etale cohomology groups that one might wantthem to. Ultimately, the basic problem is that the classical (Mazur–Ribet) style
arguments make strong use the geometry of modular curves (which is certainly
missing here) and the more modern approaches (starting with Skinner–Wiles) rely
on base change.
Notes 45.1. The optimal torsion version of Serre’s conjecture in this case is certainly open, but there certainly have been a number of developments with practicalconsequences for the modularity of elliptic curves over imaginary quadratic fields,see [AKT23] and more recently [CN23].

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46. There are non-liftable weight one forms modulo p for any pTue, 10 Jun 2014
In this post, we show:
Theorem 46.1. Let p be any prime. There exists an integer N prime to p suchthat H 1 (X1 (N ), ωZ ) has a torsion class of order p.
Almost equivalently, there exists a Katz modular form of level N and weight
one over Fp which does not lift to characteristic zero. We shall give two different arguments. The first argument will have the virtue that the torsion class is
non-trivial after localization at a maximal ideal m which is new of level N . The
second argument, in contrast, will produce torsion classes at fairly explicit levels.
Neither proof, unfortunately, implies the existence of interesting Galois representations unramified at p with image containing SL2 (Fp ). Rather, the classes will comefrom deformations of characteristic zero classes. (This post is an elaboration of mycomment in § 34.)
46.2. A first proof of Theorem 46.1. Let K/Q be an imaginary cubic extension
unramified outside p with Galois closure L/Q with Galois group S3 . There is acorresponding Galois representation:
ρ : GQ → Gal(L/Q) = S3 → GL2 (Qp ).
This representation is modular. Suppose for convenience that p ≥ 3. Associated to ρis an absolutely irreducible residual representation ρ. Let R denote the corresponding universal unramified deformation. The only characteristic zero deformations aredihedral. Let Rdh denote the corresponding universal unramified dihedral deformation ring. It’s easy to identify this ring explicitly; it is
Rdh = Zp [CE ⊗ Zp ],where CE is the class group of the imaginary quadratic subfield E of L. The ring R
will fail to be Zp -flat exactly when R ̸= Rdh . Fortunately, this can be determined
purely from the reduced tangent space of R. Note thatad0 (ρ) ≃ ρ ⊕ η,
where η is the quadratic character of E/Q. The reduced tangent space of Rdh is the
Bloch–Kato Selmer group Hf1 (Q, η), where Hf1 denotes the subring of cohomology
classes which are unramified everywhere. So it all comes down to finding K so that
Hf1 (Q, ρ) is non-zero. However, an elementary argument using inflation-restriction
shows that this is equivalent to showing that the class number hK of K is divisible
by p. So we are done provided that we can find a suitable K with class number
divisible by p. (I should mention, of course, that we are using the theorem that
R = Tm which was proved by me and David [CG18a].) The last step follows fromthe lemma below; the argument is essentially taken from this paper of Bilu–Luca(see [BL05]).
Lemma 46.3. Fix a prime p ≥ 3. There exists an imaginary cubic field K/Q of
discriminant prime to p and class number divisible by p.

90

F. CALEGARI

Proof. Consider the field K = Q(θ), where(θ2 + 1)(θ − tp + 1) − 1 = 0,
and t is an element of Q to be chosen later. Note that θ2 + 1 is manifestly a unit
in K. We may compute that(θ2 + θ + 1)θ = (1 + θ2 )tp .
Since (θ, θ2 + θ + 1) = (1) is trivial, it follows that (θ) = ap for some ideal a. We
shall show that, for a suitably chosen t, the element a is non-trivial in the classgroup. If a is trivial, then, up to a unit, θ is a pth power. On the other hand, therank of the unit group of K is one, and θ2 + 1 is a unit. Hence, it suffices to choosea t such that:×(1) θ2 + 1 generates a subgroup of OK
of index prime to p. Equivalently, θ2 + 1
is not a perfect pth power in K.
(2) None of the elements θ(θ2 + 1)i for i = 0, . . . , p − 1 is a perfect pth power
in K.
(3) The polynomial defining K is irreducible.
(4) The discriminant of K is not a square.
(5) The discriminant of K is prime to p.
By working over the function field Q(t) instead of Q, one finds that the first
four conditions hold for all t ∈ Q outside a thin set. (The discriminant ∆ is always
negative, so the signature of the field is always (1, 1).) On the other hand, thediscriminant of the defining polynomial is −3 mod t, so if one (for example) takes
t to be an integer divisible by p then the discriminant will be prime to p. Note thatthe set of integers divisible by p will contain elements not in any thin set, because
the number of integral points of height at most H in a thin set is o(H).
□√46.4. A second proof of Theorem 46.1. Let E = Q( −23), and let L =E[θ]/(θ3 − θ + 1) be the Hilbert class field of E. There is a weight one modular formof level Γ1 (23) and quadratic character corresponding to the Galois representation:
ρ : Gal(L/Q) → GL2 (Qp ).
Lemma 46.5. Let p ≥ 3. Let q = x2 + 23y 2 be a prime such that q ≡ 1 mod p.
Equivalently, let q be a prime which splits completely in L(ζp ). Then
#H 1 (X(Γ1 (23) ∩ Γ0 (q)), ω)tors
is divisible by p. More generally, for any prime q, the quantity above is divisible byany prime divisor of a2q −(1+q)2 , and aℓ ∈ {2, 0, −1} for a prime ℓ is the coefficientQof q ℓ in q (1 − q n )(1 − q 23n ).
Proof. This follows from “level–raising” in characteristic p for weight one forms.
Under the hypothesis that a2q − (1 + q)2 , we find that there is more cohomology(over Zp ) in level Γ1 (23) ∩ Γ0 (q) than is accounted for by oldforms. Assuming that
there is no torsion, this is inconsistent with the fact that there are no newforms incharacteristic zero, because weight one forms cannot be Steinberg at any place. (Theeasiest way to see this is that the eigenvalue of Uq would have to be non-integral —it also follows on the Galois side from local-global compatibility, but this is overkill.)
Note that level–raising in this context does not follow from classical level–raising —for the details I refer to you my fifth lecture in Barbados on non-minimal modularity
lifting theorems in weight one.
□

PERSIFLAGE: MATH BLOG POSTS

91

In a weak sense, the second argument is the same as the first, except one replacesclass groups with ray class groups, and every field has a ray class group of orderdivisible by p if one is allowed to choose the conductor.
Comment 46.6 (Akshay Venkatesh). I still don’t understand why there is theexperimental “suppression” of torsion classes here by characteristic zero classes.
Hopefully the Arakelov version of analytic torsion will clarify the situation eventually — at first glance, the regulator seems negligible in this case, but somethingelse must be going on.
Notes 46.7. An echo of Akshay’s comments are make by Dick in Comments 34.1.
Since the Barbados lectures are not available in any form, one can also refer
to [Cal18].

47. An obvious claim
Sun, 06 Jul 2014
It’s been a while since I saw Serre’s “how to write mathematics badly” lecture,but I’m pretty sure there would have been something about the dangers of using
the word “obvious.” After all, if something really is obvious, then it shouldn’t be
too difficult to explain why. It is especially embarrassing when someone asks you
to clarify a remark/claim in one of your papers which you claim is “obvious” andyou find yourself having no idea what the implicit argument was supposed to be.
Such a thing happened recently to me, when Toby asked me to explain why thefollowing was true:
Lemma√47.1. Let N ≡ 3 mod√ 4 be prime, and let ϵ be the fundamental unit of
K = Q( N ). Then ϵ = a + b N where a is even and b is odd.
Proof. Between Toby, Kevin, and myself, we managed to come up with the argument below, following a suggestion of Rebecca Bellovin: It’s easy enough to see(obvious) that a and b are integers and N (ϵ) = 1. Hence, it suffices to rule out thecase that b even and a odd. Write a2 − N b2 = 1. It follows that a2 ≡ 1 mod N , andsince N is prime, that a ≡ ±1 mod N . Assuming that a is odd, write a = 2N A±1,
and b = 2B. Then the equation above becomesA(N A ± 1) = B 2 .
Without loss of generality, assume that A is positive. Then this equation implies
that A and N A ± 1 are squares, say A = d2 and N A ± 1 = c2 . But thenc2 − N d2 = (N A ± 1) − N A = ±1,√
and hence η = c + N d is a (smaller) unit (in fact, η 2 = ±ϵ), contradicting theassumption that ϵ was a fundamental unit.
□
This argument is really a 2-descent on the unit group. As Kevin remarked: “So
this is a descent argument in a completely elementary situation which I don’t think
I’d ever seen before and which proves something that I don’t think I knew . . . What’s
ridiculous is that if the equation had been a cubic and we were after rational
solutions then I would have instantly leapt on descent as one of my main tools for
attacking it :-/ We live and learn!”

92

F. CALEGARISo what was I thinking when I wrote the paper? The actual claim in the paper is
this: “If H ′ is the (2 part of the) strict ray class group of K of conductor (2), thenH = H ′ , where H is the (2 part of the) class group. The “argument” is as follows:
The proof of [the above] is even more straightforward: it follows immediately
×from a consideration of the units in OK
and the exact sequence×OK→ (OK /2OK )× → H ′ → H → 0.

Well, at least the word obvious was only implicit here. I could try to place the blameon my co-author Matt here, but honestly the phrasing of the claim does sound alittle like something I would write.
Comment 47.2 (Florent Jouve). Using the descent argument in your proof one canalso show e.g. that the negative Pell equation x2 −py 2 = −1 is soluble in integers x, y
if p is a prime congruent to 1 mod 4. In a beautiful series of paper “Higher descenton Pell conics I, II and III” (available on the arXiv) Lemmermeyer gives historicalbackground on these questions and seems to claim that the descent argument you
use goes back to (at least) Legendre. Also (if you don’t mind me self advertising abit) together with E. Fouvry we recently made crucial use of the descent argument
to study the size of the solutions to Pell’s equation and the (related question of the)size of regulators of real quadratic fields, e.g. here (see [FJ13]).
Comment 47.3 (Persiflage). The claim about the existence of a norm −1 unit,
on the other hand, is more directly obvious on the Galois side, since otherwise√Q( p) would admit a quadratic extension unramified at all finite primes, which
it does not. I’m sure there will be a similar argument for the claim above, except
now√one√has to rule out the existence of a degree eight extension E/Q containing
Q( N , −1) with limited ramification properties at two. I’m sure this is not so
hard, but, perhaps, not “obvious.”

48. Report from Luminy
Tue, 08 Jul 2014For how long has Luminy been infested with bloodthirsty mosquitoes? The combination of mosquitoes in my room with the fact that my bed was 6 foot long witha completely unnecessary headboard (which meant that I had to sleep on an angle
with my ankles exposed) did not end well.
As for the math, there were plenty of interesting talks, most of which I will not
discuss here. Jan Nekovar gave a nice talk (on [Nek18]) explaining how one could
prove that the cohomology of compact Shimura varieties of GL(2)-type were semisimple. For concreteness, imagine that X is a Hilbert modular surface associated toa real quadratic field F . Suppose that ρ is the Galois representation associated to a
cuspidal Hilbert modular form of parallel weight two. Then the Langlands-Kottwitz
method shows that the semi-simplification of ρ⊗2 should occur inside H 2 . On theother hand, this argument only ever deals with the trace of Hecke operators and socannot say anything about semi-simplifications. Nekovar’s argument is to use theEichler–Shimura relation applied to partial Hecke operators for primes which split
completely in both F and the corresponding reflex field. The point is that these
operators satisfy a quadratic relation (with distinct eigenvalues for generic elementsof the Galois group), and so act semi-simply on H 2 (imagine everything is compact

PERSIFLAGE: MATH BLOG POSTS

93here). Then, by pure group theory, if the image of ρ is large enough, the sheer
number of such elements is enough to force semi-simplicity. It is perhaps useful
to note that if V is a representation such that V = (ρ⊗2 ) and V is geometric andpure, then V should automatically be semi-simple. This follows from any number of
combinations of bits of the standard conjectures, but one way to see it is that if W isgeometric and of weight zero, then (by Bloch-Kato) one should have Hf1 (F, W ) = 0.
The relevant W in the example above is ad0 (ρ). So in fact one can give an alternate
proof of the theorem using the full power of modularity lifting theorems, providing
one is willing to omit finitely many primes p. This is really an explanation of why
Jan’s result is nice! For example, as soon as one replaces ρ⊗2 by ρ⊗n , one has tostart dealing with Hf1 (F, Sym2n (ρ)(−n)), which gets a little tricky.
Ana Caraiani talked about a very nice result (now [CLH16]) concerning the signof Galois representations associated to torsion classes for GL(n)/F + for totally real
fields F + (this was joint work with Bao Le Hung). Namely, the trace of any complex
conjugation lies in {−1, 0, 1} (in fact, the result identified the exact characteristic
polynomial, which is more general in small characteristic). The basic strategy is
to follow Scholze’s construction and “reduce” the problem to the case of essentiallyself-dual forms, where one has previous results by Taylor, Bellaïche-Chenevier, andTaïbi. However, there is a problem, which is that the regular self-dual automorphic
forms one finds congruences with need not be globally irreducible, and perhaps noteven cuspidal. Suppose one can show that they decompose into an isobaric sum
π = ⊞πi where the πi are self-dual. One runs into problems if too many of the πi
are of dimension ni with ni odd. However, by considering the weights, only one ofthe πi can be odd, because otherwise the Hodge–Tate weight zero would occur withmultiplicity which would violate the fact that π is regular. There is still something tocheck for the even ni also, because previous results required some sign assumption
on the character η such that π = π ∨ η. I believe that even getting to this point
required a further assumption on the torsion class not coming from the boundary.
In the boundary case, there was also a reduction/induction case which also required
careful handling of “odd” dimensional pieces, and some computation of a restrictionof Hecke operators from the relevant Parabolic/Levi which required a sign to comeout correctly. One clever technical step was working with the cohomology of adelic
quotients G(F )\G(A)/U K where K is a maximal compact of G(R) rather thanthe connected component K 0 . The advantage of this is that, in the odd dimensionalcase, this pins down the trace of complex conjugation to be +1 rather than ±1.
This is clear when n = 1, and that one should expect it to be true follows for n odd
by taking determinants.
Peter Scholze gave a talk on his new functor. The basic elements in the construction of this functor are as follows. The Gross-Hopkins period map allows one
to view the (infinite level) Lubin–Tate tower as a GLn (F )-torsor over the (D×Severi–Brauer variety) Pn−1
Cp . So, given an admissible representation π, one canform the “local system” Fπ on the base, and then take its cohomology. The key
technical point of this construction is to show that the result is admissible for D× ,which amounts to proving finiteness of K-invariants for suitable compact open K of
D× . The first step is to pull back to the (lowest level) part of the Lubin–Tate tower,
which one can do because the GH map splits. Now the map from infinite level to thebase of the Lubin–Tate tower is really a GLn (OF )-torsor, so one only has to consider
the restriction of π to GLn (OF ). But then using the admissibility of π, one can look94

F. CALEGARI

instead at the regular representation of GLn (OF ). Now, by some sort of Shapiro’s
Lemma, one can pull everything back up to infinite level. At infinite level, however,
one can replace the Lubin–Tate space by the corresponding Drinfeld tower. Now
taking K-invariants is something that is “easy” to do, because there is an action of
K on the space, and the quotient by K is some sufficiently nice object for which
one has (again by Peter) some nice finiteness theorems for cohomology. I should
probably have mentioned that at some point we are working with coefficients in
O+ /p, i.e. in the almost world. The main application in the talk was to show thatwhen one patches completed cohomology (a la [CEG+ 16]), then one can recover
the Galois representation from the result. This essentially amounts to showing thatwhen one patches together suitable admissible πi , one can also patch the functor.
This requires more than admissibility of the functor, but some sort of “uniform”
admissibility (which is always required for patching). I think the key point here isthat if πi is something patched with a group of diamond operators ∆, then πi has afiltration by |∆| copies of the original π, and so Fπi has a corresponding uniformlyKbounded filtration by Fπ , and so H n−1 (Pn−1
has length at most |∆| times
Cp , Fπ )the corresponding length for the (fixed for all time) version for π. On the other
hand, Peter instead pulled out a new piece of kit by patching using ultra-filters. My
own feeling about logic is that it is never really necessary to prove anything, and I
think PS agreed that it wasn’t strictly required for this particular application. Now
I understand that my prejudice may not be justified (for example, it is probably
hard to prove various identities concerning orbital integrals in small characteristic
directly), but I think it applies in this case. Plus, as a purely expositional remark,if you are going to whip out ultrafilters during a number theory talk then everyone
is just going to talk about ultrafilters rather than the beautiful construction!

49. A public service announcement concerning Fontaine–Mazur
for GL(1)Sat, 12 Jul 2014
There’s a rumour going around that results from transcendence theory are required to prove the Fontaine–Mazur conjecture for GL(1). This is not correct. In
Serre’s book on ℓ-adic representations, he defines a p-adic representation V of a
global Galois group GF to be rational if it is unramified outside finitely many
primes and if the characteristic polynomials of Frobλ actually all lie in some fixed
number field E rather than over Qp . Certainly being rational is a consequence ofoccurring inside the etale cohomology of a smooth proper scheme X, and one might
be motivated to make a conjecture in the converse direction assuming that V is absolutely irreducible. But being “rational” is just a rubbish definition (sorry Serre),a mere proxy for the correct notion of being potentially semistable at all primes
dividing p (“geometric,” given the other assumptions on V ). And the implication
A character χ : GF → Qp is Hodge–Tate ⇒ χ is automorphic
doesn’t require any transcendence results at all. One can’t really blame Serre for
not coming up with the Fontaine–Mazur conjecture in 1968. The reason for thisconfusion seems to be the proof of Theorem stated on III-20 of Serre’s book on
abelian ℓ-adic representations (with the modifications noted in the updated version
of Serre’s book), namely:PERSIFLAGE: MATH BLOG POSTS

95

Theorem 49.1 (Serre–Waldschmidt). If V is an abelian representation of GF
which is rational, then V is locally algebraic.
This argument (even for the case when F is a composite of quadratic fields, thecase considered by Serre) requires some transcendence theory. But the implication
V is abelian and Hodge–Tate ⇒ V is locally algebraic (also proved in Serre) onlyuses Tate era p-adic Hodge theory. The other ingredients for Fontaine–Mazur are
as follows: First, there is the classification of algebraic Hecke characters (due toWeil, I think). A key point here is that the algebraicity forces the unit group to beannihilated by some element in the integral group ring. However, the representation
V occurs in OF× ⊗ C with dimension dim(V |c = 1) if V is non-trivial, so thisforces the existence of representations V of G on which c = −1, corresponding toCM subfields. The final step is the theory of CM abelian varieties. So although
the result is non-trivial, you can be rest assured, gentle reader, that you are notsecretly invoking subtle transcendence results every time you twist an automorphic
Galois representation by a Hodge–Tate character and claim that the result is still
automorphic.

50. 100 Posts
Wed, 20 Aug 2014
Meaningless numerical milestones are a good a reason as any for an indulgent
post. Today, I will discuss some facts from this blog which you might not otherwise
know about. It will be in the form of an (mercifully short) interview with myself.
Question 50.1. When did you start this blog?
I originally started it when I went to the IAS for a special year in 2010–2011,
but I never ended up making the blog public at that time. The irreverence has been
toned down for the current version. Sample post from the IAS: “Who wears
short shorts? Deligne wears short shorts!”
Question 50.2. What topics would you like to blog about in the future?
No promises, but here are some thoughts:• How does an NSF panel work?
• What are letters of recommendation really like?
• Who wore it better: piano v. orchestral arrangements.
• Langlands versus the world.
• Book reviews: Frenkel, Ellenberg, Harris.
• India’s greatest mathematician: Harish–Chandra.
• The top 1%: class and privilege in academia.
Question 50.3. Does that mean you are planning to have less math in the future?
No, the math posts are not really planned in advance, they are just what I
happen to be thinking about at the time. The math posts are really the main (ifnot exclusive) focus of this blog. Although, as one of my graduate students once
remarked: “I though your post on swans was your best post ever.” Yeah, thanks
for that, I’m working hard to bring you occasional insights into the vast edifice ofalgebraic number theory, and you like the guy who can wobble around to Saint–Saëns.

96

F. CALEGARI

Question 50.4. What can your readers do for you?
More audience participation! A lot of what I write is speculative, so please don’t
refrain from giving your partially formed thoughts in the comments. As the readership of this blog went up, the number of comments has gone down. I think I
understand this phenomenon, especially when it comes to math posts. The worst
thing that can happen, however, is that you say something completely ridiculous in
front of a bunch of senior number theorists. But, if you are not occasionally saying
stupid things in front of smart people, then you are doing it wrong.
Question 50.5. Does the audience have any questions?
I’m going to take audience questions in the form of random search terms which
led to this blog. Perhaps those who came here were disappointed with their search
results at the time, but perhaps if they search again this post will provide someanswers.
• 아리조나 윈터스쿨I recommend going here.
• review my paper
No thanks!
• how can i tell how many pages my paper isForm a bijection with one of the sets defined in Part II of this volume.
• paskunas conference 2013
Sounds good! Alas, I was not invited.
• maximal unramified abelian extension of a local field
It’s procyclic, and is generated by roots of unity of order prime to the residue
characteristic. Assuming, of course, your local field is of mixed characteristic,
which all the interesting ones are.
• bush is the messiah
This seems to me to be an unverifiable claim.
• galois representations matrix
Unfortunately, no one can be told what the Matrix is. You have to see it for
yourself.
• honorarium + editor + elsevier$60 for any processed paper. It is taxable income, however.
• galoisrepresentations+blog+who?
It’s me!
• bach mit pedal schiff
Bach without pedal, surely?
• how do i find out how my paper is being reviewed
Oh, I can tell you that. If you are lucky, the reviewer has completely forgotten
about it. Otherwise, the reviewer is currently cursing you for generally ruining
his or her life.

PERSIFLAGE: MATH BLOG POSTS

97• compute the average rate of change from x = −10 to x = 10. enter your answer
as a fraction in simplest terms using a slash ( / ). do not include spaces in youranswer(f (10) − f (−10))/20.
• local even galois representations
I presume you are asking about representations of a local Galois group which
are even. For this to make sense, you should probably talk about Galois representations at the infinite prime, that is, representations of Gal(C/R) on which
complex conjugation acts trivially. Let me classify those for you: they are alltrivial!
• peter scholze gowers
I don’t think they wrote any joint papers.
• ila varma grothendieck
Same answer as above.
• ila varma galois
Same answer as above.
• danny calegari brilliant; jacob lurie genius
self-googling, I imagine.
• joel specters math thesis
I shall link to it on this blog after it has been written.
• representation galois change of characteristic
I assume you are asking about the p-q-switch? There are plenty of good expositions available online.
• affirmative motives
I guess these are motives which just need a little support. For only a $5 donation
to this blog, I will help turn a poor motive into a bold and effective one, simply
by twisting.
• kevin buzzard chessI’d be surprised if he had the time. I fancy my chances.
• xxx agol in school
Personally, I rate the way Agol schooled 3-manifolds as T18+, suitable for topologists of ages 18 and above.
• the math behind a waffle
Aah, sorry about that. I’m more an expert in the waffle behind the math. On theother hand, you can learn about the chemistry of waffles here.
Notes 50.6. Note that this is only the 50th section, which gives some indicationthat math blog posts account for approximately 50% of all posts. (That ratio persists, with 305 or so posts in total, of which a touch over 150 are “math”.) I wrote
about NSF panels in § 59. A review of Michael Harris’s Book is here, a review of
Jordan Ellenberg’s book is here. Joel Specter’s thesis is here, though you can also

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F. CALEGARI

find his thesis-adjacent papers [MS15, CS19, Spe18]. Some consider my post on
swans still my best post ever.

51. The distribution of Hecke eigenvalues, Part I
Tue, 29 Jul 2014
Here is a question I raised at the Puerto-Rico conference during one of the “problem sessions.” Toby Gee seems to remember that I had some half-baked heuristics
that predicted both A and B below, but perhaps one of my readers has a moresophisticated suggestion, or even a similarly wild guess (or even a similarly contradictory collection of guesses).
Fix a pair of distinctPodd primes p and l. Now consider a random normalized
Hecke eigenform f =an q n ∈ Z of weight two and level Γ0 (N ), where N issquarefree and prime to both p and l. Now take the Hecke eigenvalue al and reduce
it modulo a random prime p above p.
Question: As one ranges over all newforms of conductor ≤ X, what is theresulting distribution — if it even exists — of al ∈ Fp ?
Let me not be too precise about what random means — for example, there’s aquestion about whether one wants to normalize in some way for Galois conjugatesof eigenforms, but none of this will really matter for the very weak questions I havein mind. For example, consider the following two possibilities:(1) A: The element al lies in Fp at least 100% of the time.
(2) B: The element al lies in Fp ∖ Fp at least 100% of the time.
Here by 100% I mean as a proportion of all forms as X → ∞, although I confessthat I can’t even rule out the extreme version of A where 100% really means every
single form.
The specifications on the level are designed to rule out some “trivial” examples.
At level Γ0 (N ) with N squarefree there will be no CM-forms, which is one cheap way
to generate large coefficient fields. The level also prevents twisting by characters (aneven cheaper trick). Finally, in general, it is possible to generate large coefficientfields for Galois representations by imposing a local condition at some auxiliary
prime q. For example, one can impose some supercuspidal condition so that the
local residual representation at q does not land in GL2 (Fpm ) for any m not divisible
by an arbitrary fixed integer chosen in advance. However, this too is not possible if
πq is forced to be either unramified or special (up to unramified quadratic twist).
Note that there do exist infinitely many semi-stable modular elliptic curves, so
al will lie in Fp at least infinitely often. This disproves the “extreme” version of B,
but doesn’t go very far towards disproving the asymptotic version of B. As for A,
every single time you write down a normalized eigenform with coefficients in somefield E ̸= Q, you disprove the extreme version of A for a positive density of pairs(p, l). But no finite collection of such forms can disprove A even for a single l andvarying p, because there will always be (many) primes which split completely in
any finite collection of number fields.
Here are three questions:PERSIFLAGE: MATH BLOG POSTS

99(1) Can you disprove the extreme version of A for all p and l?
(2) Can you disprove the super-extreme version of A, namely, show that for all
primes p, there exists a newform of squarefree level N prime to p such thatthe residual representation is not defined over Fp ? (equivalently, replace alby the collection of all al with l prime to N p.)(3) Can you give any heuristic that suggests that either A or B (in the weakerform) is either strong or true?
(4) Do you have any guesses as to the distribution of the al ?
Right now, as you read this, Kevin’s computer is churning away in sage generatingsome data, which will be the topic of Part II. But until then, I would like to hear
your opinions/guesses. For me, I think that A is probably false, but I honestly haveno feeling for B.
Comment 51.1 (Toby Gee). I’m not sure why you’re saying “at least 100%”. My
emails do suggest that you originally conjectured (during the problem session?) thatmost of the time they were in Fp , and possibly 100% of the time asymptotically,
but that you then flipflopped on that. I think the second conversation took placein the sea, though, so unfortunately I have no notes to back this claim up.
Comment 51.2 (Akshay Venkatesh). Wow, what heuristic supports (A)? I would
have thought that, most of the time the eigenvalues live in bigger and bigger extensions, so to speak, and therefore no limiting distribution? I’m sure you’ve thoughtof both of these, but both thinking about charpoly(Tl ) like a random polynomial,and thinking on the Galois side seems to point against (A). On the Galois side it
is a bit unclear, perhaps, how much a fixed residual representation deforms, but
without thinking carefully I’d imagine that Cohen–Lenstra predicts “not too much”.
Comment 51.3 (Persiflage). To be fair, my flirtation with (A) was relatively
short. (In other words, I was for it before I was against it, or something like that.)
Suppose you just count ρ. Then is the expected number of level Γ0 (N ) forms withimage containing SL2 (Fq ) something like a constant Cq that decreases rapidly with
q? I remember you told me the heuristics here, but I can never quite remember thenumbers. If you are correct, though, then surely it’s embarrassing that one can’t
disprove (A)?
Comment 51.4 (Akshay Venkatesh). I think actually Cq doesn’t decrease with q.
You are right, it is embarrassing. One thing we could try is this: if the strong formof (A) holds, then the trace of (Tlp − Tl )T ′ will be zero mod p for any other Hecke
operator T ′ , and we can try to show this doesn’t happen (at least for many N )via trace formula. At the least, the class numbers that show up here don’t dependon N , and thus we could at the least hope to show this way that a fixed (l, p)
doesn’t satisfy extreme-A for most N with a finite amount of computation. There
are also terms in the trace formula like the genus of X0 (N ), which we could arrange
to be indivisible by p even if we know nothing about class numbers, so one might
optimistically hope to get more this way.

52. The Abelian house is not closed
Sun, 03 Aug 2014

100

F. CALEGARI√
97 + 26 13= 7.064604 . . .
27
For an algebraic integer α, the house α is the absolute value of the largest
conjugate of α. Kronecker proved the following:
Today I will talk about(1) If α ≤ 1, then either α = 0 or a root of unity.
(2) If α ≤ 2 and α lives in a CM field, then |α| = 2 cos π/N .
The first claim is well known. The second claim follows from the first: the CM
condition implies that the conjugates of the squares of the absolute value are thesquares of the absolute values of the conjugates. Hence, if ζ 2 + ζ −2 = α 2 − 2, thenζ must be a root of unity by part one. On the other hand, beyond these two results,
the respective values of α are dense in [1, ∞) (general case) and [2, ∞) (CM case).
There are a number of ways to modify this problem. One way is to replace thelargest conjugate α by the d-power mean of the absolute values:

1/n
X
1
Mn (α) =|σα|n.
[Q(α) : Q]
For such a construction, it makes the most sense to assume either that α is totallyreal or lives in a CM field, so that |σα| = σ|α|. For example, if one lets
Mn = {x ∈ (1, ∞)|x = Mn (α), σcα = cσα},then Chris Smyth shows [Smy84] that, for all n ≥ 0, the smallest elements of Mn
are isolated, whereas Mn is dense for sufficiently large x. In this post, we shall beinterested in what happens when one restricts to the class of cyclotomic integers.
Namely, letabMabn = {x ∈ (1, ∞)|x = Mn (α) α ∈ Q }.
In particular, when n = ∞, we obtain the set Mab∞ consisting of the values αtheAbelianHouse. As already noted, thefor cyclotomic integers α. We call Mab∞
values of Mab∩[1,
2]
consistofelements
of
the
form
2 cos(π/N ), which includes 2∞
as a limit point. However, the spectrum of Mabforashort while beyond 2 is once∞
again discrete. For example, the main theorem of [RW13] (previously discussed here)
1/2] — it has a second limit pointcompletely
computes Mab∞ in the interval [0, (5.04)√
at 5 = 2.2360679 . . . and is once again discrete beyond this point. The case n = 2
was studied in Cassels [Cas69] and in [CMS11]. In particular, [CMS11, Theorem
9.1.1] is equivalent to:ab
Proposition 52.1. The set Mab
2 = M2 ⊂ R is closed.

Note that M2 (α)2 =: M(α) ∈ Q (the notation M being used in ibid, so
this closed subset is countable and is thus very far from being dense. MoreoverM2n (α)n = M2 (αn ), so the theorem above implies that the closure Mab
2n is alsocountable and lives inside Q1/n ∩ R ⊂ Q ∩ R. The main goal of the current post is
to generalize this result to the abelian house.
Theorem 52.2. The closure of the abelian house Mab∞ is a subset of Q ∩ R. IfabS ⊂ Mabisbounded,
then
lim
inf
S
∈
M
.
However,lim sup S is not necessarily∞∞
in Mab
,
that
is,
the
abelianhouseitselfisnotclosed.
∞One application of this is to the possible index of subfactors (see here and here
for an overview of the problem):PERSIFLAGE: MATH BLOG POSTS

101

Corollary 52.3. Let α ∈ R ∖ R ∩ Q be a real transcendental number. Then there
does not exist a finite depth subfactor A ≤ B of index in the range (α − ϵ, α + ϵ)
for some ϵ ≥ 0.
Corollary 52.4. Let α ∈ R ∩ Q be an algebraic number. Then there does not exist
a finite depth subfactor A ≤ B of index in the range (α, α + ϵ) for some ϵ ≥ 0.
Corollary 52.5. The set of indices of finite depth subfactors is a well-ordered
subset of R of ordinal type ω ω .
Remark 52.6. Just like volumes of 3-manifolds according to Thurston and Jørgensen.) I’m assuming here that it is easy enough to construct subfactors of indexnY

4 cos2 (π/pi )i=1

for distinct odd prime numbers pi .
Since the main context here is that such indices arise as the spectral eigenvalueof graphs,pit might be helpful (for contrast) to note that this latter spectrum is√dense in [ 2 + 5, ∞) (see [She89]).
This theorem came from my bag of thesis problems. I actually expected it to be
the case that Mab∞ was closed, but this turns out to be completely false. On the other
hand, the argument I had in mind to prove this theorem was roughly correct. On
the third hand, it turns out that the solution to this problem was almost entirely
included in a paper of Antonia Jones from the ’70s [Jon75] (using the method I
roughly had in mind).
52.7. Rational Linear subspace of (R/Z)k . Consider the standard torus T :=(R/Z)k with coordinates (x1 , . . . , xk ). We define a rational linear subspace V of T
to be the subspace cut out by any number of equations of the form:Xai,j xi = cj
for integers ai,j and elements cj ∈ Q/Z. Topologically, V is finite disjoint union of
tori. Any connected component of V is also a rational linear subspace. If all the
cj = 0, then we call V a rational linear subgroup. Call a point x ∈ V rational if
x = (x1 , . . . , xk ) where xi ∈ Q/Z. Given V , the rational numbers cj have a common
denominator; let M denote some integer divisible by this common denominator.
The map [m] : T → T given by multiplication by m preserves V whenever m ≡ 1
mod M .
Definition 52.8. For any rational point x on V and an admissible integer M , let
L(x) = LM (x) denote the (finite) set of rational points of the form [m]x ∈ V for
all m satisfying the following two conditions:(1) m ≡ 1 mod M ,(2) m is prime to N , where x = (x1 , . . . , xk ) are elements of Z[1/N ]/Z.
Of course, this definition comes from looking at the exponent of the conjugatesof root of unity which fix an M th root of unity. We call L(x) the line through x.
The notion of line depends on a choice of integer M , although replacing M by amultiple only (at worst) decreases the size of L(x). Our main technical lemma is
the following:

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F. CALEGARI

Lemma 52.9. Let S ⊂ V be anyS set of rational points, and let M be admissible
for V . Then the closure of W = L(x) of all lines L(x) = LM (x) for x ∈ S is aunion of connected rational subspaces W 0 of V .
We shall apply this theorem to V = T with M = 1. However, in order to prove
the result (by induction), it is easier to prove this more general statement.
Example 52.10. Suppose that n = 2, V = T, and M = 2. If x = (1/2, 1/q), thenthe line L(x) consists of points of the form (1/2, p/q) with p odd and prime to q.
The closure of all such points is the rational subspace x1 = 1/2.
Proof. We proceed by induction on the dimension of V . We may first claim that we
can assume V = V 0 is connected. The connected components of a rational subspace
are obtained by replacing the linear equations by their saturation. However, thisrequires introducing numerators into the constants cj , and so for this step (as well as
several others) we must allow the auxiliary integer M to increase. If V is connected,
it suffices to show that if the closure is not dense, then the points all lie on a (finite
union of) co-dimension ≥ 1 rational subspaces W of V , and then apply the inductive
hypothesis.
Choose a rational base point v ∈ V . After increasing M again if necessary, we
may assume that M v = 0. Under this assumption, translation by v preserves lines
and sends V to a connected rational linear subgroup of T. After an integral changeof basis, any connected rational linear subgroup is linearly equivalent to one of theform ai xi = 0 for ai either zero or one, and thus, again without loss of generality,
we may assume that V = T. Now suppose that v ∈ V is a point which is not in
the closure of the set of lines. Because the complement of the closure is open, we
may assume that v is rational. Hence, once more translating by v and increasing
M if necessary, it suffices to show that either 0 is in the closure of the set of lines,or the points are all contained in a subvariety defined by a linear equation. Let
x = (x1 , . . . , xk ) ∈ S, where one may think of the xi as being lifted to Q. The
problem is to construct an integer n with n ≡ 1 mod M and (n, N ) = 1 such thatif ∥x∥ denotes the nearest integer to x, then ∥nxi ∥ ≤ ϵ for all i, or to show that
all the xi satisfy some linear relation in Q/Z. Without the congruence condition
on M , this is exactly a lemma proved by Davenport and Schinzel in [DS67]. Their
proof does not obviously extend to this case, however. I had an idea to replace thisanalytic argument by using an idea of Cassels using the geometry of numbers. Write
xi = ai /N where N is the smallest common denominator (so the greatest common
divisor of the ai is one). Let Λ ⊂ Zk denote the latticeΛ := {λ ≡ m(a1 , a2 , . . . , ak )mod N,m ∈ Z}.

The basic idea is to break up the problem into two steps: first, find an elementof Λ of small length. If this element reduces under the natural map to Z/N Z toa multiple m of (a1 , . . . , ak ) which is prime to N and 1 mod M , then one wins.
If not, deform the element both other small vectors, and use the fact that (in anarithmetic progression) one doesn’t have to go very far to find elements prime to N
(by Iwaniec, log(N ) or so will suffice). In the end, it turns out that this improved
version is essentially proved by Jones in [Jon75]. (For me, it is easiest to modify
her proof of Theorem 1 than read the notation in some of the latter theorems, but
all of the required content is here.) In fact, the application Jones had in mind was
almost identical to the topic of this post, namely, to study the higher derived sets

PERSIFLAGE: MATH BLOG POSTS

103

of Mabn . For some reason, however, she did not seem to notice (or mention) the
implication that Mab
n was a subset of Q, possibly because her formalism was lessalgebraic than what we consider below (or she wasn’t interested!)□
Consider an infinite set S of roots of k-tuples of roots of unity (x1 , . . . , xk ) which
is closed under the action of Gal(Q/Q), and view it as a subset of Gkm . Say that a
set of k-tuples of units are constrained by a k-tuple of integers h = (h1 , . . . , hk ) if,
for all such tuples,(x1 )h1 (x2 )h2 . . . (xk )hk ∈ ζ Z ,for some fixed root of unity ζ. Since this property is preserved under taking dth
roots for any fixed d ∈ Z, we also insist that each constraining k-tuple consists ofco-prime integers. A constraint cuts out a subvariety Zh of Gkm , which in general is
not geometrically connected. The intersection of any finite number of subvarieties
Zhi is determined by the saturation of the subgroup of Zk generated by the hi . In
particular, there exists a maximal finite set of hi such that Z := ∩Zhi .
Theorem 52.11. The supremum of the elements|y1 + y2 + . . . + yk |2
in S is equal to the supremum of the quantity|z1 + z2 + . . . + zk |2 ,over (z1 , . . . , zk ) ∈ (S 1 )k ∩ Z 0 , where Z 0 ⊂ Z is some geometrically connectedcomponent of Z 0 , and Z is a variety cut out by constraints for finitely many ktuples. The infimum of the houses|y1 + y2 + . . . + yk |is realized by an element of S.
Proof. Pulling back under the isomorphism exp : T → (S 1 )k ⊂ Gkm , the preimage of any geometrically connected component Z 0 is a connected rational linear
subspace of T, and conversely any connected rational linear subspace gives rise
to such a Z 0 . Write the pre-image of y ∈ S as x = (x1 , . . . , xk ). Suppose thaty = (y1 , . . . , yk ) where each yi is a roots of unity in Q(ζN ) (with N divisible by M ),so that the denominators of the xi divide N . The action of G := Gal(Q(ζN )/Q(ζM ))on x via exp∗ sends x to mx for some m with (n, m) = 1. In particular, theconjugates on V precisely cut out the line LM (x) of T (with M = 1). It follows
from Lemma 52.9 that the`closure of exp∗ (S) consists of a finite union of connectedrational subspaces W =W 0 , and hence the closure of S is the finite union of
1 k
0
the sets (S ) ∩ Z for a finite number of geometrically connected Z 0 . This proves
the claim concerning the supremum. For the infimum, we argue as follows. Thereexists a component Z 0 such that the infimum of the largest conjugate of y1 + . . . yk
on Z 0 is equal to the infimum of the houses of elements of S. Let the supremum
of |z1 + z2 + . . . + zk |2 on this space be β. If the desired infimum is equal to β,then all elements must the same house, and the result follows immediately. If not,there exists a subset of S whose largest conjugates are bounded by β − ϵ. But sucha set can no longer be dense in (S 1 )k ∩ Z 0 . Hence, replacing S by this smallest
set, we may reduce the dimension of Z 0 . Continuing this process, we reduce tothe case when either Z 0 is a point or all the houses of elements are the same, andin either case the result follows. Note that the supremum of an algebraic function

104

F. CALEGARI

on (S 1 )k ∩ Z will automatically be algebraic — essentially by a rigidity argument.
Alternatively, one can write down the equations required for a point to be a local
minima, and observe that they are algebraic. To finish the proof of the main claim(except for the claim that Mab∞ is not itself closed) it suffices to note, following a
result of Loxton, that any cyclotomic integer of absolute value at most B can be
written as a sum of (at most) L(B) roots of unity, so, when dealing with the closureof Mab□∞ , it suffices to consider sums of k roots of unity for a fixed k.
Returning to what Jones does, her main result is to consider the sets Mab∞ (k)of cyclotomic integers which are the sum of k roots of unity, and then prove thatthe k − 1st derived set consists precisely of the element {k}. In our context, it may
seem as though the n-th derived set should consist precisely of the maxima of thenatural function on the sets (S 1 )k ∩ Z where Z has codimension n. However, there
is an extra degeneracy coming from the fact that multiplication by a root of unity
doesn’t change the house — so we may insist from the start that 1 is always one of
the roots of unity of S, imposing the condition x1 = 1.
ab52.12. The abelian house is not closed. We now prove that Mab∞ ̸= M∞ byababconstructing an explicit element of M∞ not in M∞ . Indeed, the corresponding
element will neither be cyclotomic nor an algebraic integer (although it will be
algebraic). Consider the set of cyclotomic algebraic integers:

β = ζ 2 + ζ − ζ −1
γ = ζ 2 + ζ + ωζ −1where ω is a cube root of unity, and ζ is (say) and pth root of unity for prime p.
For large p, the Galois conjugates of ζ become dense in the unit circle. It follows
that the supremum of β 2 is the square of the maximum of the quantity|X 2 + X − X −1 |over |X| = 1, and similarly the supremum of γ 2 is the maximum of the twoquantities|X 2 + X + ωX −1 |, |X 2 + X − ω −1 X −1 |,over the same region. One can compute this maximum, and it turns out, perhaps
surprisingly, that it is equal to the value√
97 + 26 13= 7.064604 . . .
27
in the first case, which is not an algebraic integer and so not in Mab∞ , and is equalin the second case to1· θ = 8.096242 . . .
27
whereθ5 − 446θ4 + 62377θ3 − 3023244θ2 + 57168180θ − 351065988 = 0,
and K = Q(θ) has discriminant 22 · 35 · 15619 and Galois closure S5 . These are,perhaps, surprisingly ugly numbers for fairly simple looking maximization problems.
It is clear, of course, that neither of these numbers lies in Mab∞ , so this proves
ab
Mab
=
̸
M
.
Moreover,
I
think
it
quite
likely(andquite
provable,
perhaps with∞∞

PERSIFLAGE: MATH BLOG POSTS

a certain amount of computational effort) that105√
97 + 26 13= 7.064604 . . . is the27

absmallest number in Mab∞ ∖ M∞ .

√97 + 26 13
Notes 52.13. I think it’s still open that
= 7.064604 . . . is the smallest
27
abnumber in Mab∞ ∖ M∞ . More generally, I think some of the ideas here would be
worth writing down in more detail.

53. The distribution of Hecke eigenvalues, part II
Sat, 02 Aug 2014
Here are some numbers from Kevin promised in 51. “For the first 61595 newformsof squarefree level coprime to 15 here’s the field extension of F3 generated by the a5
field extensions:” Actually, I have suppressed most of this table — but it is available
in source code downloadable from the arXiv.
[F3 (a5 ) : F3 ] Total Number

GQ -orbits

Density of forms

Density of orbits

Totals:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
..
.

61595
4623
2492
2397
2476
2600
2142
2289
2008
1962
1530
1837
1656
1612
1638
1455

10740
4623
1246
799
619
520
357
327
251
218
153
167
138
124
117
97

1
0.07505
0.04046
0.03892
0.040200.042210.03478
0.037160.03260
0.031850.02484
0.02982
0.02689
0.026170.02659
0.0236210.43040.11600.074390.057640.048420.033240.030450.023370.02030
0.014250.01555
0.012850.01155
0.010890.009032

102
103
104

0
0
104

0
0
1

0
0
0.001688

000.00009311

I’ve presented the numbers Kevin send me in various ways. The first column
simply counts the field generated by a5 . The second column normalizes by theorder of the field. This is a little like counting two representations which differ
by an automorphism of the coefficient field as being “the same.” The final twocomments are then the proportion of the first two columns overall.
I’m really not quite sure what to make of this data. It does suggest that A is
false, which is perhaps not surprising. It’s not terribly overwhelming evidence forB, but then, law of smaller numbers and all.
Akshay’s suggestion in the comments that the constants Cq could be independentof q must refer to the constants in the second last column, I believe. Of course,

106

F. CALEGARI

it might be the case that F3 (a5 ) is smaller than F3 (a2 , a5 , a7 , a11 , . . .), so thesenumbers aren’t exactly the same as the fields generated by the mod-p reductionsof the eigenforms. If you squint, the numbers in this column do look somewhat
constant for n ≤ 10 or so. One can even argue that n = 1 might be artificiallyinflated exactly because the phenomenon of “slipping into a subfield” mentioned
above. So I’m giving the points to AV. (Yes, that’s right, there were points available
and you missed out because you didn’t play the game.)

54. Horizontal vanishing conjectures.
Fri, 08 Aug 2014
Let F be a number field, and let G be a reductive group over F , and let Γ
be a congruence subgroup of G(OF ). I can hear Brian objecting that this doesn’t
make sense without extra choices; if you have such an objection, please make suchchoices. Matt and I have made various conjectures concerning the vanishing ofe n in the range n ≥ q0 , where q0 has been
the completed cohomology groups H
defined for all time by Borel and Wallach. (And what is q, you ask? Well, having
just consulted [BW00] by downloading a pirated djvu copy, I can tell you that
2q = dim(G/K) [BW00, §4.3, p.67]. What’s that, you say — q isn’t even always an
integer? Nope!) Several cases of this conjecture were proved by Peter (in particular,
in the Shimura variety context), but the general conjecture seems quite hard (notthat the Shimura variety case was a cakewalk!). For example, when G = GL(1),then q0 = 0 and the conjecture is equivalent to Leopoldt’s conjecture. To remindyou, one way of stating Leopoldt’s conjecture is that the profinite topology on
OF× × Zp coincides with the topology coming from the p-adic topology — that is,units are close if they are close modulo powers of p. In contrast, one can ask for the
weaker statement that that the profinite topology on OF× × Zp coincides with thecongruence topology, namely, the topology coming from looking at units modulo
N for any ideal N . This turns out to be unconditionally true and not too difficult,although it is not quite as obvious as it may seem (the same can be said of LC). It
motivates, however, the following conjecture:
Conjecture 54.1. Horizontal Vanishing Let n ≥ q0 . Then the following direct limit
vanisheslim H n (X(K), Fp ) = 0
K

as K ranges over all compact open subgroups of G(AfF ).
There is an equivalent formulation of this conjecture in terms of group cohomology for arithmetic lattices. Because the conjecture is known for GL(1), one canalso pass easily enough between SL and GL. For example, for SLN (Z) and n ≥ 2 it
has the following formulation: Any cohomology class in H n (SLN (Z), M ) for a finite
discrete module M capitulates in some congruence subgroup, providing that
 2
N
n≥
.
4
This vanishing is related to the concept of virtual cohomological dimension. The
virtual cohomological dimension of a group G is the smallest integer m such that
there exists a finite index subgroup H ⊂ G such that every cohomology class indegree ≥ m capitulates in H. The notion being considered here is what one gets by

PERSIFLAGE: MATH BLOG POSTS

107

reversing the quantifiers — one only insists that the classes capitulate in smaller andsmaller groups (in addition, we insist that H is a congruence subgroup, althoughthat is not too restrictive when the rank is ≥ 2). There is a trivial bound m ≥ q0 ,but this bound is not at all sharp. Since this seems an a priori interesting notion,
let’s define it:
Definition 54.2. pro-virtual cohomological dimension: Let G be a group, and let
p be a prime. Say that pvcdp (G) ≤ m if, for every discrete G-module M annihilated
by p, and every cohomology class [c] ∈ H n (G, M ) for some n ≥ m, there exists a
finite subgroup H so that the restriction of [c] to H n (H, M ) vanishes. Say that G
has pro-virtual cohomological dimension ≤ m if pvcdp (G) ≤ m for all p.
I have nothing profound to say about whether this concept is relevant beyond theexample at hand. As you can see for SLN (Z), the virtual cohomological dimension
and pro-virtual cohomological dimension are conjecturally quite different, the latterbeing given conjecturally
by the formula above (at least when N ≥ 2), and theNformer by.
2
I wanted to remark in this post that the Horizontal Vanishing conjecture is,
at least after localization at a non-Eisenstein maximal ideal m, a consequence ofmodularity lifting results (in the spirit of [CG18a]). Namely, the entire point ofthat method is that the patched complex has cohomology concentrated in a single
degree (q0 ), which amounts to saying that cohomology classes in H q0 +i (X(K), Fp )can be annihilated after passing to some auxiliary level coming from some choice of
Taylor–Wiles primes. Now many aspects of this argument are still conditional (notethat to annihilate classes of deep p-power level, one would need corresponding localglobal compatibility relating Galois representations associated to torsion classes toquotients of Kisin deformation rings, at the very least), but perhaps it is a lesshopeless task than trying to prove Leopoldt’s conjecture.
It’s instructive to consider what is possibly the simplest case of this conjecture
beyond Shimura varieties, namely, GL(2) over an imaginary quadratic field (hereq0 = 1, so the claim is that one can kill classes in H 2 ). Here at least one doesn’t haveto worry about vanishing of cohomology outside the indicated range. Local-global
compatibility is still a problem, but one possibly way to get around this is to workat all p-power levels at once, namely, to patch the completed cohomology groups.
(Matt, Toby, and I chatted over roast duck at Sun Wah what patching completed
cohomology for general groups should look like.) Since one certainly has Galois
representations, one gets “for free” the fact that the patched modules are modules
over the appropriate power series ring of the local deformation ring. On the other
hand, as Matt cautioned at the ’Pig, it is no longer so easy to do naive argumentswith codimensions, because the patched objects are not finitely generated over the
ring of diamond operators, but only over a non-commutative group algebra, whichleads into questions relating to the size of the corresponding p-adic representations,which leads back to questions concerning local-global compatibility in p-adic local
Langlands.
I wonder, however, if there are any softer arguments in any special cases.
Comment 54.3 (Summary). There was some clarification that this is quite different from the notion defined by Serre about “good” groups which it’s somewhat

108

F. CALEGARI

orthogonal to what is being considered here. The “interesting” cohomology of arithmetic groups is precisely the cohomology that is not coming from the “local” (inthe number theory sense) cohomology of the congruence completion, so the general
concept of goodness is in some sense opposite to what one wants in this case.

55. Is Serre’s conjecture still open?
Sun, 10 Aug 2014
The conjecture in [Ser87] has indeed been proven. But that isn’t the entire story.
Serre was fully aware of Katz modular forms of weight one. However, Serre was too
timid was prudently conservative and made his conjecture only for weights k(ρ) ≥ 2.
Well, perhaps I am overstating the case; we may as well quote Serre himself here:
Au lieu de définir les formes paraboliques à coefficients dans Fp par
réduction à partir de la caractéristique 0, comme nous l’avons fait,nous aurions pu utiliser la définition de Katz [Kat73], qui conduità un espace a priori plus grand . . . Il serait également intéressantd’étudier de ce point de vue le cas k = 1, que nous avons exclu
jusqu’ici ; peut-être la définition de Katz donne-t-elle alors beaucoup plus de représentations ρf ?
Instead of defining the cusp forms with coefficients in Fp by reduction from characteristic 0, as we did, we could have used thedefinition of Katz [Kat73], which leads, a priori, to a larger space. . . It would also be interesting to study from this viewpoint thecase k = 1 we have ruled out so far; Perhaps Katz’s definition gives
more representations ρf ?
In his Inventiones paper [Edi92] on the weight in Serre’s conjecture, Edixhoven
does give the correct formulation where one allows k(ρ) = 1 and correspondingly
also Katz modular forms. The bridge between the two conjectures essentially consists of two further conjectures: first, that Galois representations associated to residual weight one forms are unramified, and second, unramified modular representations come from weight one.
The first progress on this problem was actually pre-Edixhoven, namely, Gross’companion form paper [Gro90] in Duke. Gross deals with both directions in thecase when ρ(Frobp ) has distinct eigenvalues (I guess the assumption in the direction
weight one ⇒ unramified is that the eigenvalues of X 2 −ap X +χ(p) are distinct). Of
course, there was the famous matter of the “unchecked compatibilities,” (I’m not
one for checking compatibilities myself, to be honest) which have certainly been
resolved at this point (does Bryden Cais do this in his thesis? I think he does) The
next step was the work of Coleman–Voloch [CV92], who deal with the remainingcase under the additional assumption that p is odd. So this leaves the case p = 2.
Somewhat more recently, Gabor Wiese showed (see [Wie14]) that weight one Katz
modular forms do give rise to unramified representations without any assumptions.
So this leaves:
Conjecture 55.1 (Serre’s Conjecture as formulated by Edixhoven). Letρ : GQ → GL2 (Fq )PERSIFLAGE: MATH BLOG POSTS

109

be an absolutely irreducible modular representation of characteristic 2. Assume that
ρ is unramified at 2 and that the semi-simplification of ρ(Frob2 ) is scalar. Then ρis modular of weight one.
Wiese also explicitly dealt with the case when ρ was (projectively) dihedral, so
we can assume that ρ is absolutely irreducible with non-dihedral image. Supposethat the Serre level is N . Let m denote the maximal ideal of the weight two Hecke
algebra which does not include the Hecke operator T2 . Let’s imagine we are working
with Hecke algebras over some sufficiently large extension OE of Z2 with residue
field k so to include enough Frobenius eigenvalues. It suffices to prove that
dimT/m H 0 (X1 (N )/k, ω ⊗2 )[m] ≥ 2,
because then we will have found two modular forms f and g which are Hecke
eigenvalues for all Hecke operators away from p, and by the q-expansion principle,
some linear combination of f and g will have to be the square of the desired weight
one form.
Let Rloc denote the Kisin deformation ring at two for ρ|D2 for the decompositiongroup D2 at 2, (this is just the ordinary deformation ring, in the sense of Geraghty).
†
Let Rloc
denote the augmented deformation which also includes the crystalline
Frobenius eigenvalue T2 (or, to put it differently, the eigenvalue of Frobenius on the“unramified quotient” U2 , where the former is meant in a sense that can and does
make sense integrally. By Hensel’s lemma, both pieces of added data are equivalent.)
Now one uses the modularity machine, which is OK by Khare–Wintenberger for
p = 2 because we are in the non-dihedral setting. Let’s patch the Betti cohomology
of modular curves following KW, except now working with the modified global
Kisin deformation ring R† which remembers crystalline Frobenius, and the full†
Hecke algebra T† which includes T2 . Now Rloc
is a domain with formally smooth
generic fibre (this is proved in Snowden’s paper [Sno18] — the ring in question
e3 in ibid.). Hence, by Kisin-Khare–Wintenberger method, we obtainis denoted R†
an isomorphism R† [1/ϖ] = T† [1/ϖ]. However, because Rloc
is in addition Cohen–††
Macaulay, this can be upgraded to an R = T theorem. (It might be cleaner toinstead patch coherent cohomology — multiplicity one [which always holds with
T2 included] implies that the patched module is free of rank one, which makes it
easy to deduce the integral R† = T† theorem.) By considering the action of T†
on coherent cohomology, however, our multiplicity one assumption allows us todeduce by Nakayama that T = T† (more trivially: the space of modular forms
with coefficients in E/OE with OE /ϖ = k is co-free of rank one over both ofthese rings) and so R → R† = T† = T is surjective. However, there cannot be asurjection R → R† , because there is a map R† → k[ϵ]/ϵ2 which is trivial as a Galois
deformation but is non-trivial for (the Galois avatar of) T2 . For example, in thetrivial case, this just amounts to saying that the trivial representation of GQ2 to
GL2 (k[ϵ]/ϵ2 ) can be thought of as “ordinary with eigenvalue 1 + ϵ.” It follows thatmultiplicity one without T2 cannot hold. Thus Serre’s conjecture is true!

56. K2 (OF ) for number fields F
Fri, 05 Sep 2014

110

F. CALEGARI

Belabas and Gangl have a nice paper [BG04a] where they compute K2 (OE ) fora large number of quadratic fields E. There main result is a method for proving
upper bounds for K2 (OE ) in a rigorous and computationally efficient way. Tate had
previously computed these groups for small imaginary quadratic fields by hand; —the problem is finding an efficient way to do this in general. (Brownkin and Gangl
had previously found a non-rigorous way of computing these groups using K3 (OE )and regulator maps, but more on that later.) A good analogy to keep in mind is
the problem of computing the class groups of imaginary quadratic fields. In thelatter case, however, there are rigorous ways to determine whether an element in
the class group is non-trivial, and this is missing from the computation of K2 (OE ).
To produce lower bounds, [BG04a] use theorems of Tate and Keune to relate thep-primary part of K2 (OE ) to class groups of E(ζp ), which they can then computein some cases. One nice example they give is√ 
1 + −491= Z/13Z.
K2 Z
2Akshay and I used this as one of the examples in our paper; in our context, it
implies that the order of the group
H1 (Γ0 (p), Z)
is always√divisible by 13 where Γ0 (p) is the congruence subgroup of PGL2 (OE ) for
E = Q( −491) and p is any prime — even though the group H1 (Γ, Z) = (Z/2Z)26
is not so divisible. (Because we are talking about PGL rather than PSL, thecusps are quotients of tori by involutions, so only contribute 2-torsion to H1 . This
group is occasionally infinite; we use the convention that ∞ is divisible by 13.) It’s
always nice to see a theoretical argument come to life in an actual computation— fortunately, Aurel Page was kind enough to compute a presentation for Γ inorder for us to do this. Now that I think about it, this and many other interesting
things didn’t make it into the submitted version of the paper; you’ll have to readthe “directors cut” to learn about it.
Alexander Rahm pointed out to me that the computation of K2 (OE ) we used
was annotated with an asterisk in [BG04a], meaning that what was proved was
only an upper bound. The issue is as follows. Let p = 13, and let F = E(ζp ), let
G = Gal(F/E) = (Z/pZ)× , and let Cl(F ) denote the class group of F . What isrequired is to show, in light of Tate’s work on K2 , is that
(Cl(F )[p])G=χ

−1

̸= 0,

where χ : G → F×
p is the cyclotomic character. The problem is that F has degree
24, and it is difficult to compute class groups explicitly in such cases. Let H =Gal(F/Q), so there is a canonical decomposition H = G × Z/2Z. There are two
extensions of χ to H, given (with some abuse of notation) by χ and χη, where ηis the non-trivial character of Gal(F/Q). The main conjecture of Iwasawa Theory(Mazur–Wiles) allows one to easily compute minus parts of class groups in termsof L-values without actually computing with explicit number fields. However, weshould not expect this to help us here. Namely, it’s not hard to show that there−1−1
is an isomorphism (Cl(Q(ζp ))[p])G=χ ≃ (Cl(F )[p])H=χ . However, the former istrivial by Herbrand’s theorem, because B2 = 1/6 is not divisible by 13. That leaves−1
us with the problem of proving that (Cl(F )[p])H=χ η ̸= 0, which is a statement
about the class group of a totally real cyclotomic extension. Since χη −1 is an even

PERSIFLAGE: MATH BLOG POSTS

111

character, we get some savings by working in the totally real subfield F + of degree
12. Now pari happily tells me via bnfinit and bnfclgp that the class group of thisfield is Z/13Z, so it looks like we are in good shape. However, pari has the habit
when computing class groups of assuming not only GRH but something stronger.
What information does bnfinit actually contain? It certainly gives, inter alia:(1) The Galois automorphisms of F + , using nfisisom(nf,nf).
(2) A finite index subgroup V of the unit group U := OF×+ , using bnfinit[8][5]Let me show how, just with this data, one can prove that the relevant part ofthe class group we are interested in is non-zero. BTW, if you tell pari can you
confirm this answer is really correct? (using bnfcertify) it complains, and says
the following:*** bnfcertify: Warning: large Minkowski bound: certification willbe VERY long. *** bnfcertify: not enough precomputed primes,
need primelimit 59644617.
A rough guess (in part) as to what it might be doing: to compute all the invariants
necessary for class field theory, one needs to know the full unit group. To do this, one
can take the units V found so far and saturate them in the entire unit group U . For
each prime q, one can do this by taking representatives in V /qV and determining
whether or not they are perfect qth powers. By taking enough primes, on either
rules out the existence of such an element, or finds a candidate v ∈ V and thenchecks whether it is a qth power. On the other hand, from V , one can compute a
pseudo-regulator RV , which is related to the actual regulator RU by the unknown
index. So to make this computation finite, it suffices to have some a priori boundon the regulator (to give an upper bound on the index), which will ultimately comedown to some a priori bound on an L-value at one, which GRH probably tells you
something useful about.
One can identify the automorphisms of F + computed by pari with the elementsof the Galois group given by the corresponding quotient of H = G × Z/2Z by(−1, 1). This group is generated by the image of σ = (2, 1) = Frob2 , so it is enoughto find the automorphism σ such that σθ − θ2 ≡ 0 mod 2. View χη, a character ofdegree 12, as being valued in F×
13 . Now choose a random unit, say bnf[8][5][6](Warning! I have a feeling that bnfinit does something different each time you
run it, which means you might have to tweak the choice of index 6 above if you aredoing this at home. And by “you,” I really mean “me” in six months time. I guess I
should also tell myself that the relevant pari file is\~fcale/Zagier/BG491
We may write down a second unit as follows:
ϵ=

12
Y

i

(σ i (u))χη(σ ) ∈ (OF× /OF×p )H=χ

−1

η

i=0

What we have done is apply the appropriate projector in the group ring F13 [H] tou. Naturally enough, we can lift ϵ to an actual unit in F + .
Now choose an auxiliary prime q which splits completely in F , say q = 38299.
I chose this because it actually splits completely in Q(ζ13·491 ), which will make acomputation below slightly easier. We reduce ϵ modulo a prime q above q in OF +and we find thatϵ(q−1)/13 ̸≡ 1 mod q.

112

F. CALEGARI

What this last computation proves is that ϵ actually generates(OF× /OF×p )H=χ

−1

η

,

which has dimension one by Dirichlet’s theorem. Note also that the inequality
above does not depend on the choice of q — any other choice is conjugate to q
which replaces ϵ by σϵ and the latter is a non-zero scalar multiple of the former
modulo 13th powers by construction.
On the other hand, let ζ be a primitive 13 · 491th root of unity. Then we may
consider the projection of 1 − ζ modulo 13th powers to the χ−1 η eigenspace (thelatter is naturally also a character on F (ζ491 )). Remember this eigenspace is generated by ϵ. Take q = 38299 again, so q − 1 = 13 · 491 · 6. Then 26 is a primitive
13 · 491th root of unity modulo q. On the other hand,(q−1)/13
Yn
≡ 1 mod q(1 − 26n )n( 491 ) (Z/13·491Z)×(The exponent of (1−26n ) is just the value of χη(n) — remember that the character
gets inverted in the projection formula — and that η −1 = η.) This implies that the
projection of (1−ζ) to the χ−1 η-eigenspace of units modulo p = 13 is trivial, because
the image of ϵ(q−1)/13 computed above was not 1 mod q. The same is trivially true
for the units in Q(ζ491 ) and Q(ζ13 ), because the projection of any unit in a subfieldof F can only be an eigenvalue for a character of the corresponding quotient of
the Galois group. In particular, if C denotes the group of circular units, we haveshown that the map−1−1(C/13C)χ η → (OF× /OF×13 )χ η
is the zero map. This proves that the index of the circular units in the entire units
is divisible by 13. This is enough to prove that 13 divides h+
F , but even better, by
the Gras conjecture (also proved by Mazur–Wiles, following Greenberg) it follows
that the χ−1 η-part of the class group is non-zero, and hence, given the previous
upper bound, this gives a proof that
K2 (OE ) = Z/13Z.
56.1. Further Examples. Let’s now look at other examples in [BG04a]. Consider
the following example:√ 
1 + −755
K2 Z=? Z/41Z ⊕ Z/2Z.
2
Let F = E(ζ41 ), and let F + be the totally real subfield of F of degree 40. Well
we certainly won’t be able to say so much about the class group of F + . On theother hand, we can do the latter part of the computation, namely, testing that theχ−1 η-eigenspace in the circular units looks like it has index divisible by p in theentire units. For example, if q = 123821 = 1 + 41 · 755 · 4, we can compute that(q−1)/13Yn
(1 − 2n(q−1)/(41·755) )n( 755 ) ≡ 1 mod q(Z/41·755Z)×

For good measure, the same congruence holds for the next seven primes which
split completely in F (ζ755 ). (One also has to check that the multiplicative orderof 2 for all these primes is co-prime to 41 · 755.) But, although this is compellingPERSIFLAGE: MATH BLOG POSTS

113−1

numerically, it doesn’t prove anything. If ϵ is a generator of (OF× /OF×p )χ η , it might
be the case that ϵ(q−1)/p ≡ 1 mod q for q above the first thousand primes of normq ≡ 1 mod p. This would simply correspond to a certain ray class group being
divisible by p. By Cebotarev, we know that we can find some prime q for which
this congruence does not hold, but explicit Cebotarev bounds tend to be rubbish
in practice.
If we re-think our original computation, what we really want is a “generic” unitof F + in order to project. Since F is abelian, we actually know how to compute a
finite index subgroup of the unit group, namely, by projecting (via the norm map)the group of circular units from some cyclotomic overfield. Of course, this exactly
won’t be good enough to find a candidate unit ϵ. One approach is to take our latticeV ⊆ U = OF×+ and saturate it. Now we only have to saturate it at p = 41. In fact,
we only need to saturate the χ−1 η eigenspace, which is one dimensional. That is,
it suffices to show thateχ−1 η NF (ζ41 )/F + (1 − ζ)is a pth power in F + . (Before taking the norm, the element is already in F + up topth powers, and [F (ζ41 ) : F + ] has order prime to 41.) But if I ask pari to computethe following:(q−1)/13Yn
mod Φ41·755 (ζ)(1 − ζ n )n( 755 ) (Z/41·755Z)×

it complains and conks out. Well, probably René Schoof could do this computation,
but let’s think about these things a little differently.
56.2. Higher Regulators. So far, we’ve been relying on the fact that the fields
E we are considering are abelian, in order to be able to explicitly write down somefinite index subgroup of the full unit group using circular units. But what if we wantto compute K2 (OE ) for non-abelian fields E? For this, I want to talk about anearlier paper of Gangl with Brownkin [BG99]. Their approach is through the studyof higher regulators. Borel constructs a higher regulator map for odd K-groups (theeven ones are trivial after tensoring with Q). For imaginary quadratic fields andK3 , this amounts to a map
K3 (OE ) → (2πi)2 · R,
where the co-volume of the image is a rational multiple of ζE (2). The Quillen–
Lichtenbaum conjecture predicts that the covolume differs exactly from ζE (2) bya factor coming from the torsion in K3 (OE ), which has order dividing 24, someslightly mysterious powers of 2 which I will ignore, and — the most relevant term
for us — the order of K2 (OE ). Now the Quillen–Lichtenbaum conjecture is true. So
how does this help to compute anything? Well, first one has to ask how to compute
K3 (OE ). As an abelian group, it is easy to compute, but this is not enough tocompute the regulator map. One could give explicit classes in π3 (BGL(OE )+ ), ofcourse, but that may not be the most practical approach. It turns out that thegroup K3 is computable in a natural way because of its relation to the Bloch group
B(E), due to theorems of Bloch and Suslin. (That is, via the Hurewicz map we getclasses in H3 (GLN (OE ), Z) which turn out to be seen by GL2 .) To recall, the Bloch

114

F. CALEGARI

group is defined as the quotient of the pre-Bloch group:Xni [xi ], xi ∈ E × , such thatXni (xi ∧ (1 − xi )) = 0 ∈

2
^

E×

by the 5-term relation[x] − [y] +

hyi
x


−


 
1−y
1 − y −1
= 0, x, y ∈ E × ∖ 1.
+1−x
1 − x−1

Now the Bloch group admits a very natural regulator map
B(E) → Rr2(where E has signature (r1 , r2 )) given by (under the various complex embeddings)
the Bloch–Wigner dilogarithmD(z) = Im(Li2 (z)) + arg(1 − z) log |z| ∈ R.
Now all of this is (almost) very computable. Namely, one can replace E × by theS-units of OE for some (as large as you can) set S, compute the pre-Bloch group,
then do linear algebra to find the quotient. Since (roughly) K3 (OE ) = Zr2 ⊕ T
for an easy to understand finite group T which has order dividing 24, as soon as
one has a enough independent elements in the Bloch group (which can be detectedby computing D(z)) you can compute a group BS (E) which has finite index in
B(E). Moreover, the dilogarithm is also easy to compute numerically, and so one
can compute a regulator DS (E) coming from the Bloch group. Now this regulator
map is known to be rationally the same as Bloch’s regulator map (by Suslin andBloch). Assuming this is also true integrally, we expect there to be a formula:
3|dE |3/2· ζE (2) =? K2 (OF ),π 2 D(E)at least for primes p ≥ 3. (The 3 is coming from the torsion of K3 , and thisformula is probably only true up to powers of 2 — this formulation above comesfrom Brownkin–Gangl [BG99].) For S big enough, DS (E) should stabilize to D(E),which gives a method of computing the order of K2 (OE ). This is what Brownkin
and Gangl do. There are two issues which naturally one has to worry about. The
first is that it’s not known that the regulator map coming from dilogarithms is thesame on the nose as Bloch’s map. However, even granting this (and it should betrue), this algorithm will not certifiably end, because one can never be sure that
DS (E) = D(E). If you compare this to the computation that pari is doing withthe class group, the problem is that there is no a priori bounds on the size of thecorresponding regulators. Well, I guess this algorithm can sometimes end, namely,when one can be sure if the indicated upper bound for K2 (OE ) matches with aknown lower bound. However, we are exactly in a situationin which we are trying√
to prove a lower bound. For example, when E = Q( −755), Brownkin and Gangl
predict that |K2 (OE )| = 2 · 41 because, for a set of larger and larger primes S, theindex formula above stabilizes. So, beyond the issue of relating two different higher
regulator maps, we have the problem of determining whether a class in K3 (OE )is divisible by a prime p or not. This seems harder than our previous problem
of determining whether a unit was divisible or not! (To be fair, however, it seems
impossible to find units in E(ζp ) once E is non-abelian and p is in any sense large.)

PERSIFLAGE: MATH BLOG POSTS

115

56.3. Chern Class Maps. We want to understand whether a class in the Blochgroup B(E) or in K3 (OE ) is divisible by p or not. Instead of working over R,
another approach is to work modulo a prime q. (It may seem a little strange towork modulo q to detect divisibility by p, but bear with me.) Soulé constructedcertain Chern class maps, which include a map:
c2 : K3 (OE ) = K3 (E) → H 1 (E, Zp (2)).
These maps are the boundary map in the Atiyah–Hirzebruch spectral sequence forétale K-theory. Now compose this maps with the reduction modulo p map. Then,after restricting to F = E(ζp ), we may identify Zp (2)/p with µp , and so, by Kummerand Hilbert 90, we get a map:
c2 : K3 (OE )/p → H 1 (F, Zp (2)/p) ≃ H 1 (F, Zp (1)/p) = F × /F ×p .
Keeping track of the various identifications, the image lands in the χ−1 invariant
subspace, where χ is the cyclotomic character of G = Gal(F/E).
Lemma 56.4. Let p ≥ 3 be a prime which is totally ramified in E(ζp )/E andsuppose that p does not divide the order of K2 (OE ). Then the Chern class map
induces an isomorphism−1(Z/pZ)r2 = K3 (OE )/pK3 (OE ) → (OF× /OF×p )χ .
That is, the image of c2 in F × /F ×p may be taken to land in the unit group, andthe ranks of all the groups are the same and equal to r2 , the number of complex
places of the field E.
This lemma follows from Quillen–Lichtenbaum, but it can also be proved directlyfrom the surjectivity of the Chern class map as proved by Soulé, the known rankof K3 ⊗ Q by Borel, and some knowledge of the torsion of K3 proved by Merkuriev
and Suslin. It turns out that the hypothesis on K2 (OE ) is necessary not only forthe proof but for the lemma to be true.
To detect whether a class in K3 is divisible by p, it suffices to “compute” theChern class map above and see whether it is zero. If one ever wants to compute
anything, it makes sense to work with the Bloch group B(E) instead. On the other
hand, it seems hopeless to give a “concrete” map:
B(E) → F × /F ×p .
Even though one can write down elements in the first group somewhat explicitly,
it’s hard to imagine a recipe that would produce explicit elements in F × with thecorrect Galois action.
Instead, what we do is reduce modulo q for some prime q ≡ −1 mod p. That is,
we pass from the Bloch group over E (which will be generated by S units for someS) to the Bloch group of the field Fq . The construction over Fq is just the same.
By a theorem Hutchinson, this group will have order q + 1. The numerology here
is intimately related to Quillen’s result that K3 (Fq ) = Z/(q 2 − 1)Z. Now there aresome commutative diagrams one has to check commute here; I think the key point
to keep in mind is that Quillen’s computation of K3 (Fq ) can already be realized in
the cohomology group H 3 (SL2 (Fq ), Z), and so the map of Bloch groups will be thesame as the map on K-groups via comparison with the Hurewicz map.
Let’s choose a prime q ≡ −1 mod p which splits completely in E(ζp + ζp−1 ) ⊂F = E(ζp ). So we have a map B(E) → B(OE /q) ⊗ Fp = B(Fq ) ⊗ Fp = Fp . The
Bloch group can be thought of in terms of (a quotient of a subgroup of) the free116

F. CALEGARI

abelian group of elements of P1 (E), so there’s no issue about this reduction map.
Moreover, given an element of the Bloch group, we can explicitly compute its image
in the latter group. If this image is non-zero, that gives a certificate that the original
element is not divisible by p. This will be enough to compute K2 (OE ) as long as
the Bloch regulator map agrees with the dilogarithm map.
This argument is still yoked to real regular maps. Let’s try to work entirely withc2 and finite auxiliary primes q ≡ −1 mod p. Another manifestation of the map
B(E) → B(Fq ) ⊗ Fp is the map:
c2 : K3 (OE ) → OF× /OF×p → (OF /Q)× ⊗ Fp = Fp ,where Q is a prime above q in F . Let’s go back to considering the case when E is
an imaginary quadratic field. The image of a generator of K3 (OE ) will map exactly
to a non-zero multiple of the non-trivial element unit ϵ ∈ F × /F ×p . If K2 (OE ) is−1
prime to p, it will even land in (OF× /OF×p )χ . The latter map is exactly computing(up to a non-zero scalar) ϵ(q−1)/p mod q, and so, purely using the Bloch group, we
can check whether this is trivial or not. In particular, given an element of the Blochgroup B(E) which (we think) is a generator, or at least not divisible by p, we canfind a prime q ≡ −1 mod p such that the reduction to B(Fq ) ⊗ Fp is non-zero,
which will imply that the image of c2 is non-zero, which will imply thatϵ(q−1)/p ̸≡ 1

mod p.

This gives an explicit value of q for which this is true without ever having tocompute ϵ. For such a prime q, we can then check that the circular units project tothe identity in this space, which will prove unconditionally that K2 (OE ) is divisible
by p. (Part of this computation assumed that p did not divide K2 (OE ), but that’s
OK, because to prove that p does divide this group we are allowed make thatassumption anyway). Back to our example. We now want a prime q ≡ −1 mod 41,
which is also a square modulo 755. We take q = 163. Now this is not the most
attractive computation in the world, because the root of unity ζ of order 37 · 755
cuts out the extension Fq300 , as we can see by computing the multiplicative orderof q = 163 modulo 41 · 5 · 151. Let’s do it in baby steps. By choosing a suitableprime Q in E(ζ755 ), we can ensure that−1
ζ 755 + ζ −755 = ζ41 + ζ41≡ 4 mod Q.
We writeζ 1510 − 4ζ 755 + 1 = F (ζ)G(ζ) mod 163,
where F (ζ) is any of the four factors of degree 300 (there are also two factors ofdegree 150, and factors of degrees 2, 4, and 4.) Now we want to compute, with
p = 41, q = 163, and r = 755,
(q300 −1)/pYn(1 − ζ n )n( r ) mod Q = (163, F (ζ))η := 
(Z/prZ)×

Of course, one should first reduce the exponents χ−1 η(n) = n(n/r) modulo p = 41
before taking the powers. (Actually, it’s probably kind of stupid to take a productover φ(pr) = 24000 different terms, and one can surely set this up much more
efficiently, but whatever.) We find (drum roll) that:η≡1

mod 163.

PERSIFLAGE: MATH BLOG POSTS

117

To finish, we have to take an element in the Bloch group B(OE ) and show that it
doesn’t vanish in B(OE /q) ⊗ Fp = B(F163 ) ⊗ F41 . At this point, I email Herbert(Gangl), and he sends me an email with the following beautiful element of B(E),where α2 = −755:








3−α
7−α
3−α
7α + 221−8− 10−8+ ... + 6
.
10
10
100
972
(There are 114 terms in all! This should be a generator of B(E).) Into my magma
programme it goes, which cheerily reports that the image of this element is non-zeroin B(F163 ) ⊗ F41 ! So K2 is really divisible by 41. (You might question the veracityof my programme’s output, but more on that below.)
56.5. Stark and Beyond. Here are some more general remarks. Let’s still supposethat E is imaginary quadratic. Take the image of a generator [M ] of B(E), which
is defined up to torsion and up to sign. The image of the Chern class map forsome p ≥ 3 and p not dividing K2 (OE ) gives a canonical unit in OF× /OF×p , where
F = E(ζp ). Let me a be a bit more careful here: by writing F as F = E(ζp ), weare choosing a root of unity (this unit depends on this choice). There’s also an
automorphism of Gal(E/Q) which acts, but this changes the sign of [M ], so thatis the same ambiguity we had before. What is this canonical unit? It is not just acircular unit, but a canonical one (modulo pth powers). What is it? More generally,−1
when r2 = 1, both K3 and (OF× /OF×p )χ have rank r2 = 1, so if p is prime toK2 (OE ) we are generating canonical units. It’s tempting here to conjecture somerelation to Stark units here, and in particular to the special value of L(1, E, χ−1 ),but let me say no more about this. When r2 ≥ 1, one is no longer in the Stark
world, but there is still a canonical map from the Bloch group to the unit group(the group Zr2 has no canonical generator when r2 ≥ 1 — but in the manifestationof this group as a Bloch group, one does have explicit elements.)
Actually, I haven’t even explained how to compute c2 . So far, I have only
explained how to compute whether it is zero or not modulo p. To evaluate it exactly
requires a further threading of the needle through the previous maps (on the Blochgroup), and ultimately uses a test element coming from torsion in B(Q(ζp + ζp−1 )).
Although this is somewhat delicate, and I have not yet proved all of the appropriate
diagrams commute (blech), one can work with it in practice and it gives many
consistency checks on all the computations. (So, for example, once one has theimage of c2 , one can compute the reduction of the corresponding element in the
Bloch group in B(Fq ) ⊗ Fp for one prime q ≡ −1 mod p knowing its image in thecorresponding group for another such prime. Generating the same element of Fp
for p = 13 and twenty different primes q is pretty convincing.)
In fact, computing this map exactly is exactly the problem √that I was thinking
about in the first place.
I
didcomputeit
explicitly
for
K
=
Q(
−491) and p = 13√
(and also K = Q( −571) for p = 5), and the image of a generator of the Blochgroup is not a unit. Instead, it gives a generator of a13 for a non-trivial ideal in the−1class group Cl(F ) of F = E(ζp ), indeed, an element of Cl(F )[p]χ η . (In particular,
it gives, having fixed a root of unity, a canonical element of this class group, whichis also somewhat mysterious.) Let me also mention the Coates–Sinnott conjecture,higher Stickelberger elements, and work of Banaszak and Popescu which are closely
related to the topics in this post (in particular, using Chern class maps to construct
Euler systems generalizing the circular unit Euler system, although not so much

118

F. CALEGARI

question of identifying these elements in some explicit way — especially becausemuch less is known about higher analogues of the Bloch group). But perhaps thisis enough for now.
Notes 56.6. The relevant pari/gp file mentioned above can also be found here.
This post is clearly closely related to some of the material in [CGZ23].

57. The Artin conjecture is rubbish
Thu, 11 Sep 2014
Let ρ : GQ → GLN (C) be a continuous irreducible representation. Artin conjectured that the L-function L(ρ, s) is analytically continues to an entire function on C
(except for the trivial representation where the is a simple pole at one) and satisfies
a functional equation of a precise shape. Langlands later had the profound insightto link this conjecture to functoriality in the Langlands program, which would additionally imply that ρ is automorphic which implies, inter alia, that L(ρ, s) = L(π, s)for a cuspidal automorphic representation π for GL(N )(Q).
This is a beautiful and fundamental conjecture. However, it does appear to becompletely useless for any actual applications. The most natural application ofArtin’s conjecture is to prove . . . the Cebotarev density theorem. This is why Cebotarev’s density theorem is so amazing! True, one can upgrade the error estimates
if one knows Artin, but to do this one also has to know GRH. And if you know
GRH, you are not too far away from Artin anyway, because then L(ρ, s) at worst
has poles on the critical strip, and so you can (essentially) get close to optimal
bounds for Cebotarev anyway.
I thought a little bit about applications of Artin’s conjecture when I wrote a
paper about it, but I came up empty. Then recently, I had occasion to look at mypaper again, and found to my chagrin that when Springer made the final edit theylopped off a sentence in the statement of one of the main theorems. I guess that’s
why the good people at Springer get paid the big bucks. (My best ever copy editing
job, by the way, was for a paper in an AMS journal.) In a different direction, I guess
it also reflects the deep study of this paper by people in the field that nobody has
asked me about it. However, I did notice a statement in the paper that could be
improved upon, which I will mention now.
To set the context, let K gal /Q be a Galois extension with Galois group S5 ,and suppose that complex conjugation in this group is equal to (12)(34). Now
suppose that ρ is a representation of Gal(K gal /Q). We already know that L(s, ρ)is meromorphic, as proved by Brauer and Artin. One thing that can be proven isthat, in the particular case above, L(s, ρ) is holomorphic in a strip Re(s) ≥ 1 − cfor some constant c ≥ 0 which I described as “ineffective.” But looking at it again, I
realized that it is not ineffective at all, due to a result of Stark. What one actuallyshows is that if L(s, ρ) has a pole in the strip Re(s) ≥ 1 − c, then there must also
be another L-function for the same field which has a zero on the real line in thisinterval. Note that, again from by Artin, it is trivially the case that a pole of one
L-function must come from the zero of another L-function, since the product of
all such L-functions is the Dedekind zeta function. So the content here is that theoffending pole has to be on the real line. One consequence is that, in any particularcase, one can rigorously check that the L-function in question has no such zeros,PERSIFLAGE: MATH BLOG POSTS

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and hence (combined with other results in this paper) that ρ is automorphic. With
help from Andrew Booker, I was able to compute one such example (Jo Dwyer has
since gone on to compute a number of other examples.) On the other hand, back to
the general case, we do have effective results for zeros on the real line! The result in
the paper is stated in terms of the existence of a zero of ζH (s) for a certain subfield
H of K gal of degree twelve. (The definition of H was exactly what was swallowed
up by Springer, so it’s not actually defined in the paper. To define it, note that S5
has a faithful representation on six points. There is a degree six extension E whichis the fixed field of the stabilizer of a point; then H is the compositum of E and thequadratic extension inside K gal .) However, the actual argument produces a zero in
an Artin L-factor of ζH (s) which is not divisible by the Dirichlet L-function for thequadratic character of S5 . Stark shows (Some Effective Cases of the Brauer–Siegel
Theorem) (see [Sta74]) that such an L-function does not have Siegel zeros, and also
gives an explicit estimate for the largest zero on the real line. In particular, for theL(ρ, s) of interest, one deduces that they are analytic on the strip Re(s) ≥ 1 − cwhere one can take1.
1−c=1−
4 log |∆H |The result of Stark, BTW, is why one could effectively solve the class number at
most X problem for totally complex CM fields which were not imaginary quadratic
fields before Goldfeld–Gross–Zagier.
Comment 57.1 (Dick Gross). Frank, your title reminds me of a great Dan Aykroydskit on an early SNL, where he played a late night talk radio host, and kept proposing more and more outrageous topics in a (futile) effort to get someone to call in.
But I’ll bite. What “applications” do you have in mind for the Riemann Hypothesis,or the conjecture of Birch and Swinnerton-Dyer, or the Hodge conjecture?
Comment 57.2 (Persiflage). The *GRH* undisputedly has many interesting consequences (the other Artin’s conjecture, effective Brauer–Siegel, and many more).
Even the classical RH has consequences concerning the approximation of π(x) by
Li(x). BSD? It does what it says on the bottle: take an elliptic curve and you candetermine whether it has infinitely many points or not. The Hodge conjecture?
Throw in the standard conjectures and one gets a robust theory of motives. If you
wanted to give famous conjectures with absolutely zero interesting consequences,then surely additive number theory supplies the ne plus ultra of such problems: theGoldbach and twin prime conjectures.
Of course I agree with your implicit thesis that the purpose of proving theorems
is to advance understanding rather than merely as a means to proving . . . moretheorems. (It can safely be said that we learnt quite a lot from the proof of Fermat.) But doesn’t it seem a little sad that one can’t deduce anything from Artin’s
conjecture? Unlike (say) in the case of elliptic curves, the results of Cebotarev andArtin–Brauer allow one to extract all of the relevant juices out of L(ρ, s) without
knowing that it is holomorphic. In part, this is because the critical values for Artin
motives are on the edge of the strip, and one can happily talk about the special
value L(ρ, 1) without having to know modularity.
I guess one consequence of a particular case of Artin’s conjecture is . . . morecases of Artin’s conjecture! More precisely, if one knows that ρ and ϱ satisfy Artin’s
conjecture in the strong Langlands sense (is automorphic), then ρ ⊗ ϱ satisfies
Artin’s conjecture in the weak sense, by Rankin–Selberg.

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F. CALEGARI

Comment 57.3 (Matthew Emerton). When I think of applications of modularityof elliptic curves, I not only think of all the theoretical applications (Gross–Zagier,Kato, Skinner–Urban, . . . ) but also of the fact that the tables of modular elliptic
curves are complete tables of elliptic curves, ordered by conductor.
Similar, I think of modularity for odd two-dim’l Artin reps. of Gal(Q/Q) as
meaning that the table of weight one forms gives a complete list of certain number
fields. Maybe this doesn’t count as an application, but it means something. It
doesn’t seem that much less effective as a GL2 class field theory than classical CFT
does for (say) totally real fields, when one has no idea how big/small the various
ray class groups are.
Comment 57.4 (Persiflage). Threading the needle slightly, I both agree and disagree with you here. You are talking about a a very special case when you selectodd 2-dimensional Artin representations. In almost every other context, the automorphic forms corresponding to Artin representations can’t actually be computed
at all in an exact way. Using class field theory, for example, you can really computewhether any particular field has an extension with any particular solvable Galois
group G unramified outside a given finite set of primes. And yet we still don’t know
Artin for solvable extensions of Q. So while the existence (conjectural or otherwise) of a bijection between finite representations of Galois groups of number fields
and certain automorphic forms is a beautiful one, it’s a stretch to say that it is“computable.”
Comment 57.5 (Minhyong Kim). Regarding BSD, I’ve always felt that the mostimportant portion is the finiteness of Sha, which implies that the standard algorithm used to compute the Mordell–Weil group actually terminates. From this pointof view, the importance of the L-function can be questioned in the elliptic curve
context as well. However, people far wiser than I seem to believe that trying to getat Sha forces the L-function on you, one way or another.
Comment 57.6 (Persiflage). Dear MK, I both completely agree and disagreewith what you say. First, I think that the finiteness of Sha is a really fundamental
problem. On the other hand, I think your resulting corollary (that the Mordell–
Weil group of E can be computed algorithmically) is not particularly interestingat all! I mean, as a practical matter, one can compute E(Q) anyway, and knowingthat Sha is finite doesn’t help. It seems a little like saying that the finiteness ofthe class group is a fundamental problem because it allows one to algorithmically
perform explicit computations in number fields (rather than just saying that it is afundamental structural result). I’ll also push back slightly on other aspects of BSD— proving BSD modulo finiteness of Sha would be a brilliant result.
On a different matter, I also think I don’t necessarily agree with your experts.
It seems possible that one could prove Sha is finite without automorphy — and I
even have my own wise expert to back me up. To discuss a related result, I don’t
think that the Leopoldt conjecture will necessarily be proved using automorphic
methods either.

58. The nearly ordinary deformation ring is (usually) torsion overweight space
Sat, 20 Sep 2014

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121

Let F/Q be an arbitrary number field. Let p be a prime which splits completely
in F , and consider an absolutely irreducible representation:
ρ : GF → GL2 (Qp )which is unramified outside finitely many primes. If one assumes that ρ is geometric,
then the Fontaine–Mazur conjecture predicts that ρ should be motivic, and theLanglands reciprocity conjecture predicts that ρ should be automorphic. This isprobably difficult, so let’s make our lives easier by adding some hypotheses. For
example, let us assume that:• A: For all v|p, the representation ρ|Gv is crystalline and nearly ordinary,• B: The residual representation ρ has suitably big image (Taylor–Wiles typecondition.)
Proving the modularity of ρ under these hypotheses is still too ambitious — it still
includes even icosahedral representations and Elliptic curves over arbitrary number
fields. Natural further hypotheses to make include conditions on the Hodge–Tateweights and conditions on complex conjugation.
We prove the following:
Theorem 58.1. Assume, in addition to conditions A and B+, that
• C: The Hodge–Tate weights [av , bv ] at each v|p are sufficiently generic,• D: If F is totally real, then there exists at least one infinite place such that
ρ is even.
Then ρ does not exist.
The condition B+ (which will be defined during the proof) is more restrictive
than the usual Taylor–Wiles condition — we shall see from the proof exactly what
it entails. Condition C will also be explained — but let us note that, for any suitable
method of counting, almost all choices of integers are generic, even after imposing
some condition on the determinant (say av + bv is constant) to rule out stupidities.
One should think of this theorem as follows. If F is totally real, then condition D
should be sufficient to rule out the existence of any automorphic ρ in regular weight,
because (for motivic reasons) such representations should be totally odd. On theother hand, if F is not totally real, then the weights of any motive (with coefficients)
should satisfy a certain non-trivial symmetry property with respect to the action ofcomplex conjugation. So, for example, if F has signature (1, 2), then either condition
C or D should be sufficient, but we will require both. In fact, even condition C isstronger than what should be necessary. In addition to assuming regularity at allprimes, it amounts to (on the representation theoretic side) insisting that none of
the GL2 (C) weights are fixed by any conjugate of complex conjugation, whereas asingle such example should be enough for a contradiction.
Perhaps a useful way to think about Theorem I is to make the following comparison. Hida proves the following theorem:
Theorem 58.2 (Hida). The nearly ordinary Hida family for SL(2)/F is finite
over weight space and has positive rank if and only if F is totally real and thecorresponding ρ is odd at all infinite places.
On the other hand, a consequence of Theorem 58.1 is:Theorem 58.3. The fixed determinant nearly ordinary deformation ring of a residual representation ρ satisfying condition B+ is finite over weight space and has

122F. CALEGARI

positive rank if and only if F is totally real and the corresponding ρ is odd at all
infinite places.
In both cases, I am only considering the deformation rings up to twist — thedeformation ring of the character is torsion over the corresponding weight spacewhenever OF has infinitely many units. Also in both cases, it is of interest to determine the exact co-dimension of the ordinary family — this is a difficult problem,
because strong enough results would allow you do deduce Leopoldt by considering
induced representations.
OK, so what is the argument? If you have read some of my papers, you canprobably guess.
Proof. Assume that ρ exists. Let U be the representation corresponding to ρ. Now
replace U by V = Sym2 (U ). Now replace V by the tensor induction:OW =V
GF /GQof dimension 3[F :Q] . We now let C be the condition that W has distinct Hodge–
Tate weights. To see that this is generic, it really suffices to show that there is
at least one choice of weights for which this is true. But one can let the weightsof U up to translation consist of the 2-uples [0, 1], [0, 3], [0, 9], etc. and then theweights of W are, again up to translation, [0, 1, 2, . . . , 3[F :Q] − 1]. We now let B+
be the condition that the residual representation is absolutely irreducible, and thatthe prime p ≥ 2 · 3[F :Q] + 1. This is generically true, and amounts to saying thatthe conjugates of ρ under GQ are sufficiently distinct. Since the dimension of W isodd, and because it is essentially self-dual (exercise), orthogonal (obvious), nearly
ordinary (by assumption), has distinct Hodge–Tate weights (by construction), satisfies the required sign condition (automatic in odd dimension), we deduce that it
is potentially modular by [BLGGT14]. In order to win, it suffices to show, by atheorem I made Richard prove, that the action of complex conjugation on W has
trace ±1. However, this is equivalent to condition D (see below).
□
One can relax condition B+ slightly by only inducing down to the largest totallyreal subfield of F . On the other hand, there are plenty of examples of representations ρ to which Theorem 58.3 applies. I think one can take any elliptic curve E/F
without CM and such that jE ∈ F does not lie in any subfield of F , and then take p
to be any sufficiently large ordinary prime which splits completely in F (caveat emptor, I didn’t check this). Of course, the condition that p splits isn’t really necessary
either, I guess.
The proof of Theorem |refthm:two follows along the exact same lines — theconditions are strong enough to ensure, using results of Thorne [Tho12], that thenearly ordinary deformation ring of (the now residual) representation W is finite
over weight space, which translates back into finiteness of deformations of U overweight space. The result is obvious if F is totally real and ρ is odd. Otherwise, wechoose a sufficiently generic point in weight space (in the sense of C), and then,by Theorem 58.1, we see that the specialization of the nearly ordinary deformation
ring at that point must be torsion.
It remains to compute the sign of W . This is an exercise in finite group theory,
we only recall enough of the details for our purposes. Let V be a representation of

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H of dimension d. Consider the tensor induction:
O
σV .
G/H

Let T denote a set of representatives of right cosets of H in G. Let tg ∈ T denotethe corresponding choice for the coset Htg. For g ∈ G, let n(t) denote the size ofthe ⟨g⟩-orbit which contains T . If g = c has order 2, then either n(t) = 1 or n(t) = 2
Certainly
tcn(t) t−1 ∈ H, t ∈ T.
Let T0 be a set of representatives for the ⟨g⟩ orbits on T0 . Then (proof omitted)
Yϕ(tcn(t) t−1 ).
ϕ⊗G (c) =t∈T0

We observe that:(1) If n(t) = 2, then ϕ(tcn(t) t−1 ) = ϕ(tt−1 ) = ϕ(1) = d.
(2) If n(t) = 1, then tct−1 ∈ H and ϕ(tct−1 ) is what it is. For example, it is0, ±1 if and only if V is GL-odd with respect to tct−1 .
Now suppose that G = GQ and H = GF . The elements tct−1 are exactly thedifferent complex conjugations of the representations of the conjugates of H. We
deduce:(1) If dim(V ) is even, then W is GL-odd if and only if there exists at least one
real place of F such that V is GL-odd.
(2) If dim(V ) is odd, then W is GL-odd if and only if F is totally real and V
is GL-odd at every real place.
Equivalently, a product of even integers can equal zero only if at least one of themis zero, and a product of odd integers can equal ±1 if and only if all of them are±1.

59. Applying for an NSF grantFri, 26 Sep 2014
It’s not easy to write a good grant proposal. But it can be even harder to write
one for the first time, especially if you’re not quite sure who will be reading yourproposal. So today, I want to say a little bit about how an NSF mathematics panel
is run, and give you some idea of who your target audience should be.
Before I start, I want to include a pseudo-legal disclaimer. For fairly obvious
reasons, you are not supposed to reveal that you served on any particular panel.
But I am allowed to say that I have served on some panels, and there is enoughuniformity in the process to make me confident that what I say should resemble
your reality if you decide to apply. (Let me also mention that I had some help on
this post from a friend (whom I shall refer to as the Hawk) who is much moreof an NSF pro than I am. He made various corrections and suggestions on a firstdraft of this blog, and I even included a few of his remarks verbatim in the text.)
The NSF administers many different types of grants. I’m not just talking about
graduate fellowships, postgraduate fellowships and research grants here. There are
FRG grants, RTG grants, CAREER NSF grants, REUs, conference grants, and soon. However, for the purpose of this email, I want to concentrate on research grants.

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59.1. The Mechanics. The panel is comprised of approximately 10 or so mathematicians, who consider approximately 40-50 or so proposals. About six weeks
before the panel takes place, each panelist is given the list of proposals and asked
to rank the proposals 1,2,3,C based on the following criteria:
1 = I feel comfortable reviewing this proposal
2 = I could review this proposal if necessary
3 = It would be very difficult for me to review this proposal
C = I have a conflict of interest with this proposal
Here “conflict of interest” is defined in a fairly precise way. It includes someobvious things (recent co-authors, people at your institution, family members) andsome non-obvious ones (people with whom you serve with on an editorial board,people at institutions that have paid you an honorarium for giving a recent talk).
You are also free to declare a conflict of interest which is not on that list. About amonth before the panel meets, each panelist is given 12 or so files to read (all thefiles are online, of course). It is not unusual for a panelist to be given (in their suiteof proposals) one or two grants they graded as a “3” above — it depends on how
parsimonious they were in their initial grading. For each of these files, the panelist
is asked to grade the proposal on both intellectual merit and broader impact. Many
panelists also unofficially rank the proposals that they read. In addition to grading
the proposals, the panelist writes a brief summary indicating what they feel are thestrengths and merits of each proposal. A panelist can, if they wish, also read otherproposals.
The next step is that the panel meets at the NSF headquarters in Virginia,
sometime between November and March. A typical panel may last 2.5 days. The
panel is chaired by the relevant program officer and three or so other NSF employees(usually professional mathematicians who have taken a leave of absence for a twoyear position at the NSF), so there will typically be 14-15 people in a conference
room, each with either their laptop or a supplied computer. The first 1.5 days ofthe panel consist of going through the files one by one. For each file, the three (orso) panelists who were assigned the proposal read out their review of the proposal.
During this time, other panelists (especially those with some expertise) will alsooffer their opinions. During this period, anyone who is conflicted with the proposal
has to leave the room. At the end of each discussion (which takes about 10 minutes),a yellow sticky sheet with the PI’s name has to be placed on a white board withthree columns. The columns are officially designated as “strongly recommended
for funding,” “recommended for funding if possible,” and “not recommended forfunding.” The desired outcome is to have 10% of proposals in the first column, 30%
in the second, and 60% in the third. Within the first two columns the names areordered, although, during the process, certain proposals can float up or down as
they are re-evaluated in light of other proposals. During each discussion, a panelist
who was not assigned to read the proposal is assigned to be a scribe and record thehighlights of each discussion. Each panelist is a scribe on 3-4 proposals.
The final step is for each panelist to write up a summary of the panel discussions
for which they were a scribe, highlighting what the panel thought were the strengths
and weaknesses of the proposal, indicating “which column” the panel placed thename, and reflecting the extent to which there was uniform agreement or not.
Everyone then goes over these summaries to confirm that the summary does reflect
the panel discussion. If you ever apply for a grant, you will be able to read thisPERSIFLAGE: MATH BLOG POSTS

125

summary, together with the evaluation of the three members who read your proposal
in depth. (The panelists assigned to your proposal have an opportunity to modify
their evaluations during the meeting if they change their minds in light of the
discussion.)
Then the panel ends; the panel has given the program officer a (roughly) ordered
set of names, and it is up to the NSF to decide whom to fund. I’m not sure theextent to which the recommendations of the panel exactly mirror the actual results,
although I suspect that it is quite close. I can imagine, however, that a programme
officer feels that a certain proposal suffered because nobody on the particular panel
was an expert in that area, and they may decide to send that proposal off for further
review.
The Hawk says: The actual results can deviate significantly from the adviceof the panel. I think it’s safe to say that the ’highly recommended’ proposals
always get funded. After that, there are various other objectives that the program
officers are trying to achieve – gender diversity, racial diversity, support for young
PIs, support for worthy PIs at undergraduate-only institutions. The panel list is
typically the default in cases where none of those other objectives apply, though you
can imagine reasons to deviate from it (e.g. you might not let the same person suffer
the bad luck of being the first person after the cutoff two years running, you might
support a proposal in a sub-discipline that has otherwise been shut out, etc). So in
the ’recommended’ zone there are certainly some inversions. It’s also not unheardof for a ’not recommended’ proposal to end up being funded. One way this canhappen is for the proposal to be looked at by a second (perhaps more appropriate)
panel that likes the proposal much better. But also, the program officers can simply
decide that the panel’s conclusions about a proposal were unjust for some reason,and raise the proposal up in the rankings.
59.2. How narrow is the focus of each panel. As I mentioned, there are approximately 40-50 proposals for each panel, of which maybe 15 are funded. So takethe 80 or so people who are research active and applying for grants who are closest
to you mathematically, and that gives you a rough idea. If you study Galois representations and modular forms, or Iwasawa theory, or the arithmetic of Shimuravarieties, or arithmetic geometry of some kind, your proposal may well end up in
the same panel as mine was (it can happen — as it did to me last year — that yourproposal ends up being evaluated by two panels — this is possibly done in order
to normalize the orderings in some way. Because I wasn’t there, I can’t quite tellwhat the difference was between the two panels).
The Hawk says: This is the first time I’ve heard a suggestion that normalizationis the reason that some proposals are looked at by two panels. I think this happens
either because the program officers feel that the proposal straddles two panels tosuch an extent that they feel both opinions could be useful; or because the proposal
has two very different parts that genuinely fit in separate panels; or because theassigned panel decided that there were parts of a proposal that they didn’t have theexpertise to comment on, and so they suggest getting the input of another panel.
On the other hand, I’m pretty sure that my proposal would not be on the same
panel as someone like Ken Ono or Soundararajan. Could my proposal be on thesame panel as Akshay’s? I’m not sure. I probably would have said no if my proposal
didn’t end up on two panels last time. And Akshay is a collaborator of mine! So

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it’s pretty focused. On the other hand, there are certainly areas in each field which
are smaller than others, and if you work in such a sub-field, then it’s more likelythat the panelists will not be experts in your area.
59.3. Who serves on the panel. There are no formal NSF requirements for theconstitution of any panel. Who is a typical member of the panel? Well, of course, one
goal of the program officer is to make the panel is not too uniform. But, for example,
I would expect that there would always be at least one person on the committee
who knows as much (say) about modular forms and Galois representations as I do.
So if that is what you do, then you can be pretty sure that whomever that person
is will be reading your file. But you can also be sure that someone who is not anexpert will also be reading your file, perhaps someone in Iwasawa theory, say. And
this already should give you a pretty good idea of your target audience. In otherwords, you have to do two things:• You have to explain to Iwasawa theory person why the modularity theorems
you are going to prove are interesting. When is math interesting? Well,there are plenty of ways it can be interesting. You may have an idea of how
to apply previous machinery in a novel way. You may have an interesting
application in mind. You may have a completely new approach to an oldtheorem. You may have a completely new idea on how to solve an open
problem. This is what you want to get across when you are talking to
Iwasawa theory person — to give a sense of why the general problem you
are studying is interesting, and how you are going to make a contribution
to that field.
• You have an easier job convincing me (or equivalent) why your modularity
theorems are broadly interesting, but you still have to convince me that you
particular proposal is interesting. More importantly, you have to convince
me that you can carry out your proposal successfully, or at least to the pointof producing interesting mathematics.
The Hawk says: I think it would be worth mentioning here the fineline between saying enough about how you intend to carry out your plansthat the panel is convinced you can do it, and saying so much that they
think you’ve done it already. I think new proposers often struggle on thispoint.
Of course, if you do something other than what I do, then replace “Iwasawa theory
person” above by me or equivalent and “me” by someone with expertise in yourfield.
59.4. What should I take away from this? First up, I think that an NSF grant
proposal is probably the most technical audience you will write for in a context thatis not one of your research papers. So you don’t need (beyond a cursory mention) to
say how modular forms played a role in the proof of Fermat’s Last Theorem which
you might do (say) in a job application. Nor do you need to define the class group of
a number field, or explain what a modular curve is. But, at the same time, and thisis very important, it can still be incredibly useful to place your work in a broader
context. For example, on my last NSF proposal, I started out by reminding thereader briefly how there are very general conjectures linking Galois representations
coming from geometry to automorphic L-functions. I reminded the reader thatspecial degenerate cases of this conjecture correspond to very classical objects like

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the Riemann zeta function. I then mention how the work of Wiles addresses thecase when the representation comes from the cohomology of an elliptic curve over
Q. Then I explain how all the generalizations of Wiles’ theorem share a common
assumption, namely, that the Galois representations over Q that one can study bythis method have the property that they are, up to a twist, self-dual. So already, inperhaps not much more than a half a page, I have given the context to explain how
proving that a non-self-dual Galois representation is modular is “interesting.” Of
course, then I have to go on and explain how I am going to say anything interesting
about non-self-dual representations.
59.5. Do fat cats just get their grants without trying? Every proposal isevaluated on its merits, but of course “prior success” is taken into account when
judging future chances of success, and so it should be. But if Peter Scholze (say, totake someone who is not in the US so I can use his name) sends in an applicationconsisting solely of “I am working on several projects that I decline to disclose butthat I expect to be of the same importance as my prior results,” he would not be
funded. More realistically, I have heard that it has been the case that fields medalists
have been turned down for grants, but because all grants that are turned down arenever officially acknowledged, this is just hearsay. My feeling is that, on the whole,
the panels do a pretty good job, and (apart from the occasional controversial case)there is more of a uniform agreement than you might guess. The Hawk brings up
the key point that this opinion only concerns number theory panels. It may be thecase (and I occasionally hear rumours to this effect) that other areas are not run as
well. I would also say that the fat cats (on the whole) seem to put as much effort
into writing their NSF proposals as everyone else.
59.6. How can I compete with the fat cats given I’m only just starting
out? This is taken into account. If you are at most 6 years from your PhD, yourproposal is evaluated in that context; an effort is made to fund promising young
people, and also people who have never received prior NSF support. That said, it’s
not easy to get a grant the first time you apply coming straight out of a postdoctoral
position.
59.7. What about broader impact? This is hard for younger people. But everyone on the panel realizes this and so the expectations are lower. You probably
don’t have any grad students yet, so what can you say? Perhaps you have given
expository talks at a workshop? Perhaps you have written up detailed notes onotherwise hard to access topics? Perhaps you have gone into the public schools insome hardscrabble inner suburban neighbourhood and taught calculus? (Not thelast one? Then don’t suggest that you might if there’s no reason to suspect thatyou have any previous inclination to do so.)59.8. Don’t Imagine. that you are going to be held account for what you say you
are going to prove in future proposals. Future proposals will be evaluated on theirown merits (as well as prior research), and nobody is going to know or remember
what you said in your previous NSF grants. It’s expected that some of problems
you are working on might not work out, and that you will have new ideas whileworking on the proposal.
Two further suggestions from the Hawk:

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59.9. When will I hear back? Answer: who the hell knows. Usually within six
months from the deadline, but not always, especially these days when the federal
government is funded from continuing resolution to continuing resolution. If you
hear in January, either you are Peter Scholze or it’s bad news. By May, no news isgood news: you probably weren’t in the ’not recommended’ pile, and they’re waiting
to see how far they can stretch the money in the ’recommended’ pile.
59.10. If I get the grant, how much money will I get? Answer: probably lessthan what you asked for in your budget, and if not, you probably didn’t budget
enough. Less glib answer: the program officers do adjust the award sizes in order tohit their target funding rates. You shouldn’t fret that if you ask for too much and theperson who’s next on the list asks for a lower number, that could hurt your chances.
The natural followup: “If program officers have that kind of discretion, wouldn’t it
be better if they gave smaller awards to more people?” You can certainly argue
that in theory that might be better, but in practice the answer is emphaticallyno. DMS’s (DMS = division of mathematical sciences at the NSF) funding rateis already much higher than that of other divisions, as high as can politically besustained within NSF. If the funding rate went up, DMS’s budget would be cut,and the rate would go back down again.
59.11. Do you have any other thoughts? The fact that approximately 30%
percent of proposals get accepted is a fairly immutable law of nature. It is no
doubt depressing to be continually rejected by the NSF, and good people simply
stop applying, in some sense making it then harder for everyone else. If, for somereason, the number of applications suddenly doubled, it wouldn’t be the case thatthe success rate would halve, but more proposals would be awarded. So, there is areal sense in which the more people who apply the more grants are awarded.

60. In brief
Wed, 22 Oct 2014
The start of the academic year has a habit of bringing forth distractions, notleast of all to someone as disorganized as me. So here are a few remarks in brief.
60.1. The class number of Q(ζ151 )+ is one. John Miller, a student of Iwaniec
at Rutgers, wrote the following nice paper (see [Mil15b]), which improves upon aprevious result of Schoof. (Related: here, [Mil15a]) One technique that is usefulin computing the class numbers of fields with small discriminant is to make useof the Odlyzko bounds. Here’s a typical example. If K = Q(ζ37 )+ , then the root
discriminant of K is 30 or so. However, by consulting Odlyzko, one sees that any
totally real field with this root discriminant has degree at most 40. Hence the classnumber of K is either one or two, and it is easy to rule out the second possibility
by using genus theory. More generally, whenever one has an a priori bound onh+ , one can compute h+ by relating h+ to the index of the circular units (Schoof
did this in a previous paper.) This trick only works if the root discriminant of thetotally real field is at most 60 (or so), which seems to prevent one from applying
this to real cyclotomic fields for p ≥ 67. (There’s always a bound on the class groupby Minkowski, but that is a terrible bound.) The idea behind this paper is that
Odlyzko’s bound can be improved if one in addition knows that certain primes ofPERSIFLAGE: MATH BLOG POSTS

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small norm are principal. And since one has explicit fields, it is possible to show thatthe relevant ideals are principal by exhibiting explicit elements with the appropriate
norm. I can’t quite tell how lucky the author was to find such elements (he searches
for cyclotomic elements expressible as a small number of roots of unity), but itworks! Perhaps, a postiori, it is useful that these fields do actually turn out to haveclass number one.
60.2. Stickelberger’s Theorem. I proved Stickelberger’s theorem in class theother day — well, with one caveat. I proved that all the ideals q of prime norm
are annihilated by the Stickelberger ideal. This certainly implies the result, because
the class group is generated by such ideals. This follows, for example, by the Cebotarev density theorem applied to the Hilbert class field (which was my argument
in class). But then I worried that this was an anachronistic argument, and indeed
Stickelberger’s theorem was a solidly 19th century result. So what did Stickelberger
do?
Comment 60.3 (Danny). Speaking of fields with class number one, have you seen
this preprint by Darren Long and Morwen Thistlethwaite: here? (see [LT16]).
Comment 60.4 (John Miller). Long and Thistlethwaite’s paper is quite interesting. They mention that to have even a chance of producing a prime clique big
enough to show class number 1, the root discriminant must be less than 16πe.
This raises the question: Although we believe that fields with class number 1 arequite plentiful, do we have any reason to expect there are infinitely many fields ofclass number 1 with bounded root discriminant? Towers of fields with class number
1 should have increasing root discriminant; and on the other hand most towers
with bounded root discriminant are built out of Hilbert class fields of fields with
nontrivial class groups.
Comment 60.5 (Persiflage). The only clearly convincing heuristics (to me) aboutclass numbers concern families of fields of fixed degree (so the discriminant necessarily tends to infinity). The problem of understanding fields with bounded root
discriminant (where the bound is bigger than the limit of the Odlyzko bound) on
the other hand seems very quite difficult. It’s possible that one could have a tower offields all with class number one and the same root discriminant (if the corresponding Galois groups are perfect), although it doesn’t seem particularly plausible onsome sketchy heuristic grounds.

61. Mysterious formulae
Sat, 15 Nov 2014
I’m not one of those mathematicians who is in love with abstraction for its
own sake (not that there’s anything wrong with that). I can still be seduced by
an explicit example, or even — quell horreur — a definite integral. When I wasyounger, however, those tendencies were certainly more pronounced than they are
now. Still, who can fail to appreciate an identity like the following:
e−2π

∞
Y

(1 − e−2πn )24 =n=1

Γ(1/4)24.
224 π 18

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F. CALEGARI

But man cannot live on identities alone, and ultimately one’s efforts turn in otherdirections. So it’s always nice when the old and new words coincide, and an identity
is revealed to have a deeper meaning. The formula above is a special case of the
Chowla–Selberg formula [SC67], which is, possible typos in transcription aside,XXp
log y 6 |∆(τ )| + 6h log(4π ∆K ) = 3wK
χ(r) log Γ(r/∆K ).
CM (K)

Here the notation is as you might guess — y is the imaginary part of τ , which isranging over the equivalence classes of CM points for a fixed ring of integers in an
imaginary quadratic field (there is presumably a version for orders as well). The
existence of this identity (and a vague sense that it was related to the Kronecker
limit formula) was basically all that I new about this identity, but Tonghai Yanggave a beautiful number theory seminar this week explaining the geometric ideas
behind this formula, and some generalizations (the latter being the new work). So,
just as in the Gross-Zagier paper [GZ85] on the special values of j at CM points,
one now has two proofs of this result which complement each other, one analytic,
and one geometric. (I apologize in advance for not being able to attribute all [orreally any] of the ideas, Tonghai certainly mentioned many names but I never takenotes and this was 5 days ago.) The first remark is that the RHS is essentially thelogarithmic derivative of the corresponding Artin L-function. On the other hand,
it turns out (non-obviously) that the left hand side can be related to the Faltingsheight(s) of the corresponding Elliptic curves with CM by OK . I think this relation
was discovered by Colmez in his ’93 Annals paper [Col93]. The Faltings height has
always been a slippery concept to me, and in fact the theory of heights in general
has always struck me as being connected to the dark arts. In particular, various definitions depend on certain choices of height function, although they actually don’t
depend on that choice in the end. So when actually doing a calculation, it’s always
nice if you can magically produce some choice which makes calculation possible.
And of course, when making a choice of function on some (tensor power of) ω overthe modular curve, what better choice is there (if one wants to control the zeros
and poles) than ∆. (Tonghai mentioned another version of the formula where one
instead used certain forms which are Borcherds products — of which ∆ is a highly
degenerate example. I had the sense that this formulation was more generalizable toother Shimura varieties, but I never understood Borcherds products so I shall say nomore.) Key difficulties in understanding generalizations of these formulas involve
ruling out certain vertical components in certain arithmetic divisors on Shimuravarieties, which I guess must ultimately be related to understanding the mod-p
reduction of these varieties in recalcitrant characteristics (blech).
Colmez also formulated a conjectural generalization of the CS-formula, which iswhat Tonghai was talking about, and on which he (and now he together with his
co-authors) have made some progress. The viewpoint in the talk was to re-interpretthese identities in terms of arithmetic intersection numbers of arithmetic divisors
on Shimura varieties. Of course, this is intimately related to the ideas of GrossZagier and its subsequent developments, especially in the work of Kudla, Rapoport,
Brunier, Ben Howard, and Tonghai himself (and surely others. . . see caveat above).
In light of this, one can start to see how special values of L-functions and theirderivatives might appear. I can’t possibly begin to do this topic justice in a blog
post, but I will at least strongly recommend watching Ben Howard talk about thisat MSRI in a few weeks (Harris-fest, Tuesday Dec 2 at 11:00). I’ll be there to watchPERSIFLAGE: MATH BLOG POSTS

131

in person, but for those of you playing at home, the video will certainly be posted
online. Ben is talking about exactly this problem. Since he is an excellent lecturer,
I can safely promise this will be a great talk.
Added: Dick Gross emailed me the following (which also gives me the chance
to say that Tonghai did indeed mention Greg Anderson during his talk):. . . if you want to read a nice analytic treatment of the Chowla–
Selberg formula, using Kronecker’s first limit formula, you can find
it in the last chapter of Weil’s book “Eisenstein and Kronecker”.
I found an algebraic proof of [SC67] when I was a graduate student, using the moduli of abelian varieties with multiplication by animaginary quadratic field (what we would now call unitary Shimuravarieties). Deligne figured out what I was actually doing, and generalized it to prove his wonderful theorem that Hodge cycles on
abelian varieties are absolutely Hodge.
Greg Anderson formulated a generalization of [SC67] for the periods of abelian varieties with complex multiplication. This wasrefined by Colmez, and we know how to prove all the refinements
when the CM field is abelian over Q. Tonghai and Ben have been
making progress in some non-abelian cases.

62. Harris 60
Mon, 15 Dec 2014
I’ve just returned from the excellent MSRI workshop which honored Michael
Harris’ 60th birthday, and here is a brief summary of some of the gossip [excisedfrom this version] and mathematics I picked up when I was there.
Akshay gave a very intriguing talk on integral structures in cohomology. It reminded me of a question that we discussed a long time ago near Washington SquarePark in NYC. Recall that, for tempered automorphic forms contributing to the cohomology of GL(n)/Q, a computation with (g, K) cohomology shows that each suchform occurs in cohomology in degrees q0 , . . . , q0 + ℓ0 and contributes (a multiple)
of
 
ℓ0
k
dimensions in degree q0 + k. Is there any analogue of this for torsion classes orin characteristic p? Assume here that the residual representation also occurs only
inside the relevant ranges of cohomology, which should be the case as long as thecorresponding residual representation is not Eisenstein. If ℓ0 = 0, there is nothing
to say. If ℓ0 = 1, then the result follows from an Euler characteristic argument; thatis, the cohomology in each of the two non-zero degrees over Fp will have the same
dimension. If ℓ0 = 2, then Poincaré duality shows that the cohomology groups in
the two outer degrees will have the same dimension (again we are working with
non-Eisenstein classes and trivial coefficients, so from this point of view things
look compact), and then an Euler characteristic argument shows that the space
appears in some multiplicities of dimensions (1, 2, 1), as in characteristic zero. Now
suppose that ℓ0 is arbitrary. I will assume we are in a multiplicity one situation andthat all the local deformation rings are smooth. The CG-method (under suitable

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hypotheses) produces a resolution P • of R∞ consisting of finite free S∞ -modules,
where R∞ is smooth of relative dimension q and S∞ is free of relative dimension
n := q + ℓ0 . To recover the cohomology over a finite field, one takes the quotientof this resolution by the maximal ideal of S∞ and then takes cohomology. In otherwords, what we are really computing is
Ext∗S∞ (R∞ , k).
This answer depends only on the rings involved and not on the resolution. For
example, suppose that q = 0, which is the same as saying that there are no nontrivial local infinitesimal deformations. Then R∞ = Zp , and one can compute thecohomology of this ring using the Koszul complex, and one gets the expected dimensions. Note that if R = Zp , then this recovers the automorphic computation,
but this is already slightly interesting if R = Fp or even Z/pk Z. However, it seems
a little optimistic to expect this pattern to hold in general. I spent some time trying
to prove it using commutative algebra, but one problem is that it is not true in thatgenerality. For example, suppose that ℓ0 = 3, and thatR∞ = Zp [[y1 , y2 , y3 ]],S∞ = Zp [[x1 , x2 , x3 , x4 , x5 , x6 ]],where the map from S∞ sends the generators to the six possible monomials of degree
two. Then the appropriate dimensions of the ext groups are 4, 14, 14, 4. (Thanks toDaniel Erman for this example.) Now this example actually can’t occur globally,
because the same computation implies that one the Betti numbers over Qp are also
given by these numbers, which violates the previously referenced computation with(g, K)-cohomology. However, one should easily be able to deform it very slightly tokill off any cohomology in characteristic zero, for example replacing yi by yi − ϵi forsmall constants ϵi . Of course this doesn’t disprove anything, but it does strongly
suggest that the dimensions over Fp could be all over the place, subject to the
Poincaré and Euler characteristic conditions. Akshay has also pointed out that thecase ℓ0 = 3 is interesting from a different but related perspective: the analytic
torsion will vanish in this case, which implies that, at least morally (since we have
localized at a maximal ideal), that the alternating product of the the orders of thecohomology groups over Zp should equal one. Is this a consequence of the Taylor–Wiles method? I just thought of this question right now and it may have an obvious
answer which Akshay knows, I’ll ask him today and report back. A second obvious
question is what happens if one looks only at m-torsion rather than Fp -torsion;
perhaps that is the more sensible generalization of the characteristic zero question
anyway.

63. Derived Langlands
Mon, 22 Dec 2014
Although it has been in the air for some time, it seems as though ideas fromderived algebraic geometry have begun to inform developments in the Langlands
program. (A necessary qualifier: I am talking about reciprocity in the classicalarithmetic Langlands program here.)
I want to describe a very simple instance of this which came up in Akshay’s
MSRI talk which I linked to in the post above. Start by fixing a global residual

PERSIFLAGE: MATH BLOG POSTS

133(GL-)odd Galois representation:
ρ : GQ → GLN (Fp )Let us suppose that ρ is surjective. Associated to this representation is a fixed
determinant unrestricted global deformation ring R, and a fixed determinant unrestricted local deformation ring which I will call S. (Apologies for the notation, butwordpress is not great with lots of subscripts.) I assume that the reader can makethe appropriate adjustments to these definitions if the local representation is not
irreducible by adding framings. One knows (by [AC14]), at least if p ≥ 2N + 1, thatthe map
Spec(R) → Spec(S)is finite. Let us now suppose that ρ restricted to GQp admits a crystalline lift of someregular weight; associated to this weight is a Kisin local deformation ring which I
shall call T . If you like, you can even imagine that we are working in small weight
so that T has nice properties; perhaps it is even smooth. Barry Mazur has madea conjecture for what the (relative) dimension of R should be over Zp . Namely, it
should be given by the Euler characteristic of the adjoint representation, which isequal (see Def.2.1 and the subsequent comment) todim B − ℓ0 ,where B is a Borel of SLN (R), and ℓ0 is the difference between the rank of SLN (R)and SON (R). Of course, these quantities can easily be calculated explicitly in this(or any) case; for SL(N )/Q, ℓ0 is the integer part of (N − 1)/2. On the other hand,
we may also compute the (relative) dimensions of the rings S and T , and we findthat
dim(S/Zp ) = N 2 − 1, dim(T /Zp ) = N (N − 1)/2.
(The notation here means the relative dimension.) The Fontaine–Mazur sanity
check is to see that, on the associated rigid analytic spaces, the assumption that
R and T meet transversally inside S should imply that their intersection only has
finitely many points. Indeed, we can compute that the expected dimension of theintersection is exactly:
N (N − 1)/2 + dim(B) − ℓ0 − (N 2 − 1) = −ℓ0 .
When N = 2, we have ℓ0 = 0, and everything is as expected. However, as soon
as N ≥ 2, we have ℓ0 ≥ 0, and so the expected dimension is negative. This saysthat regular algebraic automorphic forms for such N are much rarer beasts thantheir counterparts for N = 2, where modular forms are abundant. For example,
it is not known if there exists a regular algebraic cusp form for GL(N )/Q giving
rise to a ρ as above for any N ≥ 5. (Note that forms from smaller groups coming
via functorial lifts will fail to give rise to representations with such large image.)
Now all of this is a philosophy that has been known and exploited for some time.
But suppose we actually try to interpret this heuristic a little more literally. For astart, we do expect that forms of characteristic zero do exist. This means that, ingeneral, there are unlikely intersections of Spf(R) and Spf(T ) inside Spf(S). Thatis, R and T will not, in general, be transverse! This is exactly a context in where,
to understand the intersection, it makes sense to introduce the derived world (see,for example, the introduction to DAG-V).

134

F. CALEGARIIn the classical picture, to recover the usual minimal deformation ring, one considers the intersection X := R⊗S T. However, science now tells us it is more natural
to consider the derived tensor product
Y = R ⊗L
S T.
If ℓ0 = 0, then the cohomology of Y should exactly recover X in degree zero andbe zero otherwise — that is, the classical context should be related to a completelytransverse intersection, and we are still in the usual word of schemes (or evencomplete local Noetherian rings). However, this will (essentially) never happen whenℓ0 ≥ 0. Classically, the ring X may be identified with the ring of endomorphisms
generated by Hecke operators on a single extremal degree of cohomology. Moregenerally, the cohomology of Y should be identified with the ring of Hecke operators
acting on the cohomology now in degrees q0 , . . . q0 + ℓ0 , where the notation is as in
Borel-Wallach and is fixed for all time. In particular, the only context in which one
should expect the intersection to be transverse (beyond ℓ0 = 0) is the case whenℓ0 = 1 and X is a finite ring (which can happen). Indeed, in such contexts, thecohomology over Zp also occurs exactly in one degree. It might be worth noting
here that the ring S is not in general regular, and so Y , a priori, is not evenbounded.
On the other hand, science also tells us that the complex Y has more information
than its cohomology; and so one should really think of Y as the correct object.
Unfortunately, I don’t have anything clever to say about derived arguments, but
let me use the Fontaine–Mazur heuristic to extract some tiny amount of juice. Since
I am not so DAGgy, I will only use algebra that goes back 30 years or more.
Instead of looking at the intersection of R and T inside the formal spectrum ofS, let us look at their intersection over Zp . In this case, all the dimensions havebeen shifted by one, so, when ℓ0 = 0, their intersection should be infinite (that is,
the length over Zp ). This is obviously the case, because X (which by assumption
exists) is flat over Zp and so automatically infinite. Well, obvious modulo Serre’s
conjecture and Fontaine–Mazur for GL(2)/Q, at least. On the other hand, whenℓ0 ≥ 1, then the dimensions don’t add up and the intersection multiplicity shouldbe zero. Thus, assuming the dimensions of the rings are correct, we should have:(1) If ℓ0 = 1, then the intersection multiplicity is finite and non-zero;(2) If ℓ0 ≥ 1, then the intersection multiplicity is zero.
In the latter case, we are invoking the conjecture of Serre (proved by Roberts
and Gillet–Soule that the intersection multiplicity is zero when the dimensions are
too small. (Heuristically, one can “move around” classes whose codimension is sufficiently large so they don’t intersect at all, although one cannot literally do thisin a local ring!) Actually, even this is a lie, because S is not necessarily regular, as
mentioned above, but pretend for this paragraph that it is. Serre’s formula tells usthat the intersection multiplicity is given by the Euler characteristic of the complex.
Let me now suppose that Y has no characteristic zero cohomology. For example,
we could be working in a weight corresponding to a (strongly) acyclic local system(as long as ℓ0 ̸= 0). What we want to compute is the alternating product of thecohomology groups, which should be equal to the alternating product of the integral cohomology groups (everything localized at the appropriate maximal ideal)of the corresponding congruence subgroup of GLN (Z). Yet this product (without
localizing at a maximal ideal) is basically equal to the Reidemeister torsion, whichPERSIFLAGE: MATH BLOG POSTS

135

is equal to the analytic torsion, which always vanishes when ℓ0 ≥ 1! Under our
finiteness conditions, when ℓ0 = 1, all the cohomology occurs in just a single degree, and so the multiplicity is just the length of X. But when ℓ0 ≥ 1, this gives(the expected) refinement of the vanishing of analytic torsion after localizing at a
non-Eisenstein maximal ideal, which was one of the questions implicitly raised in
the last blog post. To be more precise about our assumptions and conclusions, we
have:
Proposition 63.1. Let m be a non-Eisenstein maximal ideal of the cohomology
of a congruence subgroup of SLN (OF ) for a CM number field F , and let ρ be thecorresponding Galois representation. Assume that:(1) The required assumptions in [CG18a] hold — vanishing outside degrees
q0 , . . . , q0 + ℓ0 after localization at m, the representation ρ has big image,local-global compatibility, etc.;(2) The cohomology localized at m vanishes in characteristic zero.
Then the alternating product of the orders of the cohomology groups localized at
m is non-zero if and only if ℓ0 = 1.
One issue with proving this directly using the above argument is that we don’t
actually know the dimension of R in general. So, instead of working with Y , wework instead with the output of the Taylor–Wiles method as in C-G, namely:
Y∞ = R∞ ⊗L
S∞ S∞ /aHere S∞ is an Iwasawa algebra of diamond operators of dimension q + ℓ0 , theideal a is the augmentation ideal with S∞ /a = Zp , and R∞ is a patched minimal
deformation ring of dimension q. These are relative dimensions, so the transverse
case over Zp corresponds exactly to the case when ℓ0 = 1. We note here that the ringS∞ is regular, so we are in the appropriate context of Serre’s multiplicity formula,and the result follows. (Exercise: we are only using a very special case of thevanishing claim for intersection multiplicities when one component is Zp inside S∞ ;the vanishing should be easy to prove directly in this case.) This is good, because
it gives a purely Galois theoretic proof (well, really only a heuristic because of all
the conjectures one needs to assume) of a result (vanishing of analytic torsion)
which is not at all obvious. (Well, not quite; the result is localized at a maximal
ideal — which one can’t do analytically — but it only applies to non-Eisensteinmaximal ideals.) At any rate, thinking through this example after Akshay’s talkhas convinced me that this derived perspective is a very good one.

64. Stable completed homology without Quillen–Lichtenbaum
Wed, 21 Jan 2015
Having just made (hopefully) the final revisions on my paper on stable completed
cohomology groups [Cal15], I wanted to record here a few remarks which didn’t
otherwise make it into the paper.
e 2 (SL, Zp ) = Zp for p ≥ 2, one
The first is that, in addition to the result that He
may also compute H3 (SL, Zp ) for p ≥ 3. Namely:e 3 (SL, Zp ) = 0.
H

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F. CALEGARI

This result is proved in the paper up to a finite group, so the point here is thee 2 comes from the Hurewicz isomorphism
integral refinement. The computation of He 2cont (Zp ).
π2 (SK(Z, Zp ); Zp ) ≃ H
However, the Hurewicz theorem also gives an epimorphism
e 3cont (Zp ),π3 (SK(Z, Zp ); Zp ) → H
and one finds that the first group lives in an exact sequence
H 2 (Qp , Zp (2)) → π3 (SK(Z, Zp ); Zp ) → K3 (Z) ⊗ Zp
Since both flanking groups vanish for p ≥ 3, the middle group is zero, and the claim
follows.
The second remark is that, throughout the paper, I assume the Quillen–Lichtenbaum
conjecture, which is now a theorem due do Voevodsky and others. However, I must
confess, I do not have the fine details of the argument at my fingertips. How much
can one say without it? The answer is quite a lot. Due to work of Borel, Soulé, andQuillen (see [Bor74, Sou79, Qui72], all of which is much more familiar to me, at least
relatively speaking), we know that the K-groups of number fields are finitely generated abelian groups, we know their ranks, and we know that the Chern class maps
to the appropriate Galois cohomology groups are surjective. Moreover, we understand K2 (OF ) completely in terms of Galois cohomology by work of Tate [Tat76].
(In this game, I am also giving up the results of Hesselholt and Masden on the
K-theory of local fields, and instead using the results of Wagoner [Wag76], whichsimilarly give everything in very small degree and up to a finite group in higher
degrees.) In particular:(1) The computation of H2 (ΓN (p), Fp ) for large N , where ΓN (p) is the principal congruence subgroup of GLN (Z), is unaffected. This also uses thecomputation of K3 (Z) by Lee and Szczarba [LS76].
(2) The identification of the completed K-groups with Galois cohomology groups
still holds up to a finite group.
(3) The computation of the rational stable completed homology groupse ∗ (SL, Zp ) ⊗ Q = Q[x2 , x6 , x10 , x14 , . . .]Hunder the assumption that either p is regular or ζp (3), ζp (5), ζp (7) etc. are
all non-vanishing still holds.
Something that does require Quillen–Lichtenbaum is the vanishing of the partiallycompleted K-group for very regular primes.
Regarding the computation of the rational stable completed homology groups,
the referee made a very interesting point (I will come back in a later post to therefereeing of this paper and some other of my recent papers in a post on “what agreat referee report should be”). I prove that the rational stable completed homology
groups are the continuous homology of the homotopy fibreSK(Z, Zp ) → SK(Z) → SK(Zp )(The definition of SK(Z, Zp ) is just homotopy fibre of this map.) Now SK(Z, Zp )is an infinite loop space, which under the assumption that p is regular or on the
non-vanishing of the p-adic zeta function at integral arguments, has the property
that the homotopy groups with coefficients πn (SK(Z, Zp ); Zp ) are rationally nonzero in exactly degrees 2, 6, 10, etc. The referee noted that the computation of

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137

rational stable completed homology should follow precisely from this description
using the Milnor–Moore theorem, which shows that (for simply connected H-spaces)that the homology is (rationally) the universal enveloping algebra of the rational
homotopy classes (and so, in particular, the Hurewicz map is rationally injective).
One consequence is that the rational homotopy groups are precisely the primitiveclasses in rational homology. To orient the reader, this is exactly the theorem whichallowed Borel to compute the rational K-groups of (rings of integers) of number
fields from his computation of stable homology over Q. Now I was a little worried
about this, because the Milnor–Moore theorem does not literally apply, since one iscomparing here homotopy groups with coefficients in Zp and continuous homology(the latter is just the inverse limit of homology groups modulo pn ). However, having
looked at the argument in Milnor–Moore and then having Paul Goerss explain it
to me, the argument does indeed seem to simply work in this case. (Warning, thisis a weaker statement than saying I checked the details.)
To be more precise, suppose that G is a simply connected infinite loop space,and suppose that G has the property that the groups πn (G; Z/pk ) are finite for alln and k, so πn (G; Zp ) is the inverse limit of these groups. There is a pairing[, ] :

πr (G, Z/pk ) ⊗ πs (G, Z/pk ) → πr+s (G, Z/pk ),which, after taking inverse limits in k and tensoring with Q, makes π∗ (G, Zp ) ⊗ Q
into a Lie algebra over Qp , then the Hurewicz map will induce an isomorphism
U (π∗ (G, Zp ) ⊗ Q) → H∗cont (G, Qp ) := lim H∗ (G, Z/pk ) ⊗ Q
of Hopf algebras. The key technical point required here is to define the appropriate
pairing on homotopy groups with coefficients, which is done by Neisendorfer. (If
G is simply connected infinite loop space, one doesn’t have to worry about theissue of homotopy groups with coefficients in very low degree exhibiting certain
pathologies.)
As another example of this, one can take G = SK(Zp ). In this case, the rational
continuous homology reduces, by work of Lazard, to lie algebra cohomology, andgives an exterior algebra in odd degrees ≥ 1. So SKn (Zp ; Zp ) ⊗ Q has dimension
one in odd degrees ≥ 1 and is zero for all even positive degrees. This is a result ofWagoner. In fact, Wagoner proves something slightly stronger, also capturing some
information away from p. To do this, he also proves a version of the Milnor–Moore
theorem, but his assumptions are more stringent than what we discuss above.

65. Higher direct images of canonical extensions
Sun, 04 Jan 2015
I like Kai-Wen’s talks; he gives lots of examples, writes big with big chalk, andclearly explains the key points of the argument. I’m not sure I would classify histhesis as light reading material, but if he produced a video series explaining all
the details in lecture format, I would buy the DVD. Speaking of different ideas fordisseminating mathematics, I have some thoughts on that, but they will have to wait
for another time. For now, I just wanted to make the smallest remark concerning
Kai-Wen’s lecture at the Harris conference.
As all my readers surely know (this is code for I am not going to explain why), a
key ingredient in the Harris–Lan–Taylor–Thorne [HLTT16] argument is the fact the

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F. CALEGARI

the higher direct images of the of subcanonical automorphic vector bundles under
the projection from the toroidal compactification to the minimal compactificationof quite general classes of Shimura varieties vanish. In contrast, this does not holdfor the higher direct images of the canonical extensions, and when this was firstbeing discussed, it was not entirely clear (at least to me) what was going on. But
Kai-Wen’s talk actually does make the situation very clear! That is what I want totalk about.
Let X be the open Shimura variety, let Y be a minimal compactification, andlet Z be a toroidal compactification. To avoid silliness, assume that Y ∖ X has
codimension at least two. Let W be an automorphic vector bundle on X, andlet W can and W sub denote the canonical and subcanonical extensions of W to Z.
There’s a short exact sequence
0 → W sub → W can → Q → 0.
Take the pushforward of this to Y . We know that the higher direct images of thefirst sheaf vanish, and so we obtain an exact sequence
0 → π∗ W sub → π∗ W can → π∗ Q → 0.
The last sheaf is supported on Y ∖ Z, which has fairly small dimension, so its
cohomology groups vanish in high degree by Grothendieck. Now let us assume thatthe higher direct images also vanish for W can . It follows that the Leray spectral
sequence degenerates (for both W sub and W can ), and so we obtain isomorphisms
H ∗ (Z, W sub ) = H ∗ (Z, W can )in sufficiently high degree. Now the canonical bundle on Z is also an automorphic
vector bundle, and so Serre duality relates the cohomology of W sub to the cohomology of V can for another automorphic vector bundle V , and relates the cohomologyof W can to V sub . For example, for modular curves, the Serre dual of ω k is ω 2−k (∞),
because the canonical sheaf of the modular curve is Ω1 ≃ ω 2 (∞). Hence (using theassumption on codimensions made above so the numerology works out) we end up
with the isomorphism
H 0 (Z, V sub ) = H 0 (Z, V can ).
But this formula says that all modular forms of weight V are cuspidal! So this gives
an easy proof of:Lemma 65.1. If there exists at least one form of weight V which is not cuspidal,
then at least one of W sub or W can has non-trivial higher direct images under π.
Of course, we know from [HLTT16] that it will be the second (because the higher
direct images of the first vanish), but we didn’t prove that. Now I just chatted with
Kai-Wen, who did one better than this lemma. First of all, remember that there is
an automorphic line bundle ω on X (corresponding to “parallel weight”) which is
ample, and the corresponding canonical extension to Z descends to an ample on Y ,which we also call ω. What’s nice about this is that, using the projection formula,
one can replace the question about the vanishing of the higher direct images of W
by the vanishing of W under twists by powers of this bundle. But that means one
can translate the problem of asking whether there exists a non-cusp form in the
dual weight V to whether there exists a non-cusp form in weight V ⊗ ω n for somearbitrarily large n. Now as before, we have an exact sequence:
0 → π∗ V sub ⊗ ω n → π∗ V can ⊗ ω n → π∗ R ⊗ ω n → 0.

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twisted by some arbitrarily high power of ω, where we have used the vanishingof R1 π∗ V sub and the projection formula. Here R is just V can /V sub . On the other
hand, because ω is ample on Y , we know that
(1) H 1 (Y, π∗ V sub ⊗ ω n ) vanishes for sufficiently large n,(2) π∗ R ⊗ ω n is generated by global sections for sufficiently large n, and so, forsuch n, we have H 0 (Y, π∗ R ⊗ ω n ) ̸= 0 as long as π∗ R ̸= 0.
So if one shows that π∗ R is non-zero then one is done. Certainly R is non-zero,
but analyzing π∗ R is a bit more subtle (I jumped the gun a little on the firstversion of this post, but Kai-Wen told me I needed to be a little more careful).
On the other hand, there are many classical examples where one can explicitly
construct non-cuspidal forms. For example, one can take X = Ag with g ≥ 2 to be
the Siegel moduli space, and take W to be the line bundle ω k . Then Siegel himself
constructed the so-called Siegel Eisenstein series for high enough k. Kai-Wen also
tells me the non-vanishing of π∗ R can be proved more generally for X = Ag , andso one has:Lemma 65.2 (Kai-Wen). Let g ≥ 2, let X = Ag , and let W be an automorphic
bundle. Then at least one of higher direct images Ri π∗ W can with i ≥ 0 must be
non-zero.
In fact, Kai-Wen also tells me he had a proof of (a more general version of)
this last result even before HLTT knew about the vanishing of Ri π∗ V sub , but thisargument gives a completely transparent proof of why they can’t both vanish.

66. Abelian spiders
Sun, 11 Jan 2015
This is a blog post about the thesis of my student Zoey Guo, who is graduating
this year.
Let Φ be a finite graph. Associated to Φ is an adjacency matrix M such that thelargest eigenvalue λ is totally real. Let us call Φ abelian if the extension Q(λ2 ) isabelian. For example, all the Dynkin diagrams are abelian.
Several years ago, Scott Morrison and Noah Snyder (of secret blogging seminarfame) asked the following question. Given a finite graph Γ, let Γn be the graphobtained by adjoining a 2-valent tree of length n to some fixed vertex v of Γ. Then
can one classify all n for which Γn is abelian? It turns out Γn can be abelian for
only finitely many n, unless the graphs Γn happen to be one of the two infinite
families of Dynkin diagrams. The argument was effective, although not effectively
effective. (We did, however, prove a slightly weaker theorem which was sufficientfor the intended application which was effectively effective.)
What Zoey does in her thesis is consider the following generalization. Let Γ bea finite graph, and choose k vertices vi of Γ. Now adjoin k two-valent graphs ofvarying lengths n = {ni } to latex vi . Call the resulting graph a k-spider graph. The
main result of her thesis is the following:
Theorem 66.1 ([CG18b]). For any Γ and any fixed k, only finitely many of thecorresponding spiders are both abelian and not Dynkin diagrams.
What is more, the theorem is effectively effective. One key ingredient in thefiniteness results for k = 1 was the fact that, for characteristic polynomials Pn (X) of

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the graphs Γn , one can control the factors corresponding to Chebyshev polynomials.
Or, if one writes
Qn (t) = Pn (t + t−1 ),one can control the cyclotomic factors of Qn (t). This follows from a theorem ofHironaka–Gross–McMullen [GHM09], who exploit results by Mann on vanishingsums for roots of unity. However, when k ≥ 2, this breaks down completely. In fact,
in many examples, the corresponding polynomials Pn (t + t−1 ) will be divisible bycyclotomic polynomials of arbitrarily large degree. Here’s an example which Zoey
pointed out to me. Consider the disconnected graph consisting of a two copies of the
Dynkin diagrams An−1 and a third component Am for any integer m (in fact, theconstruction is much more general, but we will be very explicit here). Since ζ + ζ −1for ζ n = 1 is an eigenvalue of each connected component corresponding to An−1 ,it will be an eigenvalue of multiplicity (at least) two of their union. Now join thethree graphs by adding a vertex which is connected to the end of all three graphs;
this will be the 3-spider on a point corresponding to the triple (n − 1, n − 1, m). By
the interlacing lemma for graph eigenvalues, ζ + ζ −1 will be an eigenvalue of theresulting connected graph. So finding all the cyclotomic factors even for the explicit
polynomials coming from 3-spiders on a point seems like a real pain, and so one
needs a new argument to deal with roots of unity.
The proof of the theorem comes down to two key steps. First, for any sequenceof spiders, all but a uniformly bounded number of eigenvalues will lie in the interval[−2, 2], and all the eigenvalues will lie in some uniform interval [−M, M ]. Of course,
this means that the squares of the eigenvalues lie always in [0, M ] for some M andmostly in [0, 4]. Now imagine the largest such number λ2 . We know that it isalgebraic, and we are assuming that it is abelian. The key quantity to control turns
out to be the normalized trace of γ := (λ2 − 2)2 . Work of Cassels shows that, if
λ2 is cyclotomic, one can classify all cyclotomic integers for which the normalized
trace of γ is small. What kind of an upper bounds do we have? Well, if the degreeof λ is very large, then the bulk of the contribution has to come from λ ∈ [−2, 2],
or λ2 − 2 ∈ [−2, 2] (this is the reason why λ2 − 2 occurs above — it is a Chebyshev
polynomial). The worst case scenario is that all of the conjugates of λ2 are near 4,
which will give an estimate on the normalized trace of γ of 4 plus a quantity that
goes to zero with the degree. However, it is hard for an algebraic number to havetoo many of its conjugates near any particular integer (in this case, 4). To exploit
this, one can note, for example, that3 − x − log(4 − x) ≥ 0,x ∈ [0, 4].

If we denote the conjugates of γ := (λ2 −2)2 by σγ and suppose that this has degreen, then we deduce thatY
3n − nTr(γ) − log(4 − σγ) ≥ O(1).
The O(1) term (which is completely explicit) comes from the fact that finitely manyof the σγ are outside [0, 4] and so one can not use the previous inequality. Let us
consider the inequality. As long as γ ̸= 4, the logarithmic factor is a norm of thealgebraic integer 4 − γ, and hence non-negative. So we get an upper bound for thenormalized trace which is now 3 plus some explicit error term which tends to zeroas the degree goes to infinity. This type of idea was first used by Chris Smyth whenstudying the trace problem of Siegel (see this post). Now 3 is still too big to apply

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the results of Cassels. So one has to also exploit that (λ2 − 2)2 is not too close to 3
either (this is the where the inequality above is an equality). In fact, one ends up
using not just 4 and 3 but 43 different algebraic integers which are all of the form
2 + ζ + ζ −1 , and where one uses logarithms weighted by various real constants. The
precise constants and algebraic integers were optimized by simulated annealing. In
the end, one gets an upper bound of the form 2.4 plus a very explicit error term,
which is enough for the Cassels machine. This gives a complete answer as long as
the degree of λ is big enough.
If λ has small degree, then one is also in good shape — given a bound for λ anda bound for the degree, there are only finitely many such algebraic integers, whichis enough to prove the theorem. However, this last step — whilst effective — is
not at all effectively effective. So another argument is required to make everything
work in practice. Note that the degree of the polynomial defining λ certainly goesto infinity, but it may be reducible, and in particular divisible by many cyclotomic(Chebyshev) factors all of whose roots are in [−2, 2]. Let’s explain how to overcome
this issue in the case of 3-spiders coming from the trivial graph (which is typicalof the general case). If you take the 3-spider with legs of length (a, b, c), then, for(a, b, c) big enough, one finds that3λ→ √ .
2(The limit of the largest eigenvalues of a sequence of infinite spiders will always be
an algebraic number.) Importantly, this convergence is exponential. However, it’seasy to see that any algebraic integer all of whose conjugates are uniformly boundedcannot be extremely close to any fixed algebraic number, and it easy to give effective
bounds to this effect. So one wins in high degree by Cassels type arguments and inlow degree by the fact that the eigenvalues converge rapidly to computable algebraic
numbers. One annoying issue is that the convergence requires all of the (a, b, c) totend to infinity, so one has to inductively reduce to the case of 2-spiders with somefinite list of possible c, which entails a certain amount of combinatorial explosion.
However, as a complete worked example, one has the following:
Theorem 66.2 ([CG18b]). The complete list of abelian 3-spiders on a point isgiven by:
e6 , Ee7 , Ee8 , whose largest eigen(1) The Dynkin diagrams An , Dn , E6 , E7 , E8 , E−1value is of the form ζ + ζ ,(2) The 3-spiders (3, 3, 3), (2, 4, 4), and (2, 3, 7) with√
5 + 132λ =,2(3) The 3-spiders (3, 3, 7), (2, 8, 8), and (2, 7, 11) with ζ 13 = 1 andλ2 = ζ 11 + ζ 10 + ζ 3 + ζ 2 + 2,
(4) The 3-spiders (4, 4, 4), (3, 5, 5), and (3, 4, 9) with√
λ2 = 3 + 2.
Now all of this is quite amusing, but you may complain that it doesn’t really haveany practical application. However, as it happens, Scott Morrison asked me whether
it was possible to find all abelian 2-spiders for some very explicit graph (omittedhere), in order to further the classification of finite index subfactors, because the

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F. CALEGARI

all the current non-number theoretic obstructions could not rule out this familyof examples as coming from subfactors. Zoey’s method could be applied to showthat every 2-spider in the corresponding family was not abelian. So Zoey’s results
have already been used outside her field (see forthcoming work of Morrison and his
collaborators)
to complete the classification of subfactors of index between 5 and√3 + 5. All this, of course, while having the thesis with the best title.
Comment 66.3 (Scott Morrison). Happily, we’ve found yet another applicationof Zoey’s results in the classification of subfactors,
and indeed our almost finished√paper on the classification up to index 3 + 5 will make two independent, andrather different uses, of this work.
The first is the example you talk about above and that Zoey used as an illustration of the method.
The second is even more exciting — and we haven’t yet had a chance to see if
this method could help pushing the classification even further.
The basic idea is that we have a family of graphs as follows: fix some finite graphΓ, and mark one vertex “x”, and name all the vertices at the maximal radius from x“Y ”. We now consider adding a chain of n edges to x, and gluing on some arbitrary
finite graph to Y .
For the Γ we’re interested in (and possibly ’often’) we can show by the theoryof “connections” that any graph in this family which is the principal graph of a
subfactor must have graph norm satisfying some particular polynomial depending
just on n (and not on what we glue to Y ).
Now we don’t know that this polynomial has anything to do with any particular
finite graph (in fact, it’s a multiple of the minimal polynomial for the norm of acertain infinite graph) so we have to work a bit harder in places to apply Zoey’s
method, but happily it all works out, and one can show that the graph norm cannot
possibly be cyclotomic, thereby ruling out all graphs in the family as principal
graphs of a subfactor.
Notes 66.4. The paper which relies on the main theorem of [CG18b] is [AMP23].
But that paper only cites the arXiv version of [CG18b], not the published ones,
despite the latter appearing in 2023 and the former in 2018.

67. Inverse Galois problems II
Tue, 13 Jan 2015
David Zywina was in town today to talk about a follow up to his previous
results (see § 17) discussed previously on this blog. This time, he talked about his
construction of Galois groups which were simple of orthogonal type, in particular,
the simple groupsΩ(V ) ⊂ SO(V ) ⊂ O(V )where V is a vector space V over Fl of odd dimension at least five. The group Ω(V )here is a simple group of index two inside SO(V ). In the special case when n = 5,
there is an exceptional isomorphismΩ(V ) ≃ PSp4 (Fl ).
In contrast to his constructions of number fields with Galois group PSp2 (Fl ), Zywina actually constructs a family of compatible families whose residual image is

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generically Ω(V ). When David told me about this construction (scribbled on apiece of paper) in Frankfurt airport on the way back from Oberwolfach, I wastroubled by something which I shall now explain. Without saying so much about
the construction (you can read about it here), the compatible families of Galois
representations of interest occur inside H 2 (XT , Q(1)) for a carefully chosen familyof non-isotrivial elliptic surfaces X. As David explained in his talk today, the zerosection and the fibres of bad reduction contribute a large Galois trivial summand to
H 1,1 , and the remaining piece is five dimensional. What disturbed me at the time
was that this construction was surely liftable to a compatible family of four dimensional representations with generalized symplectic image. After all, Tate’s result on
H 2 (GQ , C× ) guarantees that one can lift any projective representation with image
in PSp4 (Fl ) to a genuine generalized symplectic representation. This representation
should then come from a Siegel modular form, since all oddness conditions should
be automatic. On the other hand, if you want a family of Galois representations
giving rise to a family of Siegel modular forms, especially one for which the maximal
difference between any two Hodge–Tate weights in ∧2 W is two, then you expectthat they have to come from a family of abelian surfaces, or at least abelian varieties A of dimension 2n with endomorphisms by the ring of integers in a totallyreal field of degree n. However, there is an obstruction to making this work — thecorresponding Galois representations will have Hodge–Tate weights [0, 0, 1, 1], andthey will have similitude character that is an even finite order character times thecyclotomic character. It’s easy to see that for such a family, the residual representations will (at least half the time) land in PGSp4 (Fl ) and not in the simple index
two subgroup, similar to what happens for modular forms of weight two. I thought
at the time that I must have been making some group theory error, so after today’stalk we sorted out the details.
In the process of this computation, however, I realized what my error actuallywas. I was imagining that the original compatible family of Galois representations in
H 2 had Hodge–Tate weights [0, 1, 1, 1, 2], but they could equally have had Hodge–
Tate weights [0, 0, 1, 2, 2]. And in this latter case, the Galois representation (up to
twist) of the corresponding Siegel modular form in GSp4 will have Hodge–Tateweights [−1, 0, 0, 1]. In particular, we are not looking for classical Siegel modular
forms of low weight, but the nasty Siegel modular forms which do not contributeto holomorphic limits of discrete series and only occur in coherent cohomology via
H 1 or H 2 . (A reference for this fact is George Boxer’s talk in Barbados.) And now
everything makes sense! That is, if you have a family of Galois representations with
Hodge Tate weights [−1, 0, 0, 1] and quadratic similitude character, then (with somegood luck) you can really have projective representations which land in the right
simple group for all but finitely many l.
A related point: when lifting projective representations using Tate’s theorem,
one may have to increase the size of the residue field. In fact, when ℓ ≡ 3 mod 4,
it will not be possible to lift an odd PSp4 (Fl ) representation to one in GSp4 (Fl )(there is an obstruction at infinity). Indeed, the natural lift is the group Sp4 (Fl )together with a scalar matrix I with I 2 = −1. This suggests what the picture
should be motivically: there should be an eight dimensional piece of H 2 (Y ) (forsome Y ) which admits an involution breaking the representation√ up into two four
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F. CALEGARI

such a Y explicitly? This does remind me of the motivic lifting problems that Stefan
Patrikis knows about.
From this analysis, it also becomes clearer why Zywina could find a family ofcompatible families with residual image Ω(V ) when dim(V ) is odd and at least five,but only isolated examples of compatible families with residual projective image
PSL2 (Fl ) ≃ Ω(V ) with dim(V ) = 3. In the latter case, the corresponding modular
forms will be forced to have odd weight k ≥ 1, and so the Hodge–Tate weights
will differ by at least two, and so Griffiths’ theorem implies that they should notdeform in a family. On the other hand, if you want to look for Siegel modular forms
which could possibly correspond to geometric families, and you want the similitude
character to be an even power of the cyclotomic character times a finite character,then it is possible to escape the spectre of Griffiths on your shoulder, but only
barely — you will be pretty much forced to work with forms whose HT weights
are [−1, 0, 0, 1]. Of course, I’m not sure I can prove that any Siegel modular formsof this kind actually exist! (insisting the Mumford–Tate group is big, naturally).
My proposal in the previous post to look for these representations using Siegel
modular forms would also have only found sporadic compatible families, because toensure computability and the determinant condition I suggested looking in weights
where the Galois representation was regular and had HT weights something like
[0, 1, 3, 4], — the gap being necessary to make the multiplier character a square ofa Hodge–Tate character.
There is one check left on these musings (though I’m sure it must be correct),
namely, that for the surface X in 1.4 of Zywina’s paper (also see [Zyw23]), one
should haveh2,0 (X) = 2.
Proof. The Hodge diamond of a minimal elliptic surface π : X → C was computedby Miranda, see [Mir89, IV.1.1] here; I’ll try to give a self contained argument. Let
ωE be the Euler characteristic of OX . Let L−1 be the bundle L−1 = R1 π∗ OX on
X; it is a line bundle because the fibres are elliptic curves, so it makes sense totalk about L. The bundle L has positive degree if and only if the fibration is not
isotrivial (this is not so hard, but let me give the proof of III.1.6 of Miranda as areference); let us assume this is the case. From the Leray spectral sequence, there
is an exact sequence
0 → H 1 (C, π∗ OX ) → H 1 (X, OX ) → H 0 (C, L−1 ) → H 2 (C, π∗ OX )Since π∗ OX = OC , the first term has dimension g, the genus of C. Since we areassuming that L has positive degree, the third term is also zero, and hence theirregularity of a non-isotrivial elliptic surface is
H 1 (X, OX ) = g.
It follows that
χ(OX ) = h0,0 − h1,0 + h2,0 = 1 − g + h2,0 .
In our particular case, the genus of C is zero. On the other hand, as noted in 2.4
of Zywina’s paper, the degree of the minimal discriminant is 12 · χE = 12 · χ(OX ).
In the example at hand, Zywina computes (see section 8) that χE = 3, and so
h2,0 = 3 − (1 − g) = 3 − 1 = 2.
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68. H2 (ΓN (p), Z)
Fri, 30 Jan 2015
In this post (which is a follow-up to § 64, I wanted to compute the group
H2 (ΓN (p), Z), where ΓN (p) is the congruence subgroup of SLN (Z) for large enoughN and p is prime. In fact, to make my life easier, I will also assume that p ≥ 3, andin addition, ignore 2-torsion. The first problem is to compute the prime to p torsion.
By Charney’s theorem, this will come from the cohomology of the homotopy fibre
X of the map
SK(Z) → SK(Fp ).
The relevant part of the Serre long exact sequence is, using classical computationsof the first few K-groups of the integers together with Quillen’s computation ofK∗ (Fp ),0 → π3 (X) → Z/48Z → Z/(p2 − 1)Z → π2 (X) → Z/2Z → 0.
Here is where it is convenient to invert primes dividing 6; from Hurewicz theorem
and Charney’s theorem we may deduce that, where ∼ denotes an equality up to a
finite group of order dividing 48,
H2 (ΓN (p), Z[1/p]) ∼ Z/(p2 − 1)Z.
In order to deal with 3-torsion, then we also have to show that the map K3 (Z) ⊗Z/3Z → K3 (Fp ) is injective for p ̸= 3. I have a sketch of this which I will omit fromthis discussion but it is not too hard (assuming Quillen–Lichtenbaum). It remains
to compute the homology with coefficients in Zp . I previously computed that there
was an isomorphism
H2 (ΓN (p), Fp ) = ∧2 g ⊕ Fp = H2 (GN (p), Fp ) ⊕ Fp ,where GN = SLN (Zp ) and g = H1 (ΓN (p), Fp ) is the adjoint representation.
68.1. Some facts concerning the cohomology of GN (p). There are short exact
sequences:
0 → H2 (GN (p), Zp )/p → H2 (GN (p), Z/pZ) → H1 (GN (p), Zp )[p] → 0,
0 → H2 (GN (p), Zp )/p2 → H2 (GN (p), Z/p2 Z) → H1 (GN (p), Zp )[p2 ] → 0.
Since H1 (GN (p), Zp ) = g is annihilated by p, we may deduce that
H2 (GN (p), Zp )/p = H2 (GN (p), Zp )/p2
as long as|H2 (GN (p), Z/p2 Z)| = |H2 (GN (p), Z/pZ)|.
Such an equality (for any group) is a claim about the Bockstein maps having a big
an image as possible. Indeed, for any group Φ, there is an exact sequence:
H3 (Φ, Z/pZ) → H2 (Φ, Z/pZ) → H2 (Φ, Z/p2 Z) → H2 (Φ, Z/pZ) → H1 (Φ, Z/pZ)
The first and last maps here are the Bockstein maps β2 and β1 . Since p is odd,
β1 ◦ β2 = 0. On the other hand, we see that the orders of the cohomology groups
with coefficients in Z/pZ and Z/p2 Z will have the same order if and only ifker(β1 ) = im(β2 ).

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F. CALEGARI

Hence we have reduced to the following claim. Take the complex
H∗ (GN (p), Fp ) = ∧∗ g
where the differentials are given by the Bockstein maps. Then we have to show thatthe cohomology of this complex vanishes in degree two. But what are the Bockstein
map is in this case? Note that since H1 (GN (p), Zp ) = g is annihilated by p, theBockstein map β1 will be a surjective map:
β1 : ∧2 g → g.
To compute this explicitly, recall that the isomorphism H1 (GN (p), Zp ) = g comesfrom the identification of g with GN (p)/GN (p2 ). Then, computation omitted due
to laziness, we find that the Bockstein is precisely the Lie bracket. Moreover, sincethe (co-)homology is generated in degree one, the higher Bockstein maps can becomputed from the first using the cup product formula. So the Bockstein complex
above is, and I haven’t checked this because it must be true, the complex computing
the mod-p Lie algebra cohomology of g. And this cohomology vanishes in degrees
one and two, so we are done. One consequence of this computation is that
H2 (GN (p), Zp ) = H2 (GN (p), Zp )/pis annihilated by p. Moreover, the last term can be identified with the kernel of the
Lie bracket (Bockstein) on H2 (GN (p), Fp ) = ∧2 g.
68.2. Returning to the main computation. From the Hochschild–Serre spectral sequence and the computation of stable completed cohomology, one has anexact sequence:0 ← H2 (GN (p), Zp ) ← H2 (ΓN (p), Zp ) ← Zp ← H3 (GN (p), Zp ).
From known results in characteristic zero, we immediately deduce that there issome α such that there is an exact sequence
0 ← H2 (GN (p), Zp ) ← H2 (ΓN (p), Zp ) ← Z/pα Z ← 0.
we also deduce that there is an exact sequence:0 ← H2 (GN (p), Z/pn Z) ← H2 (ΓN (p), Z/pn Z) ← Z/pmin(α,n) Z ← 0,
There are spectral sequences:
Hi (SLN (Fp ), Hj (ΓN (p), A)) ⇒ Hi+j (ΓN , A)nfor A = Z/p Z and A = Zp . For both of these rings, we have
H1 (ΓN (p), A) = g,

H0 (ΓN (p), A) = A.

Moreover, for sufficiently large N, we have
Hi (SLN (Fp ), A) = 0,
this follows from and is equivalent to Quillen’s computation which implies that theK-groups of finite fields have order prime to p. Since H2 (SLN (Z), Zp ) is trivial forp ≥ 2, we deduce that
H0 (SLN (Fp ), H2 (ΓN (p), A)) = H2 (SL2 (Fp ), g) = Fp ,where the last equality was already used in my paper. The compatibility of thespectral sequence above for different A implies that we also get an isomorphism
H0 (SLN (Fp ), H2 (ΓN (p), Zp )) = H0 (SLN (Fp ), H2 (ΓN (p), Z/pZ)).

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On the other hand, the invariant class must be an element of order p in
Z/pα Z;and hence the reduction map
Z/pα Z → Z/pZ
sends an element of order p to an element of order p, and so α = 1.
68.3. Putting things back together. Assembling all the pieces, we see that wehave proven the following:
Theorem 68.4. Let p ≥ 3, and let N be sufficiently large. Let g be the Lie algebraslN over Fp . Then, up to a finite group of order dividing 48, we have
H2 (ΓN (p), Z) ∼ Z/(p2 − 1)Z ⊕ Z/pZ ⊕ ker [ , ] ∧2 g → g .
Moreover, still with p ≥ 3, then up to a group of order dividing 16, we have thesame equality with p2 − 1 replaced by (p2 − 1)/3.

69. Review of Buzzard–Gee
Wed, 04 Mar 2015
This is a review of the paper “Slopes of Modular Forms” submitted for publication
in a Simons symposium proceedings volume (see [BG16]).
tl;dr: This paper is a nice survey article on questions concerning the slopes ofmodular forms. Buzzard has given a (very explicit) conjecture which predicts theslopes of classical modular (p-stabilized) eigenforms of level prime to p, at least
under a certain regularity hypothesis. One consequence is that, under favourable
circumstances, all the slopes are integers. The current paper describes the link
between this and related problems to the p-adic Langlands program, as well as
raising several further intriguing questions concerning the distributions of theseslopes. The paper is well written, and is a welcome addition to the literature. I
strongly recommend that this paper be accepted.
Review: Buzzard’s slope conjectures live somewhere in the world between 19thand 21st century mathematics. Suppose that one considers the space of overconvergent cusp forms of level N = 1 for p = 2. Then, using nothing more thanclassical identities between modular functions, one may prove that the smallest
eigenvalue of the compact operator U2 is at most ∥23 ∥2 = 1/8. On the other hand,
it is now a “folklore” conjecture (Conjecture 4.1.1 of the paper under review) that,
if p is odd and f is a classical modular form of level Γ0 (N ) prime to p, then theresidual representation:
ρ : GQ → GL2 (k)is irreducible locally on the decomposition group at p whenever the valuation of ap
is not an integer. This problem seems to be a deep question in the p-adic Langlands
program for GL2 (Qp ). The two cases where this is known, v(ap ) ≤ 1 and v(ap )sufficiently large, both require machinery from p-adic Hodge theory — in the former
case, one needs the full local Langlands correspondence for GL2 (Qp ).
Comments: Here are some comments given in some order that bears little
relation to the actual paper.

148

F. CALEGARI• I don’t like the table on page 6, in particular, because certain ranges ofnumbers are bunched together, the output looks a little strange. Can one
improve this in some way? Perhaps finish at 310 ? Perhaps include only
selected powers bigger than 310 ? Perhaps normalize for the length of therange?
• Corollary 5.1.2. Do you want to speculate on what happens in the reducible
case? In some sense, in Buzzard’s conjecture, one doesn’t see the fact thatthe residual representations are globally reducible or not. On the other
hand, weird stuff certainly happens for p = 2, as previously mentioned
here. What happens in the reducible case for p = 3?
• Conjecture 4.1.1 demands that k is even, but that is a consequence of thelevel being of the form Γ0 (N ) — something which is noted immediately
after the statement. So why include the condition in the statement of theconjecture? Also, perhaps it’s also worth remarking upon the case when ap
is a unit.
• The authors (in Remark 4.1.3) point to the origins of this Conjecture 4.1.1
to around 2005. However, I feel like I remember some discussion of thisconjecture in the Durham symposium of 2004. There were certainly hintsof this conjecture on «la serviette de Kisin», upon which Mark gave aheuristic local argument for why the Eigencurve was proper — although theargument was slightly dodgy in that it collapsed if the napkin was rotated
90 degrees. (Of course, Mark was proved right when Hansheng Diao and
Ruochuan Liu did indeed prove this result using local methods here, [DL16])
Also, isn’t Conjecture 4.1.1 a consequence of Buzzard’s original conjecture
as modified by Lisa Clay? Somehow it seems to me that what Remark
4.1.3 is referring to is the idea that Conjecture 4.1.1 is a consequence of apurely local conjecture, and refers to the period (2005?) when Breuil wasformulating the first versions of the p-adic Langlands program.
• For Conjecture 4.2.1, wouldn’t it make more sense to normalize the valuation in terms of the coefficient field Qp (χ), so the statement once more
becomes that ap has integral valuation?
• Why is the condition on Buzzard’s conjecture different when p = 2? (Iunderstand it has to be modified in order to have a chance of being true,
but I am asking if there is any explanation for why this is necessary.)• The authors remark (p.3) that it is not known whether there are infinitely
many Buzzard irregular primes. Here is a short argument to prove that thisis a consequence of standard conjectures of prime values of polynomials. We
start with the observation that the first Buzzard irregular prime is p = 59,
and that the offending representation:
ρQ∆ : GQ → GL2 (F59 )has exceptional image in the context of Serre and Swinnerton-Dyer (Onl-adic representations and congruences for coefficients of modular forms,
Antwerp III, [SD73]). Indeed, this particular example features prominently
in that paper. I always thought this was not an entirely random coincidence,and since it seems relevant here, I thought I would finally bother to figureout what is going on. (For the next prime, p = 79, the corresponding representation has image containing SL2 (F79 ), so it is somewhat of an accident.)
The mod-59 representation above has projective image S4 . Now supposePERSIFLAGE: MATH BLOG POSTS

149

that p ≡ 3 mod 4 is a prime such that H/Q is an S4 -extension whichis unramified away from p. Such a representation will give rise (following
an argument of Tate) to a mod-p representation of Q which is unramified
away from p. The congruence condition on p implies that it will be odd,
and hence modular, by Langlands-Tunnell. Now let us suppose, in addition,that 4 divides the ramification index ep . Under this assumption, the representation cannot be locally reducible, because the ramification index of any
power of the cyclotomic character divides p − 1 ≡ 2 mod 4. Hence, if there
are infinitely many such fields, there are infinitely many SL2 (Z)-irregularprimes. Consider the following fields studied by Darrin Doud [Dou99]:
K = Q[u]/f (u),f (u) = (u + x)4 − p∗ u,

256x3 − y 2, p∗ ̸= 1 + 4x.
27
Doud shows that K has discriminant (p∗ )3 and Galois group S4 , and it
is easy to see that the splitting field H/Q has all the required properties
needed above as long as p ≡ 3 mod 4. The formula relating tame ramification and the discriminant implies that ep (K/Q) = 4. Standard conjectures
now predict that there are infinitely many such primes of this form — if y
is odd, then p∗ ≤ 0. The first few such primes which are 3 mod 4 are(−1)(p−1)/2 p = p∗ =59, 107, 139, 283, . . .
which compares with the first few Buzzard irregular primes (taken from
Buzzard’s paper):59, 79, 107, 131, 139, 151, 173 . . .
On the other hand, an unconditional proof by these means seems out ofreach, because any modular S4 -extension unramified outside p ≡ 3 mod 4√
forces Q( −p) to have class number divisible by 3, and we don’t know if
there exist infinitely many such primes.

70. Chenevier on the Eigencurve
Fri, 24 Apr 2015
Today I wanted to mention a theorem of Chenevier about components of the
Eigencurve. Let W denote weight space (which is basically a union of discs), andlet
π:E →Wbe the Coleman–Mazur eigencurve together with its natural map to W. It will dowell to also consider the versions of the eigencurve corresponding to quaternion
algebras D/Q as well.
Theorem 70.1 (Chenevier). Suppose that
(1) E has “no holes” — that is, a family of finite slope forms over the punctured
disc extends over the missing point,(2) The “halo” of E is given by a union of finite flat components whose slope
tends to zero as x ∈ W tends to the boundary of the disc.
Then every non-ordinary component of E has infinite degree.

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F. CALEGARI

In particular, since both of these theorems are now known in many cases (properness by Hansheng Diao and Ruochuan Liu [DL16], and haloness by Ruochuan Liu,
Daqing Wan, and Liang Xiao [LWX17], at least in the definite quaternion algebracase), the conclusion is also known.
The proof is basically the following. Given a component C of finite degree, thefirst assumption implies that it actually is proper and finite. One may then consider
the norm of Up on C to the Iwasawa algebra to obtain a bounded (hence Iwasawa)function F = Norm(Up ). This function cannot have any zeros (again by properness),and hence, by the Weierstrass preparation theorem, it is a power of p times a unit.
But that implies that F has constant valuation near the boundary, which contradicts
the fact that the slopes are tending to zero (except in the ordinary case).
Naturally one may ask whether E has only finitely many components, although
this seems somewhat harder to prove.
Notes 70.2. For the general halo conjecture for GL(2), see here, [DY23].

71. 144169
Tue, 12 May 2015
The space of classical modular cuspforms of level one and weight 24 has dimension two — the smallest weight for which the dimension is not zero or one. Whatcan we say about the Hecke algebra acting on this space without computing it?
Formally, the Hecke algebra T is a rank two Z-algebra, which is either an order inthe ring of integers of a real quadratic field, or a subring of Z ⊕ Z. Let’s investigate
the completion of this algebra at various primes p.
Let’s first consider the prime p = 23. The curve X0 (23) has genus two, and thecorresponding Hecke algebra in weight two is Z[ϕ], where ϕ is the Golden Ratio.
The prime p = 23 does not split in this field, and hence modulo p there is a pair ofconjugate eigenforms with coefficients in Fp2 . Multiplying by the Hasse invariant,
we see that this eigenform also occurs at level one and weight 24 over Fp . It follows
that:
T ⊗ Z23 = W (F232 ).
√In particular, T = Q( D) for some square-free integer D ≥ 0.
Now let us consider primes p ≤ 23. Any Galois representation modulo such a
prime will occur — possibly up to twist — in lower weight. Yet all the spaces inlower weight have dimension at most one, and hence it follows that the residue fields
of T are all of the form Fp . Suppose further that 5 ≤ p ≤ 23. Then, using theta
operators, we may find two distinct eigenforms in weight 24, from which it follows
that T has two distinct residue fields of characteristic p, and so, for 5 ≤ p ≤ 23, wehave:
T ⊗ Zp = Zp ⊕ Zp .
One expects at level one that a2 (f ) always generates the Hecke field. This is stilla conjecture, but we may deduce this unconditionally in weight 24 because thedimension of the cuspforms is two, and so this follows automatically from the Sturmbound! Hence we may write:
"
√ #
√
1+ D
a+b D
∈Z
T = Z[a2 (f )], a2 (f ) =2
2

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151

where b ̸= 0. Even better, using Hatada’s Theorem — giving congruences for a2
and a3 for eigenforms of level one modulo 8 and 3 respectively — we may write√
a2 (f ) = 12(a + b D), a, b ∈ Z
where b ̸= 0. This gives an upper bound on D in light of the Deligne bound |a2 | ≤2 · 223/2 . More precisely, we obtain the bound b2 D ≤ 227 /242 , and hence thatD ≤ 233017.
Let’s now think more carefully about p = 2 and 3. For these primes, there will bea unique Coleman family of slope v(−24) = 3 for p = 2 and v(252) = 2 for p = 3.
I can’t quite see a pure thought way of proving this, but at least this would be aconsequence of the strong form of the GM-conjecture as predicted by Buzzard. So
we should expect that, in these cases
T ⊗ Zp ,→ Zp ⊕ Zp .
In addition to congruences for small primes, there will also be congruences between
the unique cusp form with an Eisenstein series modulo the numerator of B24 , whichis−1
× 103 × 2294797.
B24 =
2 · 3 · 5 · 7 · 13
I claim that these primes will also have to split in T. For example, it is impossible
for b to be divisible by 2294797, because that would violate the inequality on b2 D
above, and hence it follows that p = 2294797 must also split in T ⊗ Q. The same
argument works for p = 103 having ruled out some very small D. To summarize,
we have the following:√
The primes 5 ≤ p ≤ 23, p = 103, 2294797 split in K = Q( D), but p = 23 does
not split, and D ≤ 233017. Moreover, we expect that 2 and 3 also split.
This is enough to determine D completely up to 72 possibilities, and 9 with theunproven assumption at 2 and 3. On the other hand, all of these D are quite large(the smallest are 3251 and 15791 respectively), which forces b to be very small. But
we also have the congruence√
12(a + b D) ≡ 1 + 223 mod 2294797.
For the remaining D, we can determine, with |b| satisfying the required inequality,
whether there exists such a congruence with |a| ≤ 227/2 /24 ∼ 483. A simple checkshows that is a unique solution (with the assumption on two or three or not), andhence, by (something close to) pure thought, we have shown that D = 144169, andmoreover (using Deligne’s bound again) that√√a2 (f ) = 12(45 ± 144169),
T = Z[12 144169].
One can indeed check this is the case directly, if you like. Curiously enough, thisHecke eigenvalue is quite close to the Deligne bound — the probability it is (inabsolute value) this big is, assuming a Sato–Tate distribution, slightly under 5%.
Extra Credit Problem: Hack Ken Ribet’s Yelp password by using the factthat 144169 is his favorite prime number.

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F. CALEGARI

72. Counting solutions to ap = λFri, 15 May 2015
We know that the eigenvalue of T2 on ∆ is 24. Are there any other level one cusp
forms with the same Hecke eigenvalue? Maeda’s conjecture in its strongest form
certainly implies that there does not. But what can one prove along these lines?
Conjecturally, one would certainly predict the following:
Conjecture 72.1. Fix a tame level N prime to p. If λ ̸= 0, there are finitely many
eigenforms of level N an arbitrary weight such that ap = λ. If λ = 0, there are
finitely many eigenforms with the additional condition that they do not have CM bya quadratic field in which p is inert.
I have no idea how to prove this conjecture. If one counts the number of suchforms of weight ≤ X, then the trivial bound for eigenforms with ap = λ is O(X 2 ).
When I visited Princeton a few weeks ago, Naser Sardari, a student of Sarnak,
showed me a short preprint he is writing which improves this bound by a powersaving (additionally, it gives a power saving for each individual weight as well). The
most interesting case of this result is when λ = 0, but today I want to talk about
the much easier case when λ ̸= 0, where, via some p-adic tricks, one can obtain asubstantial improvement on the trivial bound. Let’s start from the following:
Proposition 72.2. Let Sλ (X) denote the number of cuspforms of level N andweights ≤ X such that ap = λ. Assume that λ ̸= 0. ThenSλ (X) = O(X).
Proof. Since λ ̸= 0, the p-adic valuation of λ is finite. However, all forms withbounded slope belong to one of finitely many Coleman families, so the number ofsuch forms in any weight is bounded. Using Wan’s explicit results, one can evengive an explicit bound here that depends only on N, p, and the valuation of λ. □The point of this post, however, is to give an improvement on this bound.
Theorem 72.3. Let Sλ (X) denote the number of cuspforms of level N and weight≤ X such that ap = λ. Assume that λ ̸= 0. Then, as X → ∞,Sλ (X) ≪λ log log log log log log log X.
The argument will (obviously) allow for an arbitrary number of logs. But thenthe statement would become more cumbersome.
Proof. As in the proof of the previous result, we may reduce to the case where
we are considering a single Coleman family F. Over this family, the function Up
is continuous, and hence so is Up (Up − λ). More importantly, over a small enoughdisc, it is an Iwasawa function. Let Σ denote an infinite set of integral weight suchthat, for the relevant points of F, we have Tp = λ, orUp (Up − λ) = −pk−1 .
If s is a limit point of Σ, then certainly Up (Up −λ) will vanish at s. Since this function
is a non-zero bounded function on a disc, it has only finitely many zeros, and so
the set of weights Σ will have only finitely many limit points. Thus, we may reduceto the case where the set of weights has a single limit point. In particular, if Sλ (X)is not bounded, we may imagine that the set Σ consists of a sequence of integers

PERSIFLAGE: MATH BLOG POSTS

153(which we may assume to be increasing in the Archimedean norm): k0 , k1 , k2 , . . .
which converge p-adically to k∞ , and, at the relevant point of F, correspond to an
eigenform which satisfies the equation
Up (Up − λ)(ki ) = −pki −1 .
Around a zero η, any Iwasawa function F has an asymptotic expansion of the form
F (η + ϵ) ≃ A · ϵm + . . .
where the LHS has the same valuation as the leading term of the RHS for sufficiently
small ϵ. If F = Up (Up −λ) and η = k∞ , we deduce that, for sufficiently large integers
ki in our list,−pkn −1 = F (kn ) = F (k∞ + (kn − k∞ )) = A(k − k∞ )m + . . . ,and so, taking p-adic valuations,kn − 1 = v(A) + m · v(kn − k∞ ),where m > 0 is constant. This certainly implies that v(kn − kn+1 ) = v(kn − k∞ )by the triangle inequality, and sov(kn − kn+1 ) = akn + b,
for constants a, b with a > 0. But two integers whose valuation are very close arevery far apart, and indeed we deduce thatkn+1 − kn ≥ Cpakn
for some a > 0 and some constant C > 0. This iterated exponential growth proves
the result.
□
The argument also shows that if the set Σ is infinite, the limit roots of Up −λ = 0
will be transcendental Liouville numbers, which seems unlikely. The result also
applies if one replaces λ by a sufficiently continuous function without zeros, say
a2 = 24(1 + 2(k − 12)2 ). On the other hand, I don’t think these analytic methods
will ever be enough to prove the conjectural bound, which is O(1).
Notes 72.4. I still think that for λ ̸= 0 this is the best known bound. But the λ = 0case now has a non-effective but asymptotically optimal result by [CTS21].

73. Hilbert modular forms of partial weight one, Part III
Sat, 17 Oct 2015
My student Richard Moy is graduating! Richard’s work has already been discussed in § 3 and § 8 on this blog before, where we discussed his joint work with
Joel Specter showing that there existed non-CM Hilbert modular forms of partial
weight one. Today I want to discuss a sequel of sorts to that paper, which also
forms part of Richard’s thesis (I should note that he already has five publications
and will have 7 or 8 papers by the time he graduates.) The starting observation is
as follows. Fix a real quadratic field F . From the perspective of Galois representations, the Hilbert modular forms of partial weight one fall under the case ℓ0 = 1 in
the notation of my paper with David Geraghty (this is in the context of coherent
cohomology). To orient the reader, let us discuss three classes of such forms:(1) Hilbert modular forms of weight [2k + 1, 1] for a real quadratic field F.

154

F. CALEGARI(2) Regular algebraic cuspidal automorphic forms for GL(3)/Q.
(3) Regular algebraic cuspidal automorphic forms for GL(2)/F for an imaginary quadratic field F.
Suppose one fixes a tame level N and then looks at the space of such forms as
the weights vary. In both of the latter cases, the problem has been raised (or evenconjectured, for N = 1 and GL(3) by Ash and Pollack here [AP08]), of whether
all but finitely many such forms arise via functoriality from a smaller group. Moreexplicitly, one can ask whether:(1) If G = GL(2)/F, then all but finitely many cuspidal regular algebraic formsof conductor N either arise (up to twist) via base change from GL(2)/Q,
or are induced from a quadratic CM extension E/F.
(2) If G = GL(3)/Q, then all but finitely many cuspidal regular algebraic formsof conductor N arise up to twist as the symmetric square of a form from
GL(2)/Q.
Naturally enough, one can make the same conjecture whenever ℓ0 > 0, appropriately formulated. There does not seem to be any case of this conjecture which is
known, although there are analogous results (where one fixes the weight and varies
the level) in both weight one (where it is almost trivial) and for imaginary quadratic
fields (in the work of Calegari–Dunfield [CD06] and Boston–Ellenberg [BE06]). Still,
the conjectures in varying weight seem pretty hard even for N = 1. In that context, Richard
proves the following nice complementary pair of theorems below. Let√
F = Q( 7). The field F has narrow class number 2 and there is a unique√ oddeverywhere unramified quadratic character χ of GF with fixed field E = F ( −1).
Theorem 73.1 (Moy). Let F and χ be as above. Every Hilbert modular form over
F of weight [2k + 1, 1] and level N = 1 is CM, and in particular is induced from E.
Theorem 73.2 (Moy). Let F and χ be as above. Let M be a strongly compatible
family of two dimensional Galois representations of F with determinant χ, level
N = 1, and Hodge–Tate weights [0, 0] and [k, −k]. Then M is induced from E.
The first theorem is an almost immediate corollary of the second, with the caveatthat one doesn’t quite have complete local-global compatibility for partial weight
one modular forms (though results and methods of Luu, Jorza, and Newton get
close). Theorem 73.2 on the other hand is a consequence of the following:
Theorem 73.3 (Moy). Let F and χ be as above. Letρ : GF → GL2 (Q3 )be a continuous irreducible representation with determinant χ that is unramified at
all finite places except for one prime v|3. Then ρ is induced from a character ofGE .
The argument in this case is (roughly) the following. Using a Tate-style argument(with discriminant bounds), one proves that the residual representation ρ must havesemi-simplification χ ⊕ 1. The restriction of ρ to GE then has the property that its
image is pro-3 and unramified outside the fixed prime v|3. Yet one shows by a classfield theory computation that the largest abelian 3-extension unramified outside
v|3 is cyclic, which (by consideration of the Frattini quotient) immediately implies
that the image of ρ restricted to GE factors through a cyclic quotient as well, andone is done.

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155

Note that to deduce Theorem 73.1, one first has to prove (using a congruenceargument) that at the other prime w|3, either:(1) The representation ρ is unramified at w,(2) The representation ρ restricted to Dw has unramified semi-simplification.
In particular, the generalized eigenvalues of Frobw for ρ are both the same.
To finish, one rules out the second possibility by computing all the modular
residual representations explicitly by doing computations in low weight (this canultimately be reduced to a computation on the definite quaternion side, although
Richard had to write his own programs to do this since the current magma implementation required trivial character for non-parallel weight.)
It is true that these arguments will not suffice for the more general conjecture,
but then, I haven’t seen a viable strategy to prove those conjectures either!

74. Ventotene, Part II
Fri, 18 Mar 2016
I promised to return to a more mathematical summary of the conference in
Ventotene, and indeed I shall do so in the next two posts.
One of the themes of the conference was bounding the order of the torsion
subgroup in arithmetic lattices. Tsachik Gelander gave a number of talks (in part)on the seven author paper. One nice result was a uniform bound of the shape
log |H ∗ (Γ, Z)tors | ≤ C · Vol(Γ),
where Γ ranges (say) over all lattices in SLn (R) for a fixed n ≥ 3. (The key result
here is the uniformity — this result is much easier for covers of a fixed manifold.)
Two natural questions that came up (in conversation at least) during the conference
are as follows:(1) Can one do better in low degree?
(2) What is the true expectation for the size of this group for (say) congruencesubgroups of SLn (Z)?
Let’s consider the first question. For (congruence) subgroups of SLn (Z), one
can certainly say quite a bit more. For example, H 1 is essentially trivial, by the
congruence subgroup property. However, in the stable range of cohomology (inparticular, when the completed cohomology groups become stable), the groups H ∗are finite over Zp , and so contribute very little. One does, at least, have the following
soft arguments for general groups.
Proposition 74.1. Let G be a semi-simple group over Q with Q-rank r = rQ .
e i is a torsion Λ = Zp [[G(Zp )]]-module for i ≤ rQ .
Then H
Proof. The proof is as follows: the boundary terms are also torsion, so it suffices to
e BM in the appropriate range are also torsion, where we consider
show that all the HiBorel-Moore homology. Assume otherwise. Let dim G(R)/K(R) = d. From thee BM , Λ) ⇒ H
e d−i−j , we deduce that there is at least one
spectral sequence Exti (Hjei ≤ rQ such that Hd−i ̸= 0. Yet the homological dimension of Γ\G(R)/K(R) is, byBorel-Serre, equal to d − rQ , and so all the homology in these degrees (and hence
certainly the completed homology) vanishes.
□

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F. CALEGARI

One can do better in certain algebraic cases, where one can deduce vanishingof the completed cohomology in certain degrees by perfectoid technology (as inCorollary 4.2.3 of [Sch15b]).
The answer to the second question, even conjecturally, is more mysterious. There
are some speculations related to this question in Bergeron–Venkatesh. But it seems
a little tricky to formulate a precise guess (for a good upper bound).

75. Tensor products
Tue, 29 Mar 2016
Let W be an irreducible representation of a finite group G. Say that W is tensor
indecomposable if any isomorphism W = U ⊗ V implies that either U or V is acharacter. In conversations with Matt and Toby (which permeate the rest of thispost as well), the following problem came up:
Problem 75.1. Let G be a finite group. Let V be an irreducible representation of
G. Is there a unique decomposition
V = V1 ⊗ V2 . . . ⊗ Vk
of V as a tensor product of tensor indecomposable representations (up to re-orderingand twist)? I don’t think this can be too hard, but I confess I don’t see how to doit. (Since we didn’t really need this, we didn’t think about it too hard.)(edit: when I say I don’t think this can be too hard, I don’t mean to imply that
I think it is true; just that I think either a proof or counterexample should not be
too hard to find — hopefully not both.)
One can ask an analogous problem for Lie groups. Actually, the problem for Lie
algebras is actually quite simple (and the answer is positive). It is related to thefollowing:
Lemma 75.2. Let V and W be irreducible non-trivial representations of a simple
Lie group g. Then V ⊗ W is reducible.
Proof. Assume that V ⊗ W is irreducible. In particular, it is determined by its
highest weight. Let the highest weights of V and W be λ and µ respectively. Thenthe highest weight of V ⊗W must be λ+µ. But now, by the Weyl character formula,
we deduce thatdim(V ) dim(W ) Y ⟨ρ, α⟩⟨ρ + λ + µ, α⟩1==.
dim(V ⊗ W )⟨ρ + λ, α⟩⟨ρ + µ, α⟩+ΦThe product term can also be written as:⟨λ, α⟩⟨µ, α⟩1+.
⟨ρ + λ, α⟩⟨ρ + µ, α⟩
In particular, since the pairing is non-negative between positive roots and highest
weights, we deduce a contradiction unless⟨λ, α⟩⟨µ, α⟩ = 0+

for all α ∈ Φ . The assumption that g is irreducible, however, is equivalent tosaying that Φ+ has a maximal root β, and for such a maximal root, we have⟨λ, β⟩ = 0 ⇒ ⟨λ, α⟩ = 0, ∀α ∈ Φ+ ⇔ λ = 0.

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157□
Note that this lemma is actually a special case of a theorem of Rajan [Raj04],
who proved that, for simple g, the factors of a (not necessarily irreducible) tensor
product are determined by the representation. In particular, the tensor product of
two non-trivial irreducible representations cannot be irreducible.
The problem with the initial question is that it’s hard to construct tensor products of irreducible representations which are irreducible. Or rather, it is easy, simply
by taking U ⊗V where U is an irreducible representation of H and V is an irreducible
representation of G and the tensor is irreducible for H × G. Yet the interesting caseis something closer to assuming that U and V are faithful. Actually, this motivates
the following question:
Problem 75.3. Do there exist irreducible non-trivial representations U and V ofa finite simple non-abelian group G such that U ⊗ V is irreducible?
The argument above for Lie groups suggests that this may not happen for Chevelley groups (although it certainly doesn’t prove this). It also suggests (relating
the representation theory of An and Sn to GLn ) that it doesn’t happen for thealternating groups either. It almost surely doesn’t happen for the sporadic groups
either. So my guess that the answer to the problem above is no, and that this isprobably known, and probably requires classification. (Please comment if you know
the answer.) Actually, this also reminds me of a similar problem which I think is
open.
Problem 75.4. Fix N . Does there exist a non-trivial representation V of a finitegroup G of dimension N such that Hom0 (V, V ) (of dimension N 2 −1) is irreducible?
This question came up in my paper with Barry, where I was surprised to findvery few examples. I seem to remember that the Mathieu group M12 has an 11dimensional representation whose corresponding 120 dimensional adjoint is irreducible. Can one classify all such examples coming from simple groups?
Comment 75.5 (Persiflage). I’m was confused why the list doesn’t contain M12 ,and then checked that it is false for M12 , and I must have read the character tablewrong, oops! For those playing at home, Katz’ paper can be found here (see [Kat04]).

76. Report from Berkeley
Fri, 22 Apr 2016
My recent trip to Berkeley did not result in a chance to test whether the Cheeseboard pizza maintained its ranking, but did give me the opportunity to attend thelatest Bay Area Number Theory and Algebraic Geometry day, on a (somewhat disappointingly) rainy Saturday in Evans Hall. The weather was somewhat better on
Sunday, however, allowing myself to make the trip to Mint Plaza for the following
cup, which should bear some resemblance to the banner picture on this site. (Unfortunately, they were no longer serving their mini-Brioche buns.) But now on to
the good stuff, a report on some of the talks:
Jaclyn Lang gave a talk on her work concerning the image of big Galois representations (see [Lan16]). The setup is roughly as follows. Letρ : GQ → GL2 (Fp )158

F. CALEGARI

be an absolutely irreducible odd Galois representation over a finite field (hence
modular). Suppose this Galois representation was the residual representation associated to an ordinary modular form that lived inside a Hida family that was smoothover weight space. Then one might expect that the corresponding representation
ρ : GQ → GL2 (Zp [[T ]])to have image which was as big as possible, namely, containing SL2 (Zp [[T ]]). This
can’t always be the case, however; for example, the residual representation (or theentire family) could be dihedral. However, if the residual representation contains
SL2 (Fp ), and one additionally assumes that the image of inertia at p is sufficiently
large, then this is indeed the case (probably this assumes that the residue characteristic is at least 5). There have been a number of generalizations of this result due
to Hida and others which improves the result by weaken the various hypotheses; forexample, allowing coefficients, allowing the residual representation to be dihedral,and weakening the ramification hypothesis at p. For these results, one can’t expectthat the image is full, but rather that the image contains an appropriate congruencesubgroup of SL2 . I like to think of this as follows: at classical specializations, one
knows that the image (if it is not CM or weight one) will have open image; the results of this talk and of previous work show that index can be controlled in families.
Actually, this is not quite true, because another obstruction to having open image
even classically is the existence of inner twists. The main result of the talk was todeal with this issue of inner twists, and hence also allow for a generalization of theresults not only to smooth Hida families but to any irreducible component of any
Hida family. (More details to be found here.)
A natural question: one output of Lang’s result is to give an ideal b of the Hida
family for which the image of these Galois representations contains the b-congruencesubgroup (after accounting for inner twists). In characteristic zero, my impression
from the talk was that one can identify the support of this ideal as coming fromCM points and classical weight one modular forms. On the other hand, apparently
there is also a version of this result in the reducible case (due to Hida and withextra hypotheses); in that case the zeros should correspond to the reducible locus,
or equivalently, the zeroes of the p-adic zeta function. However, a stronger result istrue, namely, that b can essentially be identified with this p-adic zeta function. So,
returning back to the residually irreducible case, the natural question is: can the
support of b contain the prime p?
Kęstutis Česnavičius gave a talk on the Manin–Stevens and Manin constantsfor elliptic curves, with emphasis on the prime p = 2. He raised the following
question: Suppose that N is odd. Is there a surjection from the space of weight
2 classical modular cusp forms of level Γ0 (N ) with coefficients in Z2 to the spaceof weight 2 Katz cusp forms of the same level over F2 ? The issue here is thatthe latter space is really the cohomology of the associated stack, not the course
moduli space. Unfortunately, this question distracted me a little as I tried to finda counter-example (I failed). A result of Serre and Carayol basically implies thatthe result can only fail after localizing at a non-Eisenstein maximal ideal m √of theQ(−1).
Hecke algebra T if the corresponding representation ρm is induced from√
(Analogously, for p = 3, when the representation is induced from Q( −3).) Thisis related to the classic failure of the first version of Serre’s conjecture for p = 3 atlevel Γ1 (13). However, as Serre quickly realized, this failure ultimately comes from a
failure to lift mod-p forms as above, except in this case from the intermediate curvePERSIFLAGE: MATH BLOG POSTS

159

XH (13), not from X0 (13). I ultimately convinced myself that lifting was always
possible unless m was not only Eisenstein but also the ideal containing Tp for all
odd p not dividing N. I think this must be related to Ken’s result on component
groups of Neron models, and how the non-Eisenstein parts arise for XH (N ) but not
for X0 (N ) or X1 (N ). (More details here and [Č18])
The final speaker of the day was Daquin Wan. The key question that arose in
his talk was the following. Suppose that D(k, T ) is the characteristic power seriesof the U operator on the space of overconvergent p-adic modular forms in integral
weight k. Can one show thatD(k + 2, T )L(k, T ) =D(k, pT )has infinitely many zeros and infinitely many poles? One actually has to assume
that k ̸= 0 here, since otherwise the result is false, as this will be a polynomial ofdimension the space of weight two forms. One feels that p-adic Langlands shouldbe able to say enough about slopes in these weights to obtain a contradiction, but
I don’t unfortunately see how to do it. The main point of the talk was two-fold.
There is an argument of Coleman that shows that D(k, T ) is not itself a polynomial.
This argument can be generalized to prove that L(k, T ) is not a rational function.
Second, the product L(k, T )L(−k, pk T ) is actually a rational function because of
the properties of the theta operator. So one deduces that at least one of thesefunctions had infinitely many poles and the other had infinitely many zeroes. This
also relies on a previous result of Wan that these functions are meromorphic. (Oh,I should mention that this was joint work with . . . and here I didn’t take notes fora talk two weeks ago . . . Liang Xiao? Please correct me if I’m wrong)(Romyar Sharifi also talked, but since I am actively trying to understand something about that talk on a more technical level, so I will have to return to a discussion of it later.)

77. Central extensions and weight one forms
Mon, 23 May 2016
As mentioned in the comments to a previous post (not on math), Kevin Buzzard
and Alan Lauder have made an extensive computation of weight one modular forms
in characteristic zero (see also [BL17]). Thinking about what that data might contain, I wondered about the following question: what are the images of the Galois
representations associated to the weight one forms of type A5 ?
Let us take a step back. Consider a projective representation
ψ : GQ → PGL2 (C)with image A5 , and assume that it is odd. (That is, complex conjugation has order2.) According to Tate, there exists a lift
ρ : GQ → GL2 (C).
This lift is unique up to twisting. Since the Schur multiplier H2 (A5 , Z) of A5 is
Z/2Z, there is a unique minimal lift up to twist whose image is a central extension
e5 by a cyclic group ∆ of 2-power order. Note that ∆ is not trivial, since A5 doesA
not have any two-dimensional representations. If |∆| = 2, then the determinant of
e5 is trivial, which contradicts
the corresponding 2-dimensional representation of A

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the assumption that ψ is odd. (Equivalently, there is an obstruction at ∞ to lifting
to the central extension by Z/2Z.) Hence 4 divides |∆|. What is the expecteddistribution of ∆ as one runs over all odd A5 -extensions?
My first guess (without any prior thought or computation) was that this mightobey some form of Cohen–Lenstra heuristic, suitably interpreted.
Note that the image of the determinant has order |∆|/2. The corresponding
determinant representation is a character of Q of 2-power order. Since Q has trivial
class number, the order |∆|/2 is equal to the maximal ramification degree ep of thisrepresentation over all primes p.
Over Q, Tate’s lifting theorem has the following stronger form: one may choosea lift ρp of ψp := ψ|Dp and insist that ρ|Ip = ρp |Ip ; that is, they agree on inertia.
This is essentially a consequence of the fact that Q has trivial class group. For
convenience, suppose that ψ is unramified at 2 and 3. Suppose that ψ is ramified
at p. There are three possibilities:(1) The image of ψp at a ramified prime p is cyclic of order 2, 3, or 5.
(2) The image of ψp at a ramified prime p is D6 or D10 .
(3) The image of ψp at a ramified prime p is Z/2Z ⊕ Z/2Z.
For a fixed p, let ϵ denote the Teichmuller lift of the mod-p cyclotomic character.
(Fix an isomorphism of C with Qp for all p.)
Let us consider the three cases in turn.
In the first case, the image factors through a cyclic quotient. One may thus
take ρp to be a direct sum which, on inertia, has the shape χ ⊕ 1 up to twist. By
comparing this to the projective representation, we see that χ has order 2, 3, or 5,
and so, after finding the twist such that the determinant has 2-power order, we seethat ep = 1 or ep = 2.
In the second case, the lift on inertia is (up to twist) of the form:ω2r ⊕ ω2prfor some r. Since the order of ω2 is p2 − 1, the order of the ratio is
p+1.
gcd(r, p + 1)
which must be equal to 3 or 5. It follows that r is even. Yet the determinant isequal to(p+1)rω2= ϵr ,

Since r is even, we see that, after twisting, we may take ep = 1.
Finally, in the third case, the lift is of the form:ω2r ⊕ ω2prfor some r. We now find that the order of the ratio of these characters is
p+1.
gcd(r, p + 1)
which must be equal to 2, and the determinant is ϵr . If r is even, then, as above, we
may twist so that ep = 1. Hence, the only way that the image after minimal twist
does not have |∆| = 4 is if we are in this third situation with p ≡ 1 mod 4, withr odd, and then (after twisting) we find that ep is the largest power of 2 dividingp − 1.

PERSIFLAGE: MATH BLOG POSTS

161(I confess that I originally forgot the fact that the third possibility could occur,and was only after noticing that this seemed to imply the inverse Galois problem
was false thought a little bit more about the possibilities.)
To summarize:
Lemma 77.1. Assume that ψ is unramified at 2 and 3 and has projective image
e5 with minimal kernel. Then order of ∆ is 4 unless there
A5 , and a lift with image A
exists a prime p ≡ 1 mod 4 such that the image of the decomposition group at punder ψ is Z/2Z ⊕ Z/2Z. In this case, we have ∆ to be twice the largest power of
2 dividing p − 1 for all such primes p.
Let ∆(ψ) denote the corresponding power of 2.
We see that ∆(ψ) is determined by purely local phenomena. This still doesn’t
e5 will be. However, I imagine
quite answer what the distribution of the extension A
that Bhargava style heuristics should certainly be able to predict the ratio of A5
with ∆(ψ) = 2n . Does anyone have a sense of how easy this might be to prove?
Or, much more modestly, how easy it would be to compute from these quite precise
heuristics the exact predicted distribution of the central extensions of A5 comingfrom weight one modular forms?
(I confess, it is not even obvious to me from this construction how to prove thate5 occur as Galois groups — but I presume this is known,
all central extensions A
and hopefully one of my readers can provide a reference.)(According to Kevin, BTW, all the A5 representations with N ≤ 1500 have
∆ = 4.)

78. Prime divisors of polynomialsSun, 29 May 2016
A heuristic model from the last post § 77 suggests that the “expected” orderof the Galois group associated to a weight one modular form of projective type
A5 is infinite. And when one tries to solve the inverse Galois problem for central
extensions of this group, one is lead to problems concerning the prime divisors ofpolynomials and their properties modulo 2. But I don’t know how to answer thistype of problems! Here is an analogous question that seems a little tricky to me:
Question 78.1. Show that there are infinitely many integers n such that all the
odd prime divisors of n2 + 1 are of the form 5 mod 8.
To make the problem slightly easier, one can ask:
Question 78.2. Show there exists an integer m and infinitely many integers nsuch such that all the odd prime divisors of n2 + 1 are of the form ̸≡ 1 mod 2m .
Is this an open problem?
Comment 78.3 (Ben Green). Frank, I believe this is a half-dimensional sieve
problem and so one expects an answer of the form: the number of n ≤ X for whichn2 + 1 has all its odd prime factors 5 (mod 8) is asymptotic to cX(log X)−1/2 forsome c, which you can specify explicitly. Details of the half-dimensional sieve canbe found in Friedlander and Iwaniec, Opera Cribro [FI10], which I don’t have to
hand.

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The reason this is a half-dimensional sieve problem is that you can locate suchn by “sieving out” the conditions n ≡ ±ap (mod p) for all p ≡ 3 (mod 8), where
ap is one of the roots of a2p ≡ −1 (mod p). So that’s two congruences classes mod
p for 41 of the primes, i.e. on average you’re sieving by half a residue class for eachprime.
Another example of a half-dimensional sieve problem is counting sums-of-twosquares, where now you sieve by the condition n ≡ 0 (mod p) for all p√≡ 3 (mod 4).
To find such n up to X, you only need to sieve by those p with p ≤ X.
Your problem is more difficult because you need to sieve by all p up to about
X, which is very large. I’d have to look at the details of the half-dimensional sieve
to figure out whether this is fatal in terms of finding infinitely many n with yourproperty — or, better, ask an expert on the subject.
Comment 78.4 (Sound). As Ben says, this is a semi-linear sieve problem, and thebest reference for that is probably the book of Friedlander and Iwaniec (chapter
14). They explain that the semi-linear sieve is capable of producing asymptotic
formulas (as in the “sums of two squares” problem), under two basic conditions:(1) A parity restriction: one must sieve integers in a sequence A by primes in a
set P , with the property that elements of A have an even number of divisorsfrom P — this holds for sums of two squares if A is taken to be integers= 1 mod 4 and P the primes which are 3 mod 4, but not in your case if one
starts with A ={values of n2 + 1 } and P = {primes = 1 mod 8} [which arethose to remove, Ben has a typo with a 3 instead of 1] . (See [FI10, 14.4]).
(2) a level of distribution condition: the sum of error terms in the sieve, up to
the bound on p that you impose, must be under control.
Condition (1) seems very intrinsic to the method (e.g., I don’t see how to get alower bound with the sieve for the integers where all prime factors are 3 mod 4).
Concretely, if instead of your problem one wants to have n2 + 1 only be divisible
by primes that are 1 mod 8, one could take A = { n2 + 1 with n divisible by 4}
2
and P = { primes = 5 mod 8}; the parity condition is then true (since√ n + 1 willbe 1 mod 8). Moreover, you only need to sieve n ≤ x by primes up to x; the levelof distribution of A is just a bit smaller (by a log or so), so there’s a chance that
it works by applying the semilinear sieve up to x1/2 /(log x), and then separately
counting the integers that remain unsieved but are not of the type you want, which
are very restricted, and should be fewer in number I think.
Notes 78.5. See this paper (also [BS21]) for a solution. See also § 84

79. Serre 1: Calegari 0Tue, 18 Oct 2016
I just spent a week or so trying to determine whether Serre’s conjecture about
the congruence subgroup property was false for a very specific class of S-arithmetic
groups. The punch line, perhaps not surprisingly, was that I had made an error.
I should note that I was pretty skeptical during the entire endeavor, so the final
resolution was not a surprise, but there were still a few interesting twists along theway. (Thanks to Matt for some informative chats along the way.)PERSIFLAGE: MATH BLOG POSTS

163

Let’s start by recalling Ribet’s proof of (what is one of many statements knownas) Ihara’s lemma. Let Γ be a congruence subgroup of SL2 (Z) of level prime to q.
There is a congruence subgroup Γ0 (q) defined in the usual way, where q|c. However,there is also a second copy Γ0 (q) of this group inside Γ with q|b. (Well, there are
q + 1 copies of this group, but let’s just consider these two for the moment.) The
two groups are conjugate inside GL2 (Q), but not inside Γ. An argument of Serre
now shows that the amalgam of Γ with itself along these groups (identified by
conjugation by [q, 0; 0, 1]) is the congruence subgroup Γ[1/q] of SL2 (Z[1/q]). Thatis, the congruence subgroup where the local conditions away from q are the sameas Γ. The Lyndon long exact sequence associated to an amalgam of groups showsthat there is an exact sequence:
H1 (Γ0 (q), A) → H1 (Γ, A)2 → H1 (Γ[1/q], A) → 0,
for any trivial coefficient system A. Now the group Γ[1/q] satisfies the congruencesubgroup property, so the group on the right is easily seem to be finite and Eisenstein. By duality, there is also a map
H1 (Γ, A)2 → H1 (Γ0 (q), A),
and the composition of this map with the projection above is a matrix with determinant Tq2 − (1 + q)2 . A bookkeeping argument now gives Ribet’s famous levelraising theorem (taking coefficients A = Fp .)Fred Diamond and Richard Taylor [DT94] generalized this theorem by replacing
the modular curve with both definite and indefinite quaternion algebras. The actual
theorem itself at this point is probably quite easily to prove by the K-W method,
but that’s not relevant here. Instead, let’s think a little about the proof. The moredifficult and interesting case is when Γ comes from the norm one units in an indefinite quaternion algebra, which we consider from now on (the case of Shimura curves
over Q.) Morally, the proof should be exactly the same. The only wrinkle is thatthe corresponding group Γ[1/q] is notoriously not known to satisfy the congruencesubgroup property, although Serre conjectures that it does. Diamond and Taylor
instead argued in the following way. (Let us specialize to the case of trivial weight,
which is the only relevant case here.) Suppose that p is a prime greater than two anddifferent from q. Then instead of working with Betti cohomology, one can instead,
via a comparison theorem, use de Rham cohomology. The Hodge filtration consistsof two pieces, one of which is H 0 (X, Ω1 ), and the other is H 1 (X, OX ). They theninvestigate the kernel of the map:
H 1 (X, Ω1 )2 → H 1 (X0 (q), Ω1 )where everything is now over Fp . Here the two maps are the two pullbacks under
the two projections X0 (q) → X. They now show that element in the kernel gives
rise to a differential ω which vanishes at all the supersingular points or does notvanish at all. The first is impossible by a degree argument when p ≥ 3, and thesecond is always impossible. They conclude that, returning to etale cohomology,
any kernel of the map
H 1 (X, Fp )2 → H 1 (X0 (q), Fp )must lie entirely in one filtered piece, from which they deduce it must be Eisenstein.
But let’s look at this argument a little more closely. Even in Ribet’s case, theconclusion is really much stronger than level raising for non-Eisenstein primes; there
is a very precise description of the kernel (or cokernel in homology) in terms of the

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homology of Γ coming from congruence quotients, which one can compute quite
explicitly. So Ribet’s theorem also gives level raising for Eisenstein representations
in some contexts. In particular, for a suitable choice of congruence subgroup (withp ≥ 2) one can make the group H1 (Γ[1/q], Fp ) vanish identically. Let’s now returnto the argument of Diamond and Taylor when p = 3. All the comparison theorems
are still valid, so the only issue is that the map
H 1 (X, Ω1 )2 → H 1 (X0 (q), Ω1 )does have a kernel, namely, if one takes the “Hasse Invariant” A which vanishes
to degree one at all supersingular points, then the two pullbacks of A to X0 (q)coincide up to a scalar, and so the kernel is at least one dimensional. In fact, theargument of Diamond-Taylor shows that the kernel is at most one dimensional. But
what does this mean in the proof of Ihara’s Lemma? It means that, assuming X
has good reduction at the prime p = 3, the level raising map always has a kernel,
and thus H1 (Γ[1/q], F3 ) is always non-trivial.
This now seems suspicious: all we need to do is find a quaternion algebra which
doesn’t have any congruence homology of degree 3. If the quaternion algebra D/Q
is ramified at a prime r, then the congruence homology coming from this prime (forp ̸= 2r) is a subgroup of the norm one elements of F×r 2 , which has order r + 1. So it
makes sense to take a quaternion algebra ramified at 7 · 13, since these are the two
smallest primes different from 3 which are congruent to 1.
Because this seemed to contradict Serre’s conjecture, I decided for fun to explicitly compute a presentation for the amalgam Γ[1/2] to help work out what wasgoing on. To first start, one needs a presentation for Γ. John Voight (friend of theblog) has written a very nice magma package to do exactly this. (More precisely,
it’s trivial to write down a presentation — Γ is torsion free, and hence a surface
group π1 (Σg ) for a genus g one can compute via other means to be g = 7; the pointis that one also wants an explicit representation as well as an explicit identification
with the norm one units of the corresponding quaternion algebra.)
I then took an embarrassingly long time to compute the subgroup Γ0 (2). The
main issue was finding a suitable element in D to play the role of η = [2, 0; 0, 1] inM2 (Q). There certainly exists such a unit in D ⊗ Q2 , so in real life one just has tofind an actual norm 2 unit which is sufficiently close 2-adically to this. However, I
am absolute rubbish at mathematica and so repeatedly made the following error:
when you define suitable quaternions i, j, k in D ⊗Q E for some quadratic splitting field E/Q, and then compute with the matrix a + bi + cj + dk, mathematica
helpfully interprets “a” here as [a, a; a, a] rather than a multiple of the identity, aprogramming decision which makes a lot of sense, said no one ever. I did this moretimes than I care to admit. Then, using John’s program, one can find the subgroupΓ0 (2), and then write down a presentation for the amalgam by conjugating thissubgroup by η and identifying the corresponding elements via a solution to theword problem as words in the original generators, and then substitute the names
for these generators for the second copy of Γ. The result is a group with 14 + 14
generators and 2 + 38 relations (corresponding to the 2 surface relations and thefact that Γ0 (2) has 3(14 − 2) + 2 = 38 generators.) Finally, one takes this group,
plugs it into magma, and finds:
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There are known congruence factors coming from 7 + 1 and 13 + 1, but here one
sees that the factor of three survives!
And then, shortly after this point, I realized that SL2 (F3 ) has a quotient of order
3, because it is A4 . So that degree three quotient is congruence after all. . . Oops!
Still, it’s nice to see that mathematics is consistent.
However, at this point one might just ask why can’t one replace the quaternion
algebra D/Q by (say) a real quadratic field in which 3 is unramified and inert.
Serre got away with it above because SL2 (F3 ) is solvable, but SL2 (F9 ) has thegood manners not to have any such quotients. So why can’t one now run the same
argument as above and disprove Serre’s conjecture? That’s a good question, and theentire argument works, up to the issue of defining the Hasse invariant. Quaternion
algebras over fields other than Q are a bit of a disaster, because they don’t have nicemoduli theoretic descriptions. That doesn’t mean they don’t have Hasse invariants,
however. But now what happens, which at this point in the game I suspected but
was confirmed and explained to be by George Boxer (Keerthi also suggested acomputation which would lead to the same conclusion): the Hasse invariant is no2
longer a section of Ω1 = ω ⊗2 = ω p−1 , but rather a section of ω p −1 , and this has
1
too large a degree to contribute to the cohomology of Ω . Since 22 − 1 ≥ 2, it still
has too large a degree when p = 2, which is good, because otherwise working at
this prime could have given rise to a counter-example to Serre’s conjecture because
SL2 (F4 ) = A5 is perfect. (One would have to be slightly more careful with p =2 about comparison theorems, but at least one is dealing with curves.) So theconclusion is that Serre’s conjecture still stands, but only because various Hasse
invariants in low weight are exactly accounted for by the solvability of SL2 (F)when |F| = 2, 3.
(Also, completely randomly and apropos of nothing, this link is now the top hiton the web to the search “Fred Diamond’s Beard.”)Notes 79.1. It still is (the first hit). The youtube link in the blog no longer works,
here’s a direct link to Fred’s talk.

80. Zp -extensions of number fields, Part I
Thu, 24 Nov 2016
In the next few posts, I want to discuss a problem that came up when I wrote a
paper with Barry Mazur. We had a few observations and remarks that we discussed
as part of a possible sequel but which we never wrote up; mostly because we never
could quite prove what we wanted to prove. But some of those remarks might beworth sharing.
The basic problem is as follows. Let E/Q be a number field of signature (r, s).
Let p be a prime that splits completely in E (this is not strictly necessary, but it
makes things cleaner). Let S be a set of primes above p. If S includes all the primes
above p, then the Leopoldt Conjecture for E and p is the statement thatrS := dimQp Gal(E S /E)ab ⊗ Q = 1 + s.
The question is then to predict what happens when S is a strict subset of the primes
above p. This leads to the following minimalist definition:

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Definition 80.1. The field E is rigid at p if(
#S − (r + s − 1), #S ≥ r + s − 1,SabrS := dimQp Gal(E /E) ⊗ Q =0,otherwise.
Note that, for any field E, the right hand side is always a lower bound. So rigid
pairs (E, p) are those which have no “unexpected” Zp -extensions. If E is totallyreal, the Leopoldt Conjecture at p is equivalent to E being rigid. However, one does
not predict that all fields E are rigid. The following is elementary:
Proposition 80.2. If E is a totally imaginary CM field, then complex conjugationacts naturally on the set S. There are inequalities rS ≥ [E + : Q] + 1 if S consistsof all primes above p, and1rS ≥ #(S ∩ cS)
2otherwise. If Leopoldt’s conjecture holds, then these inequalities are equalities.
It follows that if E is a CM field of degree at least 4, then E is not rigid forany prime p, because when S consists of two primes conjugate to each other undercomplex conjugation, thenrS ≥ 1 ≥ 2 − (r + s − 1) = 2 − s.
The “extra” extensions are coming from algebraic Hecke characters. Our expectationis that this is the only reason for a pair (E, p) to be rigid. For example:
Conjecture 80.3. Suppose that E does not contain a totally imaginary CM extension F of degree at least 4. Then (E, p) is rigid for any prime p that splits completely
in E.
(When I say conjecture here, I really mean a guess; it could be false for trivial
reasons.) Naturally these conjectures are hard to prove, since they imply Leopoldt’s
Conjecture. Even if one assumes Leopoldt’s Conjecture, this conjecture still seems
tricky. It makes sense, however, to see what can be proven under further “genericity”
hypotheses on the image of the global units inside the local units. To this end, letme recall the Strong Leopoldt Conjecture which Barry and I formulated ouroriginal paper. Let F/Q be the splitting field of E/Q. Let G be the Galois groupof F/Q. There is a G-equivariant map
Y×OF× ⊗ Qp →OF,v
⊗ Qp .
v|p

The right hand side is isomorphic as a G-module to Qp [G]. However, more is true;
for a fixed prime v|p, there is an isomorphism
Qp [G] = Q[G] ⊗ Qp
which is well-defined up to a scalar in Qp coming from a choice of p-adic logarithm
for the given place at p. It makes sense to talk about a rational subspace V of theright hand side, namely, a space of the form V = VQ ⊗Qp for some VQ ⊂ Q[G]. The
strong Leopoldt conjecture (of [CM09]) asserts that the intersection of the global
units which such a rational subspace is as small as it can possibly be subject tothe constraints of the G-action on both V and the units, together with Leopoldt’s

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167

conjecture that the map from the units tensor Qp is injective. Let H = Gal(F/E).
By inflation-restriction, there is an isomorphism
HS1 (E, Qp ) = HT1 (F, Qp )H ,where the subscript denotes classes “unramified outside S,” and where T denotes
the set of primes in F above S. By class field theory, this may be identified with
the H-invariants of the cokernel of the map
Y×OF× ⊗ Qp →OF,v
⊗ Qp .
T

The cokernel is larger than expected if and only if the kernel is bigger than expected.
In particular, rS = dimHS1 (E, Qp ) is bigger than expected only if
!H
\Y
×
×
OF ⊗ Qp
OF,v ⊗ Qp¬T

is bigger than expected. Note that the product over any subset T of primes in theright hand side is a rational subspace. Certainly the Strong Leopoldt Conjecture
determines the dimension of the intersection U ∩V of the unit group with a rational
subspace. What is slightly less clear is that the intersection (U ∩ V )H for any
subgroup H is also determined by the strong Leopoldt Conjecture, but this is true
(and we prove it). As a consequence, one has:Lemma 80.4. Assuming the Strong Leopoldt Conjecture, the dimension rS depends
only on G, H, and S.
This “reduces” the computation of rS to an intersection problem in a certain
Grassmannian. But this is a computation we were never really able to do!
This is the problem: One knows very well the structure of the unit group of F
as a G-module. So to compute the relevant intersections, one only has to computethe intersection with a “generic” rational subspace. Paradoxically, it seems very
difficult in general to give explicit examples of rational subspaces which are genericenough to obtain the correct minimal value. So while, for formal reasons, almost
any rational subspace will do, none of the nice subspaces which allow us to computethe intersection tend to be good enough.
Instead, to compute these intersections, we somewhat perversely look at actual
number fields and their unit groups. This seems like a bad idea, since even verifying
Leopoldt for a particular K and p is not so easy to do. So instead, we start witha totally real number field K of a certain form. Then, under the assumption of
Leopolodt’s conjecture we can (non-constructively) find subspaces of rational subspaces V which provably minimize various intersections dim(W ∩ V ) for various
unit-like submodules W . We then deform the field K to other fields L of different
signature, and use this construction (as well as the Strong Leopoldt Conjecture) tomake deductions about L. In the next post, we explain how this led Barry and me
to a proof of the following:
Theorem 80.5. Let E/Q be a degree n field with whose Galois closure F has
Galois group G = Sn . Assume the Strong Leopoldt Conjecture. Then (E, p) is rigid
for any prime p which splits completely in p.
I will explain the details next time. But to unwind the serpentine argument
slightly, we do not prove the result by finding rational subspaces in Qp [G] whose168

F. CALEGARI

intersection with the units of F has the a dimension which we can compute to be
the expected value, but only rational subspaces whose dimension we can computecontingent on Leopoldt’s conjecture for some auxiliary totally real number field. In
other words, we would like to compute the generic dimension of some intersection
inside some G-Grassmannian, a problem which has nothing to do with number
theory, and we compute it using Leopoldt’s conjecture. More next time!

81. Zp -extensions of number fields, Part II
Sat, 26 Nov 2016
This is continuation of § 80. We claimed there that we were going to deform a
totally real number field of degree n into a field with signature (r, s) with r +2s = n,and pass information about Leopoldt’s conjecture from one field to the other.
How does one “deform” a number field? One natural way is to think of a finite
etale map of varieties X → Y defined over Q, and then consider the fibres. Moreprosaically, write down some family of polynomials and then vary the coefficients.
Most of the time, the unit group doesn’t behave so well in such families. For example, consider the equation:x2 − D = 0.
If one varies D, even with some local control at primes dividing infinity (that is,
keeping D positive), then it is not at all clear how the fundamental unit varies.
In fact, one knows that the height of the fundamental unit is very sensitive to thesize of the class number, which changes somewhat irregularly with D. On the other
hand, consider the equation:x2 − Dx = 1.
Here one is in much better shape: as D varies, the element x will always be a unit,
and moreover always generates a finite index subgroup of the full unit group. How
might one use this for arguments concerning Leopoldt’s conjecture? The idea is toconsider (as above) a family of number fields in which some finite index subgroupof the full unit group is clearly visible, and is deforming “continuously” in terms of
the parameters. Then, by Krasner’s Lemma, we see that Leopoldt’s conjecture for
one number field (and a fixed prime p) will imply the same for all sufficiently close
number fields. To start, however, one needs to have such nice families.
81.1. Ankeny–Brauer–Chowla Fields. One nice family of number fields that
deforms nicely is the class of so-called Ankeny–Brauer–Chowla fields (from their
1956 paper [ABC56]):
Y(x − ai ) − 1 = 0
It is manifestly clear that in the field Q(x), the elements x − ai are all units, andthat (generically) there is only one multiplicative relation, namely that the productover all such units is trivial. In this way, we get a family of number fields (withgeneric Galois group) Sn and with a family of units generating a free abelian groupof rank n − 1. With a little tweak, we can also ensure that the prime p splits
completely. Concretely, consider the equationsY
Y(X − ai ) −(X − bi ) = 1,PERSIFLAGE: MATH BLOG POSTS

169

where X is a formal variable. The corresponding variety Y is connected of dimension
n, and the projection to An given by mapping to {bi } is a finite map, and so,
generically, the values of bi on Y are all distinct. In particular, for sufficiently large
primes p, Y has points over Fp where all the b2 are distinct modulo p. Fix such a
point {ai , bi } over Fp . Lift the ai in Fp to arbitrary integers in Z. Then, by Hensel’slemma, there exist p-adic integers vi congruent to bi mod p such thatYY
(x − ai ) − 1 =(x − vi ),and so p splits completely in our field as long as the ai satisfy some suitable nonempty congruences modp.
81.2. Deforming the signature. Suppose we assume that, for a fixed choice A =(a1 , .., an ), the corresponding field F satisfies Leopoldt’s conjecture. Then we see
that, in a sufficiently small neighbourhood of A, we obtain many other fields which
are totally real with Galois group Sn that also satisfy Leopoldt’s conjecture. On
the other hand, our goal is to study fields of signature (r, s) with r + 2s = n. So
we want to deform our fields to have non-trivial signature. I learnt this trick byreading a paper of Bilu: we deform the fields in a slightly different way, by making
the replacement(x − ai )(x − aj ) ⇒ (x − ai )(x − aj ) + u,
where u has very small p-adic valuation, and yet is a large positive integer. The
corresponding field no longer has n obvious units (whose product is one), but now
only n − 1 obvious units (whose product is one), where one of the units is now thequadratic polynomial above. On the other hand, one can also see that the signatureof the number field is now (n − 2, 1). So we still have a nice finite index subgroup of
the unit group. Moreover, p-adically, if our original units are written as {ui }, thenwe get (p-adically) something very close (by Krasner), except now ui and uj havebeen replaced by ui + uj . By combining other pairs of units in the same way, we canreduce the signature to (r, s) with any r + 2s = n and still have a nice p-adicallycontinuous finite index family of global units.
Proposition 81.3. Suppose that Leopoldt’s conjecture holds for the original field
K at p. Then, by deforming suitably chosen pairs of roots, we obtain a (infinitely
many) fields L with Galois group Sn and signature(r, s) with r + 2s = n such that,for a choice of r + s − 1 primes above p in L, the p-adic regulator of the units at
those r + s − 1 primes is non-zero.
As a consequence, for that choice of r + s − 1 primes, the corresponding maximal Zp -extension has rank zero. This proves that (L, p) is rigid for this choiceof S. However, since Sn is n-transitive, the same result applies for any such choiceof r + s − 1 primes. It’s an elementary lemma to see that this also implies the result
for sets S which are either larger or smaller than r + s − 1.
Proof. The argument is exactly as you expect: Given the original field K, the assumption of Leopoldt’s conjecture for K implies that at least one of the corresponding (r + s − 1) × (r + s − 1) minors must be non-zero. We then deform the field
globally so that the corresponding units in L of signature (r, s) are related to thisminor, which (by Krasner) will still be non-zero.
□170

F. CALEGARI

Question 81.4. The starting point of this construction was the assumption that
K satisfied Leopoldt’s conjecture. Can one prove this directly? That is, can one finda choice of ai such that the fieldY(x − ai ) − 1 = 0.
satisfies Leopoldt at p?
This seems quite plausible, after all, we have seen above that there are n niceunits of finite index in the unit group whose regulator varies p-adically. So, it suffice
to show that the regulator is not zero in the entire family. This certainly seems likean easier problem, because it’s easier to prove a function is non-zero rather than
the special value of a function (for example, by looking at the derivative). Still, I
confess that I don’t know how to prove this.

82. Artin no-go Lemma
Tue, 13 Dec 2016
The problem of constructing Galois representations associated to Maass forms
with eigenvalue 1/4 is, by now, a fairly notorious problem. The only known strategy,
first explained by Carayol, is to first transfer the representation to a unitary groupover an imaginary quadratic field, where one can realize the corresponding transfer
in the coherent cohomology of a related “Griffiths–Schmid” variety X. Then one
hopes to study the action of Hecke operators on this space and relate it to some(hopefully existing) rational structure on the cohomology. The wrinkle is that X
is not algebraic but merely a complex manifold, so it’s not so easy to see how to
impose any rational structure on the (higher) coherent cohomology. I have nothing
intelligent to say about whether this approach will work. However, suppose one is
as optimistic as possible, and thinks about what one might hope to be true — not
only to construct Galois representations but also prove the converse (Artin). Then,following [CG18a], one might hope to find an integral structure on this cohomology(with interesting torsion) on which to study congruences and then glue together
torsion classes using Taylor–Wiles to produce a patched complex of the right length.
What is the invariant l0 in this case? One might (generally) hope in this context tostudy conjugate self-dual representations
ρ : GE → GL3 (A)for an imaginary quadratic field E (in which p splits) for local Artinian rings (A, m)
with A/m of characteristic p which are unramified at p. The difference in dimension
between the ordinary local deformation ring and the unramified deformation ringappears to be 3, and thus we expect l0 = 3. Correspondingly, we expect cohomology
to occur in a range of cohomological degrees [q0 , q0 +3] for some q0 . Moreover, in thepresence of cohomology in characteristic zero, we expect to see cohomologies in all
such degrees. Yet this doesn’t happen for X; in fact, the cohomology only occurs (incharacteristic zero) in degrees 1 and 2 (according to Richard). This suggests not onlythat we won’t be able to prove modularity using integral cohomology of X, but eventhat — in the most naive sense — we should not expect an integral structure at least
with the usual properties. Namely, if we patch a complex of integral cohomologyof length 1, then the corresponding patched modules in cohomology will be too big

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for any unramified deformation ring to act. So it appears that the best possible
scenario is too good to be true.
On a different matter, there is another pressing issue I would like to bring to myreaders. In the papers I have written with Matt and David (and some by myself), we
have used the notation l0 — which has its origins in the book of Borel and Wallach.
There is, however, a competing notation in some of Akshay’s papers, namely δ. One
argument for the latter is that l0 specifically comes from a particular calculation in(g, K)-cohomology, and is not compatible with other situations in which one mightwant to consider the problem of cohomology in various degrees. (For example, forweight one modular forms, the Galois l0 = 1 whereas GL(2)/Q has discrete series.)
My argument is that there will never be any confusion when using l0 , and that it has
the property of being unlikely to every conflict with any other notation. Moreover,the phenomenology in both coherent and Betti cohomology both depend on l0 inexactly the same way. Dear reader: what is your opinion?
Notes 82.1. I am happy to say that l0 has won the battle of the notations.

83. Correspondance Serre–Tate, Part I
Sun, 25 Dec 2016
Reading the correspondence between Serre and Tate (see [Col15a, Col15b, Col17])
has been as delightful as one could expect. What is very nice to see — althoughperhaps not so surprising — is the utter delight that both Serre and Tate find in
discussing numerical examples. One of the beautiful aspects of number theory isthat there is an abundance of examples, each of which exhibit both special cases√ of avast general theory and yet each delighting with their own idiosyncrasies: Q( −23),
X0 (11), 691, 144169, etc. (It is precisely the absence of such examples, or at leastany discussion of them, why geometric Langlands tends to leave me completely
cold.) Take, for example, the following:
Letter from Tate to Serre, Dec 8, 1958:
Are you aware that the class number of the field of 97th rootsof 1 is divisible by 3457 and 118982593? And that 3457 = 36 * 96
+ 1 and 118982593 = 1239402 * 96 + 1?
If reading that doesn’t give you just a little thrill, then you have no soul. Does it
have any significance mathematically? The class number is large, of course, which
relates to the fact (proved by Odlyzko) that there are only finitely many Galois CM
fields with bounded class number. (The reason why one can access class numbersof CM fields F/F + is that the unit group of F and F + are the same up to finite
index, so the ratio of zeta values ζF (1)/ζF + (1) is directly related to the minus partof the class group h− uncoupled from any regulator term, so one can access thisanalytically.) Alternatively, one might be interested in the congruences of the primes
q dividing the class number. In this case, we see a reflection of the conjectures ofCohen and Lenstra. Namely, we expect that there is a strong preference for the classgroup to be “more cyclic,” especially for larger primes. The class group also has anaction of (Z/97Z)× which is cyclic of order 96. Since one expects the plus parth+ to be very small (and indeed in this case it is trivial), this means that complex
conjugation should act non-trivially, which means that the group of order 96 should(at least) act through a quotient of order at least 32. So if the class group is actually

172

F. CALEGARI

cyclic, this forces the prime divisors q of hF to be 1 mod 32, and even 1 mod 96 if
the class group of F doesn’t secretly come from the degree 32 subfield of F (which it
doesn’t). (Not entirely irrelevant is Rene Schoof’s nice paper [Sch03] on computingclass groups of real cyclotomic fields.)
Both Serre and Tate are unfailingly polite to each other. As a running joke, theexpression “talking through one’s hat” occurs frequently, as for example the letter ofb A C is discussed.
Nov 14, 1961, where the subtle issue of the failure of B ⊗A C → B ⊗(Another amusing snippet from that letter “Even G. himself makes mistakes when
he thinks causally.”) The correspondence is also fascinating from the perspective of
mathematical history — one sees the progress of many ideas as they are created,
including the Honda-Tate theorem and the Tate conjecture over finite fields. The
first time the latter appears (as a very special case) it actually turns out to be anargument of Mumford, who shows Tate an argument (using Deuring) why when
two elliptic curves have the same zeta function they are isogenous. This elicits thefollowing reaction from Tate:
Letter from Tate to Serre, May 9, 1962:“Damn! The result is certainly new to me, and it frankly makes
me mad that I never noticed it”
We have all been there, although, to be fair, most of us have the excuse of not being
Tate!

84. Central extensions, updated
Mon, 16 Jan 2017
I previously mentioned in § 78 a problem concerning polynomials, whose motivation came from thinking about weight one forms and the inverse Galois problem for
finite subgroups of GL2 (C). I still like the polynomial problem, but I realized that I
was confused about the intended application. Namely, given a weight one form withprojective image A5 , there is certainly a unique minimal lift up to twist, but theimages of the twists also automatically have image given by a central extension A5 .
So, just by twisting, one can generate all such groups as Galois groups by startingwith a minimal lift. More prosaically, every central extension of A5 by a cyclic groupe5 × Z where A
e5 is the Darstellungsgruppe of A5
is either a quotient of A5 × Z or of A
(which is SL2 (F5 )). So, to solve the inverse Galois problem for central extensionsof A5 , it suffices to solve it for SL2 (F5 ). That is not entirely trivial, but it is true.
I still think it’s an interesting problem to determine which extensions of A5 bycyclic groups occur as the Galois groups of minimally ramified up to twist extensions, but that is not the same as the inverse Galois problem.

85. The class number 100 problem
Thu, 19 Jan 2017
Some time ago, Mark Watkins busted open the “class number n” problem forsmallish n, finding all imaginary quadratic fields of class number at most 100 (theoriginal paper is here, see also [Wat04]) Although the paper describes the method
in detail, it does not actually give the complete list of imaginary quadratic fields
which occur (for fairly obvious reasons given the size of the list). I’ve occasionallyPERSIFLAGE: MATH BLOG POSTS

173

wanted to consult the actual list, and most of the time I have just emailed Markto find out the answer. But now it is available online! Here is the link. (Maybesomeone could put this on the LMFDB?)
Consulting the table one immediately notices a number of beautiful facts, such as
the fact that (Z/3Z)3 does not occur as a class group. (Our knowledge of p-parts ofclass groups, following Gauss, Pierce, Helfgott, Venkatesh, and Ellenberg, is enoughto show that (Z/2Z)n and (Z/3Z)n for varying n only occur finitely often [similarly
these groups plus any fixed group A], but those results are not effective.) One alsosees that D = −5519 and D = −1842523 are the first and last IQF discriminantsof class number 97. It’s the type of table that immediately bubbles up interestingquestions which one can at least try to give heuristic guesstimates. For example,
let µ(A) denote the number of imaginary quadratic fields with class group A. Can
one give a plausible guess for the rough size of µ(A)? One roughly wants to combine
the Cohen–Lenstra heuristics with the estimate h ∼ ∆1/2 . To do this, I guess one
would roughly want to have an estimate forX
1.
|Aut(A)|
1/2−ϵ
1/2+ϵ
x

≤|A|≤xI wouldn’t be surprised if someone has already carried out this analysis (thought I
don’t know any reference). As a specific examples:
Question 85.1. What is the expected growth rate of µ(Z/qZ) over primes q?
Question 85.2. Is there a finitely generated abelian group which provably does
not occur as the first homology of a congruence arithmetic hyperbolic 3-manifold?
At any rate, this is a result that Gauss would have appreciated.
Comment 85.3 (Emmanuel Kowalski). I vaguely remember some work of Soundon the asymptotic number of imaginary quadratic field with given class number,that might have discussed also specifying the group. . . Searching leads to this paper(see [Sou07]) and then to the follow-up here (see [HJK+ 19]) by Holmin, Jones,
Kurlberg, McLeman and Petersen that seems to address exactly the question withthe group structure taken into account.
And in fact there’s a new preprint this morning by Y. Lamzouri that’s also
related (counting imaginary quadratic fields with odd class number < H): here
(see [Lam17]).

86. Virtual coherent cohomology
Wed, 22 Feb 2017
I gave a talk yesterday where I attempted to draw parallels between the cohomology of (arithmetic) 3-manifolds and weight one modular forms. It was naturalthen to think about whether there was an analogue of the virtual Betti number
conjecture. Recall the following:
Theorem 86.1 (Agol, [Ago13]). Let M be a compact hyperbolic 3-manifold. Then
dimH 1 (N, Q) is unbounded as N ranges over all finite covers N → M.

174

F. CALEGARI(There’s an analogous version for finite volume hyperbolic manifolds with cusps.)
What is the corresponding conjecture in coherent cohomology? Here is a first attempt at such a question.
Question 86.2. Let X be a proper smooth curve of genus g ≥ 2 defined over Q.
Let L denote a line bundle such that L ⊗2 = Ω1X . As one ranges over all (finite
etale) covers π : Y → X, are the groups
H 0 (Y, π ∗ L )of unbounded dimension?
One might ask the weaker question as to whether there is a cover where this space
has dimension at least one (and in fact this is the first question which occurred tome). However, there are some parity issues. Namely, Mumford showed the dimensionof H 0 (X, L ) is locally constant in (X, L ), and this dimension is odd for precisely
2g−1 (2g + 1) choices of L (there are 22g such choices and the choices are a torsor
for 2-torsion in the corresponding Jacobian). But I think this means that one canalways make π ∗ L effective for some degree 2 cover, and thus produce at least
one dimensions worth of sections. For example, when g = 1, then Ω1X = OX , and
L = OX has global sections whereas the other square-roots correspond literally
to 2-torsion points. But those sections become trivial after making the appropriate
2-isogeny.
Another subtlety about this question which is worth mentioning is that I think
the result will have to be false over the complex numbers, hence the deliberateassumption that X was defined over Q, or at least over a number field. Specifically,
I think it should be a consequence of Brill–Noether theory that the set of X in Mg
such that
dimH 0 (Y, π ∗ L ) ≥ 1
for any choice of L and any cover π : Y → X of degree bounded by a fixed
constant D will be a finite union of proper varieties of positive dimension. And now
the usual argument shows that, as D increases, any countable union of varieties
cannot exhaust Mg . But it can, of course, exhaust all the rational points, and evenall the algebraic points.
There’s not really much evidence in favor of this question, beyond the following
three very minor remarks.
(1) The only slightly non-trivial case one can say anything about is when X
is a Shimura curve over Q, and then the answer is positive because there
exist lots of weight one forms (which one can massage to have the right
local structures after passing to a finite cover).
(2) The analogy between H 0 (X, L ) and H 1 (M, Q) is fairly compelling in thearithmetic case, so why not?
(3) There doesn’t seem to be any a priori reason why the virtual Betti number
conjecture itself was true, and it is certainly false in for related classes ofgroups (groups with the same number of generators and relations, word
hyperbolic groups), so, by some meta-mathematical jiu-jitsu, one can viewthe lack of a good heuristic in the hyperbolic case as excusing any real
heuristic in the coherent case.
Comment 86.3 (Felipe Voloch). If X is defined over a field of characteristic two,then there is a natural line bundle like in your question that you can take. Namely,PERSIFLAGE: MATH BLOG POSTS

175

in terms of divisors, the divisor of an exact differential dx is of the form 2D. Use
the line bundle corresponding to D and consider first etale covers of X with degree
a power of 2. If X is ordinary your H 0 ’s will be trivial and if X is not ordinary
they will grow linearly with the degree. The proof is not difficult but is too longfor a blog comment. Now, if X is ordinary but has a non-ordinary etale cover, thenyou get unbounded H 0 by taking covers of that cover. Does that work for all X?
Maybe, I don’t know. Unfortunately, this does not help in characteristic zero even
if X has good reduction at 2, as the inequalities for H 0 go the wrong way.
Comment 86.4 (Jordan Ellenberg). And of course the comment about what goes
wrong over C reveals that this is yet another example of a charming kind of question,“when can a “natural” countable union of proper subvarieties cover all the Q-pointsof a variety” I scratched my chin about this one too here but I think your example
goes even beyond the class of examples I discussed there!

87. A non-liftable weight one form modulo p2
Fri, 10 Mar 2017
I once idly asked Richard (around 2004ish) whether one could use Buzzard–
Taylor arguments to prove that any representation:
ρ : Gal(Q/Q) → GL2 (Z/p2 Z)
which was unramified at p and residually irreducible (and modular) was itself modular (in the Katz sense). Galois representations of this flavour are obviously something I’ve thought about (and worked on with David Geraghty) quite a lot sincethen. But I have never actually seen any examples of mod p2 forms which didn’t lift
to characteristic zero. I asked George Schaeffer about it once, but his computations
were only set up to detect the primes for which non-liftable forms existed rather
than to compute the precise structure of the torsion in H 1 (X, ω). But just today I
stumbled across an example in relation to a pairing I learned about from Akshay(which I will tell you all about some other time).
The particular form (or rather pair, since it comes with a twist by the nebentypus
character) occurs at level Γ0 (103) ∩ Γ1 (3), and is defined over the ring Z/112 Z. It
doesn’t lift to a weight one form mod113 . The nebentypus character is the only
one it could be at this level and weight: the odd quadratic character of conductor 3.
When I looked again at Schaeffer’s thesis, he does indeed single out this particularlevel as a context where computations suggested there might exist a mod p2 form.
(Literally, he says that a computation “seems to imply the existence” of such aform.) I guess this remark was not in any previous versions of the document I had,so I hadn’t seen it. Here are the first few terms of the q-expansion(s):g = q + 16q 2 + 20q 3 + 15q 4 + 58q 5 + 78q 6 + 22q 7 + . . . + 91q 11 + . . . + 104q 103 + . . .
f = q + 105q 2 + 115q 3 + 15q 4 + 63q 5 + 96q 6 + 22q 7 + . . .
Some remarks. Note that the coefficients of g and f satisfy a(g, n) = χ(n)a(f, n) forall (n, 3) = 1 and where χ is the quadratic character of conductor 3 (the nebentypus
character). On the other hand, at the prime 3, we have
 −1



χψ
0
χψ
0
ρf =
,
ρ g = χ ⊗ ρf =,0
ψ0 ψ −1

176

F. CALEGARI

and so the eigenvalue of U3 is the image of Frobenius at 3 under ψ or ψ −1 , andhence satisfies the equalitya(g, 3)a(f, 3) = ψ(Frob3 )ψ −1 (Frob3 ) = 20 · 115 = 1

mod 121.

I was temporarily confused about the fact that aq = 1 + q for the Steinberg primeq = 103 rather than aq = ±1, and thought for a while I had made an error or
mathematics was wrong. But then I realized this was weight one not weight two,and so one should have instead that (aq )2 = q −1 (note that χ(103) = 1.) And it
just so happens that the equation(1 + q)2 = q −1
in a weird coincidence has a solution very close to 103 (this is a solution mod 113 ,in fact). It’s easy enough to see that the image of rho and its twist contains
SL2 (Z/112 Z) with index two, and so has degree 3513840. (At this level, the only
real alternative is that the form is Eisenstein, which it isn’t.) The root discriminant
is not particularly small, it is21031−1/11 · 31/2 = 171.6970 . . .
Finally, the Frobenius eigenvalues at the prime p = 11 are distinct, which is easy
enough to see because otherwise the coefficient of q 11 would have to be twice thesquareroot of χ(11) = −1, which isn’t even a square mod 11.

88. Pseudo-representations and the Eisenstein Ideal
Wed, 29 Mar 2017
Preston Wake is in town, and on Tuesday he gave a talk on his recent joint workwith Carl Wang Erickson [WWE20]. Many years ago, Matt and I studied Mazur’s
Eisenstein Ideal paper from the perspective of Galois deformation rings. Using some
subterfuge (involving a choice of auxiliary ramification line at the prime N following an idea of Mark Dickinson), we proved an R = T theorem. One satisfactory
aspect (to us, at least) of our paper was that we were able to reconstruct from apurely Galois theoretic perspective some of the thorny geometric issues in Barry’s
paper, particularly at the prime 2. Another problem of Barry’s that we studied
was the question of determining for which N and p the cuspidal Hecke algebra wassmooth (equivalently, whether the cuspidal Hecke algebra completed at a maximal
Eisenstein ideal m of residue characteristic p was equal to Zp ). Our theorem showed
this was equivalent to the existence of certain Galois deformations to GL2 (F[e]/e3 ).
Although we were able to give a precise account of what happens for p = 2, forlarger p we could only prove the following:
Theorem 88.1. Let p ≥ 3 be prime, and let N ≡ 1 mod p be prime. If the rank of
the cuspidal Hecke algebra of level Γ0 (N ) localized at the Eisenstein prime is greaterthan one, then
K = Q(N 1/p )has non-cyclic p-class group.
Note that there is always trivial p-torsion class in the class group of K coming from the degree p extension inside the N th roots of unity. In our paper, we
speculated that this was actually an equivalence. To quote the relevant passage:PERSIFLAGE: MATH BLOG POSTS

177

We expect (based on the numerical evidence) that the condition
that the class group of K has p-rank [at least] two is equivalent to
the existence of an appropriate group scheme, and thus to [the rank
being greater than one].
Not a conjecture, fortunately, as it turns out to be false, already for p = 7
and N = 337. Oops! In fact, this had already been observed by Emmanuel Lecouturier in [Lec18]. Wake and Wang Erickson, however, give a complete characterization of when the rank is greater than one, namely
Theorem 88.2 ([WWE20]). Let a ∈ H 1 (Z[1/N p], Fp (1)) be the Kummer classcorresponding to N. Let b ∈ H 1 (Z[1/N p], Fp (−1)) be the (unique up to scalar) nontrivial class which is unramified at p. Then the rank of the Hecke algebra is greaterthan one if and only if the cup product a ∪ b vanishes.
They prove many other results in their paper as well. The main theoretical improvement of their method over the old paper was to work with pseudo-representations
rather than representations. On the one hand, this requires some more technicalmachinery, in particular to properly define exactly what it means for a pseudorepresentation to be finite flat. On the other hand, it avoids certain tricks that
Matt and I had to make to account properly for the ramification at N as well as
to make the deformation problem representable. Our methods would never work assoon as N is not prime, whereas this is not true for their new results. In particular,
there is real hope that there method can be applied to much more general N .
Let me also note that Merel in the ’90s found a completely different geometriccharacterization of when the cuspidal Hecke algebra had rank bigger than one;
explicitly, for p ≥ 3 and N = 1 mod p, it is bigger than one when the slightly
terrifying expression:(N −1)/2

Y

ii

i=1

is a pth power modulo N . So now there are a circle of theorems relating three
things: vanishing of cup products, ranks of Eisenstein Hecke algebras, and Merel’sinvariant above. It turns out that one can directly relate Merel’s invariant to thecup product using Stickelberger’s Theorem. On the other hand, Wake and Wang
Erickson also have a nice interpretation of the expression above as it relates to
Mazur–Tate derivatives (possibly this observation is due to Akshay), and they also
prove some nice results in this direction. And I haven’t even mentioned their otherresults relating to higher ranks and higher Massey products, and many other things.
Lecouturier’s paper is also a good read, and considers the problem from anotherperspective.
In Preston’s talk, he sketched the relatively easy implication that the vanishingof the cup product a ∪ b above implies that the class group of Q(N 1/p ) has noncyclic p-part. The main point is that the vanishing of cup products is exactly what
is required for a certain extension problem, and in particular the existence of a
Galois representation of the form:


1
a
c
0 χ−1 b  ,
0
0
1

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F. CALEGARI

where χ is the mod-p cyclotomic character. The class c gives the requisite extension (after some adjustment). Curiously enough, both the classes a and b exist forprimes N ≡ −1 mod p. On the other hand, the corresponding H 2 group vanishes
in this case, and so the pairing is always zero. Hence one deduces the following very
curious corollary:
Theorem 88.3. Let p ≥ 3, and let N ≡ −1 mod p be prime. Then the classnumber of Q(N 1/p ) is divisible by p.
Question 88.4. Is there a direct proof of this theorem? In particular, is there aneasy way to construct the relevant unramified extension of degree p for all suchprimes N ? I offer a beer to the first satisfactory answer.
Notes 88.5. The beer question has been answered by Lang and Wake in thispaper [LW22], although both authors are yet to claim their prize.

89. Who proved it first?
Wed, 26 Apr 2017
During Joel Specter’s thesis defense, he started out by remarking that the qexpansion:∞YX(1 − q n )(1 − q 23n ) =an q nf =q
n=1

is a weight one modular forms of level Γ1 (23), and moreover, for p prime, ap is equal
to the number of roots ofx3 − x + 1
modulo p minus one. He attributed this result to Hecke. But is it really due to
Hecke, or is this more classical? Let’s consider the following claims:(1) The form f is a modular form of the given weight and level.
(2) If p is not a square modulo 23, then ap = 0.
(3) If p is a square modulo 23, and x3 − x + 1 has three roots modulo p, thenap = 2.
(4) If p is a square modulo 23, and x3 − x + 1 is irreducible modulo p, thenap = −1.
At when point in history could these results be proved?
Let’s first start with Euler, who proved that
∞
∞
Y
X
2
(1 − q n ) =q (3n +n)/2 (−1)nn=1−∞Using this, one immediately sees that
XX 1
2
2
f=
q 24 ((6n+1) +23(6m+1) ) (−1)n+mThis exhibits f as a sum of theta series. With a little care, one can moreover show
that
XX 2
XX
2
2
2
2f =
q x +xy+6y −q 2x +xy+3y .
This is not entirely tautological, but nothing that Gauss couldn’t prove using facts
about the class group of binary quadratic forms of discriminant −23. The factthat f is a modular form of the appropriate weight and level surely follows from

PERSIFLAGE: MATH BLOG POSTS

179

known results about Dedekind’s η function, which covers (1). From the descriptionin terms of theta functions, the claim (2) is also transparent. So what remains?
Using elementary number theory, we are reduced to showingthat a prime p with√
(p/23) = +1 is principal in the ring of integers of Q( −23) if and only if p splits
completely in the Galois √closure H of x3 − x + 1.
Suppose that K = Q( −23) ⊂ H. What is clear enough is that primes p with
(p/23) = +1 split in K, and those which split principally can be represented by theform x2 + xy + 6y 2 in essentially a unique way up to the obvious automorphisms.
Moreover, the class group of SL2 (Z) equivalent forms has order 3, and the otherGL2 (Z) equivalence class is given by 2x2 + xy + 3y 2 . In particular, the primes
which split non-principally in K are represented by the binary quadratic form
2x2 + xy + 3y 2 essentially uniquely. From Minkowski’s bound, one can see that
H has trivial class group. In particular, if x3 − x + 1 has three roots modulo p,then the norm of the corresponding ideal to K is also principal and has normp = x2 + xy + 6y 2 . This is enough to prove (3).
So the only fact which would not obviously be easy to prove in the 19th centuryis (4), namely, that if p = x2 + xy + 6y 2 , then p splits completely in H. The mostgeneral statement along these lines was proved by Furtwängler (a student of Hilbert)
in 1911 — note that this is a different (and easier?) statement than the trivialityof the transfer map, which was not proved until 1930 (also by Furtwängler), after
other foundational results in class field theory had been dispensed with by Tagaki(another student of Hilbert!). Yet we are not dealing with a general field, but themuch more specific case of an imaginary quadratic field, which had been previously
studied by Kronecker and Weber in connection with the Jugendtraum. I don’tknow how much Kronecker could actually prove about (for example) the splittingof primes in the extension of an imaginary quadratic field given by the singularvalue j(τ ). Some of my readers surely have a better understanding of history than I
do. Does this result follow from theorems known before 1911? Who proved it first?

90. Elementary class groups updated
Sun, 11 Jun 2017
In § 88, I gave a short argument showing that, for odd primes p and N such thatN ≡ −1 mod p, the p-class group of Q(N 1/p ) is non-trivial. This post is just to
remark that the same argument works under weaker hypotheses, namely:
Proposition 90.1. Assume that N is p-power free and contains a prime factorof the form q ≡ −1 mod p, and that p is at least 5. Then the p-class group of
K = Q(N 1/p ) is non-trivial.
The proof is pretty much the same. If N has a prime factor of the form 1 mod p,then the genus field is non-trivial. Hence we may assume there are no such primes,
from which it follows that HS1 (Fp ) has dimension one and HS2 (Fp ) is trivial, where
S denotes the set of primes dividing Np. The prime q gives rise to a non-trivial classb ∈ HS1 (Fp (−1)) which is totally split at p (this requires that p be at least 5), andthe field K itself gives rise to a class a ∈ HS1 (Fp (1)). But now the vanishing of H 2

180

F. CALEGARI

implies that a ∪ b = 0 and hence there exists a representation of GS of the form:


1
a
c
ρ : GS → 0 χ−1 b  ,001where χ is the mod-p cyclotomic character. The class c gives the requisite extension(after possibly adjusting by a class in the one-dimensional space HS1 (Fp )). The main
point is that the image of inertia at primes away from p is tame and so cyclic, but
any unipotent element of GL3 (Fp ) has order p if p is at least three. This ensures cis unramified over K away from the primes above p. On the other hand, the class b
is totally split at p. This implies that the class c is locally a homomorphism of the
Galois group of Qp , and so after modification by a multiple of the cyclotomic classin HS1 (Fp ) may also be assumed to be unmarried at p. The fact that b ̸= 0 ensuresthat c ̸= 0, and moreover the fact that p is at least 5 implies that the kernel of c isdistinct from that of a, completing the proof. (This result was conjectured in thepaper Class numbers of pure quintic fields by Hirotomo Kobayashi [Kob16], which
proves the claim for p = 5.)
Comment 90.2 (Franz Lemmermeyer). The proposition follows, as Iimura remarked in [Iim86], from results due to Jaulent [Jau81]. I guess that it can be constructed classically by taking a subfield of the ray class group modulo q in the fieldof p-th roots of unity.

91. New results In modularity, Part I
Fri, 23 Jun 2017
I usually refrain from talking directly about my papers, and this reticence stems
from wishing to avoid any appearance of tooting my own horn. On the other hand,
nobody else seems to be talking about them either. Moreover, I have been involved
recently in a number of collaborations with multiple authors, thus sufficiently diluting my own contribution enough to the point where I am now happy to talk aboutthem.
The first such paper [ACC+ 23] I want to discuss has 9(!) co-authors, namely
Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard Taylor, and Jack Thorne. The reason for such a large
collaboration is a story of itself which I will explain at the end of the second post.
But for now, you can think of it as a polymath project, except done in a style moresuited to algebraic number theorists (by invitation only).
In this first post, I will start by giving a brief introduction to the problem. Then I
will state one of the main theorems and give some (I hope) interesting consequences.
In the next post, I will be a little bit more precise about the details, and explain
more precisely what the new ingredients are.
Like all talks in the arithmetic of the Langlands program, we start with:The Triangle:
Let F be a number field, let p be a prime, and let S be a finite set of places
containing all the infinite places and all the primes above p. Let GS denote theabsolute Galois group of the maximal extension of F unramified outside S. In manytalks in the Langlands program, one encounters the triangle, which is a conjectural
correspondence between the following three objects:

PERSIFLAGE: MATH BLOG POSTS

181• A: Irreducible pure motives M/F (with coefficients) of dimension n.
• B: Continuous irreducible n-dimensional p-adic representations of GS (forsome S) which are de Rham at the places above p.
• C: Cuspidal algebraic automorphic representations π of GL(n)/F.
In general, one would like to construct a map between any two of these objects,
leading to six possible (conjectural) maps, which we can describe as follows:• A →B: This is really the only map we understand, namely, etale cohomology. (I’m being deliberately vague here about what a motive actually is,
but whatever.)• B →A: This is the Fontaine–Mazur conjecture, and maybe some parts ofthe standard conjectures as well, depending on exactly what a motive is.
• B →C: This is “modularity.”• C →B: This is the existence of Galois representations associated to automorphic forms.
• A →C: We really think of this as A →B →C and also call this modularity.
• C →A: Again, this is a souped up version of C →B. But note, we still
don’t understand how to do this even in cases where C →B is very wellunderstood. For example, suppose that π comes from a Hilbert modular
form with integer coefficients of trivial level over a totally real field F ofeven degree. We certainly have an associated compatible family of Galois
representations, and we even know that its symmetric square is geometric.
But it should come from an elliptic curve, and we don’t know how to provethis. The general problem is still completely open (think Maass forms). On
the other hand, often by looking in the cohomology of Shimura varieties,
one proves C →A and uses this to deduce that C →B.
This triangle is also sometimes known as “reciprocity.” The other central tenet of
the Langlands program, namely functoriality, also has implications for this diagram.
Namely, there are natural operations which one can easily do in case B which should
then have analogs in C which are very mysterious.
91.1. Weight zero. For all future discussions, I want to specialize to the caseof “weight zero.” On the motivic/Galois side, this corresponds to asking that the
representations are regular and which Hodge–Tate weights which are distinct andconsecutive, namely, [0, 1, 2, . . . , n−1]. The hypotheses that the weights are distinct
is a restrictive but crucial one — already the case when F = Q and the Hodge–Tateweights are [0, 0] is still very much open (specifically, the case of even icosahedral
representations). On the automorphic side, the weight zero assumption corresponds
to demanding that the π in question contribute to the cohomology of the associatedlocally symmetric space with constant coefficients.
For example, if n = 2, then we are precisely looking at abelian varieties of GL(2)type over F (e.g. elliptic curves). This is an interesting case! We know they aremodular if F is Q, or even a quadratic extension of Q. More generally, we knowthat if F is totally real, then such representations are at least potentially modular,that is, their restriction to some finite extension F ′ /F is modular. This is often goodenough for many purposes. For example, it is enough to prove many cases of (someversion of) B →A. In this case, we have quite complete results, although still shortof the optimal conjectures, especially in the case when the residual representationis reducible.

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F. CALEGARI

There are many other modularity lifting results generalizing those for n = 2,
but they really involve Galois representations whose images have extra symmetry
properties. In particular, they are either restricted to representations which preserve (up to scalar) some orthogonal or symplectic form, or they remain unchanged
if one conjugates the representation by an outer automorphism of GF (for example when F/F + is CM and one conjugates by complex conjugation). There werebasically no unconditional results which applied either in the situation that n ≥ 2or that F was not completely real, and the representation did not otherwise havesome restrictive condition on the global image. Our first main theorem is to prove
such an unconditional result. Here is such a theorem (specialized to weight zero):
Theorem 91.2 ([ACC+ 23]). Let F be either a CM or totally real number field,
and p a prime which is unramified in F . Letρ : GS → GLn (Qp )be a continuous irreducible representation which is crystalline at v|p with Hodge–
Tate weights [0, 1, .., n − 1]. Suppose that
(1) The residual representation ρ has suitably big image.
(2) The residual representation is “modular” in the sense that there exists anautomorphic form π0 for GL(n)/F of weight zero and level prime to p suchthat r(π0 ) = ρ.
Then ρ is modular, that is, there exists an automorphic representation π of weightzero for GL(n)/F which is associated to ρ.
One could be more precise about what it means to have big image. In fact, I cando this by saying that it has enormous image after restriction to the composite of
the Galois closure of F with the pth roots of unity. Here enormous is a technicalterm, of course. There is also a version of this theorem with an ordinary (rather
than Fontaine–Laffaille) hypothesis (more on this next time).
Let me now give a few nice theorems which can be deduced from the theorem
above:
Theorem 91.3 ([ACC+ 23]). Let E be an elliptic curve over a CM field F . Then E
is potentially modular.
When I had a job interview at MIT in 2006, I was asked by Michael Sipser, thechair at the time, to come up with a theorem which (in a best case scenario) I
would hope to prove in 10 years. I said that I wanted to prove that elliptic curves
over imaginary quadratic fields were modular. (Reader, I got the job . . . then went
to Northwestern.) It is very gratifying indeed that, roughly 10 years later, thisresult has actually been proved and that I have made some contribution towards
its eventual resolution. (OK, we have potential modularity rather than modularity,
but that is splitting hairs. . . ). It is also amusing to note that a number of co-authors
were still in high school at this time! (Fact Check: OK, just one . . . )In fact, one can improve on the theorem above:
Theorem 91.4 ([ACC+ 23]). Let E be an elliptic curve over a CM field F . Then Symn (E)is potentially modular for every n. In particular, the Sato–Tate conjecture holds
for E.
Finally, for an application of a different type, suppose that π is a weight zero cuspidal algebraic automorphic representation for GL(2)/F. For each prime v of good

PERSIFLAGE: MATH BLOG POSTS

183

reduction, one can associate to πv a pair of Satake parameters {αv , βv } satisfying|αv βv | = N (v). The Ramanujan conjecture says that one has|αv | = |βv | = N (v)1/2 .
An equivalent formulation is that the sum av of these two eigenvalues satisfies|av | ≤ 2N (v)1/2 . We prove the following:
Theorem 91.5 ([ACC+ 23]). : Let F be a CM field, and let π be a weight zero
cuspidal algebraic automorphic representation for GL(2)/F. Then the Ramanujan
conjecture holds for π.
If F is totally real, then the Ramanujan conjecture follows from Deligne’s theorem. One can associate to π a motive, whose Galois representation is either ρ = ρ(π)
or ρ⊗2 . Then, by applying purity to these geometric representations, one deduces
the result. (Of course, this was famously proved by Deligne himself in the case when
F = Q. The case of a totally real field, especially in cases where one has to go via
a motive associated to ρ⊗2 , is due (I think) to Blasius.) This is decidedly not theway we prove this theorem. In fact, we do not know how to prove the Fontaine–
Mazur conjecture for the representation ρ associated to π, even in the weak senseof showing that ρ or even ρ⊗2 appears inside the cohomology of some projective
variety. Instead, we prove that Symn ρ is potentially modular, then use the weaker
convexity bound to prove the inequality:|αv |n ≤ N (v)n/2+1/2 .
Taking n sufficiently large, we deduce that |αv | ≤ N (v)1/2 , which (by symmetry)
proves the result. Experts will recognize this as precisely Langlands’ original strategy for proving Ramanujan using functoriality! In a certain sense, this is the firsttime that Ramanujan has been proved without a direct recourse to purity. I say “insome sense”, because there is also the ambiguous case of weight one modular forms.
Here the Ramanujan conjecture (which is |ap | ≤ 2 in this case) was deduced by
Deligne and Serre as a consequence of showing that ρ has finite image so that αv
and βv are roots of unity. On the other hand, that argument does also simultaneously imply that the representations are motivic. So our theorem produces, I
believe, the only cuspidal automorphic representations for GL(n)/F for which weknow to be tempered everywhere and yet for which we do not know are directly
associated in any way to geometry.
Question 91.6. Suppose I’m sitting in my club, and Tim Gowers asks me to say
what is really new about this paper. What should I say?
Answer 91.7. The distinction (say) between elliptic curves over imaginary quadratic fields and real quadratic fields, while vast, is quite subtle to explain to someone who hasn’t thought about these questions. You could explain it, but the clubis hardly a place to do so. Instead, go with this narrative: We generalize Wiles’
modularity results for 2-dimensional representations of Q to n-dimensional representations of Q. If you are pressed on previous generalizations, (especially those due
to Clozel–Harris–Taylor), say that Wiles is the case GL(2), Clozel–Harris–Taylor isthe case GSp(2n), and our result is the case GL(n).
If you had slightly more time, and the port has not yet arrived, you mightalso try to explain how the underlying geometric objects involved for GSp(2n) areall algebraic varieties (Shimura varieties), but for GL(n) they involve Riemannian

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F. CALEGARI

manifolds which have no direct connection to algebraic geometry. Here is a good
opportunity to name drop Peter Scholze, and explain how this is the first time thatthe methods of modularity have been combined with the new world of perfectoid
spaces.
Notes 91.8. Of course there have been many updates since this paper. To mention
just one, there is the work of Caraiani–Newton [CN23] on the modularity of elliptic
curves over imaginary quadratic fields.

92. New results in modularity, Part II
Fri, 23 Jun 2017
This is part two of series on work in progress with Patrick Allen, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, James Newton, Peter Scholze, Richard
Taylor, and Jack Thorne.
It has been almost 25 years since Wiles first announced his proof of Taniyama–
Shimura, and, truthfully, variations on his method have been pretty much the only
game in town since then (this paper included). In all generalizations of this argument, one needs to have some purchase on the integral structure of the automorphic
forms involved, which requires that they contribute in some way to the cohomologyof an arithmetic manifold (locally symmetric space). This is because it is crucial
to be able to exploit the integral structure to study congruences between modular
forms. Let’s briefly recall Wiles’ strategy. One starts out with a residual representation
ρ : GS → GL2 (Fp )which one assumes to be modular, that is, is the mod-p reduction of a representation
associated to a modular form which is assumed to have some local properties similar
to rho. One then considers a deformation ring R which captures all deformations of
the residual representation which “look modular” of the right weight and level (someaspects of Serre’s conjecture due to Ribet are employed here, although SkinnerWiles came up with a base change trick to circumvent some of these difficulties).
On the automorphic side, one looks at the cohomology groups M = H 1 (X, Zp )m of
modular curves (X = X0 (N )) localized at a maximal ideal m of the Hecke algebra T
associated to rhobar, and proves that there is a surjective map:
R → Tm .
Already many deep theorems have been used to arrive at this point. To begin,
one needs Galois representations associated to modular forms, but moreover, one
needs to know that these representations satisfy all of the expected local-global
compatibilities at the primes in S. In the case of modular forms, all of these facts
were basically known before Wiles.
The next step, which lies at the heart of the Taylor–Wiles method, is to introduce
certain auxiliary sets Q of carefully chosen primes, and consider the spaces MQ =H 1 (X1 (Q), Zp )m which relate to spaces of modular forms of larger level. If TQ is theassociated Hecke algebra, and RQ is the corresponding deformation ring in which
ramification is allowed not only at S but now also at Q, there are compatible maps
as follows:PERSIFLAGE: MATH BLOG POSTS

RQ

TQ ⊂ EndMQR

T ⊂ EndM

185

The key point concerning how one chooses the sets Q is to ensure that, eventhough RQ may get bigger, its infinitesimal tangent space does not. Hence all
the RQ are quotients of some fixed ring R∞ = Zp [[X1 , . . . , Xq ]] (Here q is chosen so that q = |Q|.) In this process, all the rings also have an auxiliary action of aring S∞ = Zp [[T1 , . . . , Tq ]] of diamond operators, coming from the Galois group of
X1 (Q) over X0 (Q) on the automorphic side, and the inertia groups at Q on the Galois side. The action of S∞ on these modules factors through RQ by construction, by
local global compatibility. After throwing away the Galois representations almost
entirely (but keeping the diamond operators), one can patch the modules MQ /pn
for different sets of primes Q, and arrive at a patched module M∞ for R∞ and S∞
such that:• The module M∞ has positive rank as an S∞ module.
• If a is the augmentation ideal of S∞ , then R∞ /a = R, and M∞ /a = M.
The first statement may be viewed as saying that there are “lots” of automorphic
forms. On the other hand, the fact that R∞ has the same dimension of S∞ saysthat there are not “too many” Galois representations. Indeed, this friction is enoughin this context to prove that M∞ is free over R∞ , and then to deduce the same
claim for M over R, from which R = T follows. (Already included here is aninnovation due to Diamond where one deduces freeness as a consequence rather
than building it in as an assumption.) The argument I have very briefly sketchedabove is really only a proof of modularity in the minimal case. The general caserequires a completely separate argument to bootstrap from minimal to non-minimallevel using two further ingredients: Wiles’ numerical criterion, and a lower bound on
the congruence ideal necessary to apply the numerical criterion, which ultimately
follows from Ihara’s Lemma.
The “first generation” of improvements to Wiles consisted of understanding
enough integral p-adic Hodge theory to make the required arguments on the Galois side. Notable papers here include the work of Conrad–Diamond–Taylor andBreuil–Conrad–Diamond–Taylor [CDT99, BCDT01] (but let us also not forget here
the contribution of the Hawk [Sav04]). Improvements along these lines continue totoday, and are very closely intertwined with p-adic Langlands program and workof Breuil, Colmez, Kisin, Emerton, Paškūnas, and many others.
The “second generation” of improvements consisted of relaxing the assumptionthat R∞ is smooth, by allowing instead R∞ to have multiple components (but stillof the same dimension) associated to different components in the local deformation
rings at primes in S (at p and away from p). This innovation was due to Kisin, who
also introduced the notion of framing to handle this.
The “third generation” of improvements (somewhat orthogonal to the second)comes from replacing 2-dimensional representations with n-dimensional representations, but still under some very restrictive assumptions on the image of rho. One
key consequence of these assumptions is that the spaces of modular forms MQ =H ∗ (X1 (Q), Zp )m all occur inside a single cohomology group, which allows one tocontrol the growth of these spaces when patching. Here one thinks of the work of

186

F. CALEGARI

Clozel–Harris–Taylor [CHT08]. Also pertinent is that the analog of Ihara’s Lemma
is open for higher rank groups; Taylor came up with a technique to bypass it whenproving modularity lifting theorems now known as “Ihara avoidance.”(Of course there were many other developments less directly relevant to thispost, including but not limited to Skinner–Wiles and Khare–Wintenberger.)
The problem with considering general representations for GL(n) for n ≥ 2, evenover Q, is that the automorphic forms are spread over a number of different cohomology groups, in fact in some range [q0 , q0 + 1, . . . , q0 + l0 ] for specific invariants q0
and l0 . This manifests itself in two ways:(1) There are not enough automorphic forms; the patched modules M∞ will
not be free over S∞ .
(2) There are not enough Galois representations: the ring R∞ does not havethe same dimension as S∞ but rather dim R∞ = dim S∞ − l0 .
Of course these problems are related! My work with David Geraghty was precisely
about showing how to make these problems cancel each other out. The rough idea
is as follows. The cohomology groups H ∗ (X1 (Q), Zp )m which contain interestingclasses in characteristic zero occur in the range [q0 , . . . , q0 + l0 ]. Suppose one knows
this to be true integrally as well, even with coefficients over Fp instead of Zp . Theninstead of patching the cohomology groups MQ themselves, one instead patchescomplexes PQ of length l0 . The result is a complex P∞ of finite free S∞ modules oflength l0 , with an action of R∞ on the cohomology of this complex. But the only way
the cohomology of this complex can be small enough to admit an action of R∞ is if
the complex is a free resolution of the patched module M∞ of cohomology groups
in the extreme final degree, and moreover it also implies that M∞ is big enough as
in Wiles’ original argument to give an R = T theorem. Note that it is crucial herethat one work with the torsion in integral cohomology. It is quite possible that, at
all auxiliary levels Q, there are no more automorphic forms at level Q than are wereat level one. (This can only happen for l0 > 0, and the idea that torsion should be asuitable replacement is the moral of my paper with Barry Mazur.) These argument
is also compatible with the improvements to the method including Taylor’s “Ihara
Avoidance” argument.
On the other hand, there is a big problem. This argument required many inputs
which were completely unknown at the time we worked this out, so our results
were very conditional. To be precise, our results were conditional on the following
desiderata:(1) The existence of Galois representations on Hecke rings T which acted as
endomorphisms of H ∗ (X, Z/pn Z) for locally symmetric spaces X associatedto GL(n)/F .
(2) The stronger claim that the Galois representations constructed in part one
satisfied the correct “local-global” compatibility statements for all v in S
(including v|p).
(3) The vanishing of the cohomology groups H i (X, Z/pn Z)m outside the range i ∈[q0 , . . . , q0 + l0 ], for a non-Eisenstein ideal m.
A different approach to some of these questions (which Matt and I discussed,see § 26) involves first passing to completed cohomology, where one expects (orhopes!) that all the cohomology groups except in degree q0 should vanish after
localization at a non-maximal ideal.

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187

The first big breakthrough was the result of Scholze, who proved part 1 above,at least up to issues concerning a nilpotent ideal (this was discussed § 23). Another innovation appeared in Khare–Thorne, where it was observed that one cansometimes drop the third assumption under the strong condition that there existed
global automorphic forms with the exact level structure corresponding to the original representation. (Unfortunately, in the l0 > 0 setting, there is no way to produce
such forms.)
So this is roughly where we stood in 2016. The key new ingredient which led tothis project was the new result of Caraiani and Scholze [CS24] proving vanishingtheorems for the cohomology of non-compact Shimura varieties in degrees above themiddle dimension (localized at m) under the assumption of certain genericity hypotheses on m. Since the cohomology of the boundary (for suitably chosen Shimuravarieties) is precisely related to the cohomology of arithmetic locally symmetricspaces for GL(n) over CM fields, this allowed for the first time a new constructionof the Galois representations for GL(n) which directly related them to the Galois
representations coming from geometry. (I say “directly related,” but perhaps I mean
simply more direct than Peter’s original construction.) In particular, it was clear
to Caraiani and Scholze that this result should have implications for the required
local-global compatibility result above. Meanwhile, the IAS had just started a newseries of workshops on emerging topics. I guess that Richard must have had conversations with Ana about her work with Peter, which led them to choosing this as
the theme, namely:
Ana Caraiani and Peter Scholze are hopeful of extending the methods of their joint paper (see [CS17]) to non-compact Shimura varieties. This would give a new way to attack local-global compatibility at p for some of the Galois representations Scholze attached
to torsion classes in the cohomology of arithmetic locally symmetric spaces. The aim of this workshop will be to understand how
much local-global compatibility can be proved and to explore theconsequences of this, particularly for modularity questions.
So now (1) was available, there was an approach to (2), and a technique foravoiding (3). One issue with the Khare-Thorne trick, however, was that it involved
localizing at some prime ideal of characteristic zero, and so did not interact sowell with Ihara Avoidance, which was crucial for any sort of applicable theorem.
Here’s the subtlety, which can be described even in the case when l0 = 0. The
usual Ihara avoidance game is to compare deformation rings R and R′ at Steinberg
level and ramified principal series level respectively (after making a base change toensure that the prime v at the relevant prime q satisfies N (v) = 1 mod p). Let M
and M ′ be the corresponding modules. One has that M/p = M ′ /p and R/p = R′ /p.
Suppose, however, that M behaved perfectly as expected, so that M∞ was free (evenof rank one say) over S∞ and free over R∞ . What could happen, if one doesn’t′
have vanishing of cohomology outside a single degree, is that M∞/p = M∞ /p is′free over S∞ /p, but that M∞ is the cohomology of a non-trivial complex S∞ →′S∞ given by multiplication by p. So M∞
is trivial in characteristic zero, even′though M∞ /p = M∞ /p. So this is a problem. But it is exactly a problem which
was resolved during the workshop. The point, very loosely speaking, is that even′though the complexes “S∞’ and “[p] : S∞ → S∞ ” have the same H 0 after reducing
modulo p and taking cohomology, their intersection with S∞ /p are quite different

188

F. CALEGARI

on the derived level, so if one can formulate a version of derived Ihara avoidance,then one is in good shape.
So what remained? First, there were a number of technical issues, some of which
could be dealt with individually, and one had to make sure that all the fixes werecompatible. For example, it is straightforward to modify the original strategy in
my paper with David to handle the issue of only having Galois representations up
to nilpotence ideals of fixed nilpotence, but one had to make sure this would notinterfere with the more subtle derived Ihara avoidance type arguments. Relevant
here was the work of Newton and Thorne which placed some of the arguments withcomplexes more naturally in the derived category. Second, there was the issue ofreally proving local-global compatibility from the new results of Caraiani-Scholze.
A particularly interesting case here was the ordinary case. The rough problem one
has to deal with here is deducing that rho is ordinary from knowing that ρ ⊕ ρ∨is ordinary. But be careful — the latter representation is reducible and so really apseudo-representation — so it’s not even clear what ordinary this means (though see
work of Wake and Wang Erickson, as well as of my student Joel Specter). It turns
out that some interesting and subtle things turn up in this case which were foundby the “team” of people who wrote up this section. (Although we achieved quite a
lot in a week, there were obviously a list of details to be worked out, and we divided
ourselves up into certain groups to work on each part of the paper.) But I think wewere fairly confident at this point that everything would work out. What was myrole in the writing up process you ask? I was selected as the ENFORCER, who goesaround harassing everybody else to work and write up their sections of the paper
while sipping on Champagne. Presumably I was less selected for my organizational
skills and more for my ability to tell Richard Taylor what to do (see [Tay12]).
So there we have it! It was clear even during the workshop that some improvements to our arguments were possible, but since the paper is already going to bequite long, we did not try to be completely comprehensive. I expect a number of
improvements will follow shortly. I would not be surprised to see in a few years a
modularity result for regular weight compatible systems over CM fields which are
as complete as the ones (say) in [BLGGT14].
Comment 92.1 (David Loeffler). Just out of curiosity, how important is the“weight 0” hypothesis for what you’re doing? Is there some specific step that definitely breaks down when the Hodge–Tate weights are distinct but non-consecutive;
or is that restriction just imposed to keep the project manageable?
Comment 92.2 (Persiflage). The problem is that in potential automorphy results
you are always comparing to some geometric family (in this case, the Dwork family
studied in [HSBT10]), and such families have consecutive Hodge–Tate weights. You
can get around this with Hida theory (but this only works if you know that you
have lots of ordinary primes, which we don’t know in any generality), or the Harris
tensor product trick — but unfortunately (and perhaps slightly unexpectedly) there
are technical problems with getting this trick to work for CM fields.
So something new is needed to change weight in this setting; to the best of myknowledge this hasn’t been worked out yet, but people have promising ideas. If I
was a betting man, I would put money on [BLGGT14]-style theorems being proved
over CM fields in the next couple of years, but at the moment, Ramanujan is open
in weight ≥ 2.

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189

Notes 92.3. We are still some distance from [BLGGT14]-style results, but we
do know the Ramanujan conjecture in (many) higher weights by [BCG+ 23b]. See
also [Mat23].

93. Schaefer and Stubley on class groupsSun, 29 Oct 2017
I talked 88 about work of Wake and Wang-Erickson on deformations of Eisenstein residual representations. In that post, I also mentioned a paper of EmmanuelLecouturier who has also proved some very interesting theorems. Today, I wanted
to talk about some complementary results by my student Eric Stubley in collaboration with Karl Schaefer (a student of Matthew Emerton) (see [SS19]). To duplicate
slightly from that previous post, recall that Matt and I proved the following:
Theorem 93.1. Let p ≥ 3 be prime, and let N ≡ 1 mod p be prime. If the rank of
the cuspidal Hecke algebra of level Γ0 (N ) localized at the Eisenstein prime is greaterthan one, then
K = Q(N 1/p )has non-cyclic p-class group.
Using work of Merel, one can dispense with the discussion of Hecke algebras andinstead give an equivalent reformulation of the first condition, namely, e ≥ 1 if andonly if M1 is a p-th power, where
M1 =

p−1
Y

(M k)!k ∈ F×
N,

k=1

M=

N −1
p

We followed up this result with the comment:
We expect (based on the numerical evidence) that the condition
that the class group of K has p-rank [at least] two is equivalent to
the existence of an appropriate group scheme, and thus to [the rank
being greater than one].
As noted previously, there are counter-examples, already for p = 7 and N = 337.
However, there was still clearly some relationship between these quantities beyond
the one-way implication above. In particular, the numerical evidence still stubbornly
supported the hope that the converse may indeed be true for p = 5. This is the first
theorem that Schaefer and Stubley prove. More precisely, they completely determine
the rank of the class group of Q(N 1/5 ) for primes N which are 1 mod 5.
Theorem 93.2 (Schaefer–Stubley). Let N ≡ 1 mod 5 be prime. Then the 5-rank
rK of the class group of K = Q(N 1/5 ) is either 1, 2, or 3 Moreover:(1) rK = 1 if and only if the Merel invariant M1 is not a perfect√ 5th power.
5−1(2) rK = 2 if and only if M1 is a perfect 5th power, and α =is not a2
perfect 5th power modulo N.
(3) rK = 3 if and only if M1 and α are both 5th powers modulo N .
This also answers a conjecture of Lecouturier. Their argument greatly clarified (to me) the exact relationship between the class group of K and a number of other related quantities in this picture. To recall, a third reformulation of

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F. CALEGARI

whether the Hecke algebra has non-trivial deformations can be given (as in Wake–
Wang-Erickson) by whether a certain pairing between specific classes b and c−1 in
1
1
−1
HN
) vanish or not. The point is that the vanishing of a cup
p (Q, ϵ) and HN p (Q, ϵproduct ensures the existence of an extension


1
b
c0
0 ϵ−1 c−1 0 01and one can show (after some massaging) that c0 gives rise to something in the
p-class group of K. Conversely, if one starts with a class in the p-class group of K,
and then takes the Galois closure over Q, then (sometimes) one arrives with a
Galois extension M/Q with a Galois representation to GL(3) of the above form.
The problem is, in other circumstances, one arrives at a representation which has
a much larger Galois group and a map to the Borel subgroup in higher dimension,
which looks something like this:
1 ϵ−1 · b ϵ−2 · b2 /2 ϵ−3 · b3 /6
...
c0
0
ϵ−1
ϵ−2 · bϵ−3 · b2 /2
...
c−1 




..


.


1−m
−m
. . .

ϵ
ϵ
·
b
c
1−m 

. . .
ϵ−m
c−m 
...
1
Suppose one now tries to construct a representation of this form in order to finda non-trivial class in the p-class group of K. First, one can start by finding a1−m
suitable class c−m ∈ HN
) which cups trivially with lb. The vanishingp (Q, ϵof a generalized Merel invariant (under a regularity hypothesis) is exactly what
guarantees the existence of such a suitable class c−m , at least when m is odd.
However, one is then faced with an increasing sequence of obstruction problems
in order to climb the ladder and get all the way to the full representation of theform above. Here one has to deal with not only cup products, but also (implicitly)
higher Massey products. Ultimately, the relation between the quantity rK and thedeformation rings of Hecke algebras is most precise only when p = 5. It turns outthat there is still something one can say for p = 7, however. Consider the higher
Merel invariant
p−1
Y
n
N −1
Mn =
(M k)!k ∈ F×
M=
N,
p
k=1

for odd values of n. Suppose that p is a regular prime. One can show that if rK ≥ 2,then at least one of these quantities Mn is a perfect pth power for an odd n ≤ p − 4.
When p = 5, this is a weaker version of the theorem above. So an optimistic
variation on the conjecture above is that rK ≥ 2 if and only if Mn is a perfect
pth power of for at least one odd n ≤ p − 4. The description of the relationshipbetween these classes (which also come up in Lecouturier, they arise via an explicit
analysis of Gauss sums and Stickelberger’s theorem) suggests that this conjecture
is too optimistic in general, and indeed there are counter-examples for p = 11. But,
Schaefer and Stubley do prove the following:
Theorem 93.3 (Schaefer, Stubley). Let p = 7, and let N = 1 mod p be prime.
Then the 7-class group of K = Q(N 1/p ) has rank rK ≥ 2 if and only if either M1
or M3 is a perfect 7th power modulo N .

PERSIFLAGE: MATH BLOG POSTS

191

For example, consider the previous “counter-example” for N = 337 and p = 7.
Here the non-trivial class group is explained by the fact that M3 is a perfect 7th
power modulo N .
One thing I especially like about this result is that there are three groups ofpeople (Wake–Wang-Erickson, Lecouturier, and Schaefer–Stubley) are all working
around a similar problem, but their results are complementary to each other. I
believe that all five people will be at the upcoming IAS workshop, so I hope to hearmore about this then.

94. Mathieu magicTue, 17 Oct 2017
I previously mentioned (in Comment 75.5) that I once made (in a footnote) the
false claim that for a 11-dimensional representation V of the Mathieu group M12 ,the 120 dimensional representation Ad0 (V ) was irreducible. I had wanted to write
down representations W of large dimension n such that Ad0 (W ) of dimension n2 −1
was irreducible. In the comments, Emmanuel Kowalski pointed to a paper of Katz
where he discusses actual examples (including the 1333 dimensional representation of the Janko group J4 ). On the other hand, I recently learned from Liubomir
Chirac’s thesis. that it’s an open problem to determine whether there exists sucha representation for all n (although he does write down infinitely many examples
in prime power dimension). Chirac’s thesis also lead me to the paper of Magaard,
Malle, and Tiep, who do classify all such examples for (central extensions of) simple
groups. Turns out that I could have used M12 after all, or rather the 10-dimensional
representation of the double cover 2.M12 , which does have the required property(the 99-dimensional representation factors through M12 , naturally).
One reason (amongst many) that (either of the) 11-dimensional representations
V of M12 do not have Ad0 (V ) irreducible is that they are self-dual (oops). On
the other hand, if you eyeball the character table, you will find that there is anirreducible representation W of dimension 12. Moreover, let me write down thecharacters of [V ⊗ V ∗ ] − [1] and [W ]:[V ⊗ V ∗ ] − [1] :120, 0, 8, 3, 0, 0, 8, 0, 0, −1, 0, 0, 0, −1, −1;[W ] :

120, 0, −8, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, −1, −1.

These seem surprisingly close to me! So now the question is, as one ranges over(some class perhaps all) finite groups G, what is the minimum number of conjugacy
classes for whichχ = [V ⊗ V ∗ ] − [1] − [W ]can be non-zero for irreducible V and W , assuming that it is non-zero? Since V
is irreducible, by Schur’s Lemma, this virtual representation is orthogonal to [1](unless [W ] = [1] which would be silly). So ⟨χ, 1⟩ = 0, which certainly implies
that there must be at least two non-zero entries of opposite signs. I don’t see any
immediate soft argument which pushes that bound to 3. I admit, this is a slightly
silly question. But still, a beer to anyone who proves the example above is either
optimal or comes up with an example with only two non-zero terms. (To avoid
silliness, say that the dimension of V has to be at least 5.) More precisely:192

F. CALEGARI

Problem 94.1. Classify all pairs (G, V, W ) of a finite group G and irreducible
representations V and W with V faithful such that χ = [V ⊗ V ∗ ] − [1] − [W ] is zeroon all but at most 2 conjugacy classes but χ itself is non-zero.
The characters in the example above look strikingly similar to me, and it does
make we wonder if there is any reason for why they are so close. Perhaps if I knew
more about groups, I could feel more confident in just chalking up the resemblance
above to a law of small numbers.
Probably a more sensible question is to ask for how small the number of non-zero
entries of [V ] − [W ] can be for two distinct irreducibles. That question has surely
been studied!

95. Abelian surfaces are potentially modular
Sat, 11 Nov 2017
Today I wanted (in the spirit of this post) to report on some new work in progresswith George Boxer, Toby Gee, and Vincent Pilloni. (see [BCGP21])
Recall that, for a smooth projective variety X over a number field F unramified
outside a finite set of primes S, one may write down a global Hasse-Weil zetafunction:Y
1
ζX,S (s) =1 − N (x)−swhere the product runs over closed points of a smooth integral model. From the
Weil conjectures, the function ζX,S (s) is absolutely convergent for s with real partat least 1 + m/2, where m = dim(X). One has the following well-known conjecture:
Conjecture 95.1 (Hasse–Weil Conjecture). The function ζX,S (s) extends to ameromorphic function on the complex plane. Moreover, there exists a rational number A, a collection of polynomials Pv (T ) for v dividing S, and infinite Gamma
factors Γv (s) such thatYY1ξX (s) = ζX,S (s) · As/2 ·Γv (s) ·Pv (N (v)−s )
v|∞

v|S

satisfies the functional equation ξX (s) = w · ξX (m + 1 − s) with w = ±1.
Naturally, one can be more precise about the conductor and the factors at thebad primes. In the special case when F = Q and X is a point, then ζX,S (s) isessentially the Riemann zeta function, and the conjecture follows from Riemann’s
proof of the functional equation. If F is a general number field but X is still a point,then ζX,S (s) is (up to some missing Euler factors at S) the Dedekind zeta function
ζF (s) of F , and the conjecture is a theorem of Hecke. If X is a curve of genus zero
over F , then ζX,S (s) is ζF (s)ζF (s − 1), and one can reduce to the previous case.
More generally, by combining Hecke’s results with an argument of Artin and Brauer
about writing a representation as a virtual sum of induced characters from solvable(Brauer elementary) subgroups, one can prove the result for any X for which thel-adic cohomology groups are potentially abelian. This class of varieties includes
those for which all the cohomology of X is generated by algebraic cycles.
For a long time, not much was known beyond these special cases. But that is not
to say there was not a lot of progress, particularly in the conjectural understandingof what this conjecture really was about. The first huge step was the discovery andPERSIFLAGE: MATH BLOG POSTS

193

formulation of the Taniyama-Shimura conjecture, and the related converse theoremsof Weil. The second was the fundamental work of Langlands which cast the entire
problem in the (correct) setting of automorphic forms. In this context, the HasseWeil zeta functions of modular curves were directly lined to the L-functions ofclassical weight 2 modular curves. More generally, the Hasse-Weil zeta functionsof all Shimura varieties (such as Picard modular surfaces) should be linked (via
the trace formula and conjectures of Langlands and Kottwitz) to the L-functionsof automorphic representations. On the other hand, these examples are directly
linked to the theory of automorphic forms, so the fact that their Hasse-Weil zetafunctions are automorphic, while still very important, is not necessarily evidence
for the general case. In particular, there was no real strategy for taking a variety
that occurred “in nature” and saying anything non-trivial about the Hasse-Weilzeta function beyond the fact it converged for real part greater than 1 + m/2, whichitself requires the full strength of the Weil conjectures.
The first genuinely new example arrived in the work of Wiles (extended byothers, including Breuil-Conrad-Diamond-Taylor), who proved that elliptic curves
E/Q were modular. An immediate consequence of this theorem is that Hasse-Weilconjecture holds for elliptic curves over Q. Taylor’s subsequent work on potentiallymodularity, while not enough to prove modularity of all elliptic curves over all
totally real fields, was still strong enough to allow him to deduce the Hasse-Weilconjecture for any elliptic curve over a totally real field. You might ask what havebeen the developments since these results. After all, the methods of modularity
have been a very intense subject of study over the past 25 years. One problem
is that these methods have been extremely reliant on a regularity assumption on
the corresponding motives. One nice example of a regular motive is the symmetricpower of any elliptic curve. On the other hand, if one takes a curve X over a number
field, then h1,0 = h0,1 = g, and the corresponding motive is regular only for g = 0
or 1. The biggest progress in automorphy of non-regular motives has actually comein the form of new cases of the Artin conjecture — first by Buzzard–Taylor andBuzzard, then in the proof of Serre’s conjecture by Khare–Wintenberger over Q, andmore recently in subsequent results by a number of people (Kassaei, Sasaki, Pilloni,
Stroh, Tian) over totally real fields. But these results provide no new cases of the
Hasse-Weil conjecture, since the Artin cases were already known in this setting by
Brauer. (It should be said, however, that the generalized modularity conjecture is
now considered more fundamental than the Hasse-Weil conjecture.) There are a few
other examples of Hasse-Weil one can prove by using various forms of functoriality to
get non-regular motives from regular ones, for example, by using the Arthur-Clozel
theory of base change, or by Rankin-Selberg. We succeed, however, in establishing
the conjecture for a class of motives which is non-regular in an essential way. The
first corollary of our main result is as follows:
Theorem 95.2 ([BCGP21]). Let X be a genus two curve over a totally real field.
The Hasse-Weil conjecture holds for X.
It will be no surprise to the experts that we deduce the theorem above from thefollowing:
Theorem 95.3 ([BCGP21]). Let A be an abelian surface over a totally real field F .
Then A is potentially modular.

194

F. CALEGARIIn the case when A has trivial endomorphisms (the most interesting case), thistheorem was only known for a finite number of examples over Q. In each of thosecases, the stronger statement that A is modular was proved by first explicitly computing the corresponding low weight Siegel modular form. For example, the teamof Brumer–Pacetti–Tornaría–Poor–Voight–Yuen prove [BPP+ 19] that the abeliansurfaces of conductors 277, 353, and 587 are all modular, using (on the Galoisside) the Faltings-Serre method, and (on the automorphic side) some really quite
subtle computational methods developed by Poor and Yuen. A paper of Berger–
Klosin [BK20] handles a case of conductor 731 by a related method that replaces
the Faltings–Serre argument by an analysis of certain reducible deformation rings.
The arguments of our paper are a little difficult to summarize for the non-expert.
But George Boxer did a very nice job presenting an overview of the main ideas,
and you can watch his lecture online (posted below, together with Vincent’s lecture
on higher Hida theory). The three sentence version of our approach is as follows.
There was a program initiated by Tilouine to generalize the Buzzard–Taylor method
to GSp(4), which ran into technical problems related to the fact that Siegel modular
forms are not directly reconstructible from their Hecke eigenvalues. There was asecond approach coming from my work with David Geraghty, which used insteada variation of the Taylor–Wiles method; this ran into technical problems related
to the difficulty of studying torsion in the higher coherent cohomology of Shimuravarieties. Our method is a synthesis of these two approaches using Higher Hida
theory as recently developed by Pilloni. Let me instead address one or two questions
here that GB did not get around to in his talk:
Question 95.4. What is the overlap of this result with [ACC+ 23]?
Answer 95.5. Perhaps surprisingly, not so much. For example, our results are
independent of the arguments of Scholze (and now Caraiani–Scholze [CS24]) on
constructing Galois representations to torsion classes in Betti cohomology. We dogive a new proof of the result that elliptic curves over CM fields are potentiallymodular, but that is the maximal point of intersection. In contrast, we don’t provethat higher symmetric powers of elliptic curves are modular. We do, however, prove
potentially modularity of all elliptic curves over all quadratic extensions of totallyreal fields with mixed signature, like Q(21/4 ). The common theme is (not surprisingly) the Taylor–Wiles method (modified using the ideas in my paper with DavidGeraghty).
Question 95.6. What’s new in this paper which allows you to make progress on
this problem?
Answer 95.7. George explains this well in his lecture. But let me at least stress
this point: Vincent Pilloni’s recent work on higher Hida Theory (see [Pil20]) was
absolutely crucial. Boxer, Gee, and I were working on questions related to modularity in the symplectic case, but when Pilloni’s paper first came out, we immediately
dropped what we were doing and started working (very soon with Pilloni) on thisproblem. If you have read the Calegari–Geraghty [CG20] paper on GSp(4) and are
not an author of the current paper (hi David!), and you look through our manuscript (currently a little over 200 pages and [optimistically?!] ready by the end of theyear), then you also recognize other key technical points, including a more philosophically satisfactory doubling argument and Ihara avoidance in the symplectic
case, amongst other things.

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195

Question 95.8. So what about modularity?
Answer 95.9. Of course, we deduce our potential modularity result from a modularity lifting theorem. The reason we cannot deduce that Abelian surfaces are all
modular, even assuming for example that they are ordinary at 3 with big residual
image, is that Serre’s conjecture is not so easy. Not only is GSp4 (F3 ) not a solvablegroup, but — and this is more problematic — Artin representations do not contribute to the coherent cohomology of Shimura varieties in any setting other thanholomorphic modular forms of weight one. Still, there are some sources of residually modular representations, including the representations induced from totallyreal quadratic extensions (for small primes, at least). We do, however, prove thefollowing (which GB forgot to mention in his talk, so I bring up here):
Proposition 95.10. There exist infinitely modular abelian surfaces A/Q (up to
twist) with EndQ (A) = Z.
This is proved in an amusing way. It suffices to show that, given a residual
representation
ρ : GQ → GSp4 (F3 )with cyclotomic similitude character (or rather inverse cyclotomic character withour cohomological normalizations) which has big enough image and is modular(plus some other technical conditions, including ordinary and p-distinguished) thatit comes from infinitely many abelian surfaces over Q, and then to prove the modularity of those surfaces using the residual modularity of ρ. This immediately reduces
to the question of finding rational points on some twist of the moduli space A2 (3).
And this space is rational! Moreover, it turns out to be a very famous hypersurface
much studied in the literature — it is the Burkhardt Quartic. Now unfortunately— unlike for curves — it’s not so obvious to determine whether a twist of a higher
dimensional rational variety is rational or not. The problem is that the twistingis coming from an action by Sp4 (F3 ), and that action is not compatible with thebirational map to P3 , so the resulting twist is not a priori a Severi-Brauer variety.
However, something quite pleasant happens — there is a degree six cover
6:1

A−
2 (3) → A2 (3)(coming from a choice of odd theta characteristic) which is not only still rational,
but now rational in an equivariant way. So now one can proceed following theargument of Shepherd-Barron and Taylor [SBT97] in their earlier paper on mod-2
and mod-5 Galois representations.
Question 95.11. What about curves of genus g > 2?
Answer 95.12. Over Q, there is a tetrachotomy corresponding to the cases g = 0,
g = 1, g = 2, and g > 2. The g = 0 case goes back to the work of Riemann. The
key point in the g = 1 case (where the relevant objects are modular forms of weighttwo) is that there are two very natural ways to study these objects. The first (andmore classical) way to think about a modular form is as a holomorphic function
on the upper half plane which satisfies specific transformation properties under theaction of a finite index subgroup of SL2 (Z). This gives a direct relationship between
modular forms and the coherent cohomology of modular curves; namely, cuspidal
modular forms of weight two and level Γ0 (N ) are exactly holomorphic differentials
on the modular curve X0 (N ). On the other hand, there is a second interpretation of

196

F. CALEGARI

modular forms of weight two in terms of the Betti (or etale or de Rham) cohomologyof the modular curve. A direct way to see this is that holomorphic differentials canbe thought of as smooth differentials, and these satisfy a duality with the homology
group H1 (X0 (N ), R) by integrating a differential along a loop. And it is the second
description (in terms of etale cohomology) which is vital for studying the arithmeticof modular forms. When g = 2, there is still a description of the relevant forms interms of coherent cohomology of Shimura varieties (now Siegel 3-folds), but there
is no longer any direct link between these coherent cohomology groups and etale
cohomology. Finally, when g > 2, even the relationship with coherent cohomology
disappears — the relevant automorphic objects have some description in termsof differential equations on locally symmetric spaces, but there is no longer any
way to get a handle on these spaces. For those that know about Maass forms, thesituation for g > 2 is at least as hard (probably much harder) than the notorious
open problem of constructing Galois representations associated to Maass forms ofeigenvalue 1/4. In other words, it’s probably very hard! (Of course, there are special
cases in higher genus when the Jacobian of the curve admits extra endomorphisms
which can be handled by current methods.)
Finally, as promised, one can find the videos here and here.
Notes 95.13. The “over 200 pages” became 349 pages, I guess. There is also
progress on the question of modularity in genus g = 2, this is work in progressof the same four authors [BCGP24].

96. The ABC conjecture has (still) not been proved
Mon, 18 Dec 2017
The ABC conjecture has (still) not been proved.
Five years ago, Cathy O’Neil laid out a perfectly cogent case for why the (atthat point recent) claims by Shinichi Mochizuki should not (yet) be regarded as
constituting a proof of the ABC conjecture. I have nothing further to add on thesociological aspects of mathematics discussed in that post, but I just wanted to report on how the situation looks to professional number theorists today. The answer?
It is a complete disaster.
This post is not about making epistemological claims about the truth or otherwise of Mochizuki’s arguments. To take an extreme example, if Mochizuki had
carved his argument on slate in Linear A and then dropped it into the Mariana
Trench, then there would be little doubt that asking about the veracity of the argument would be beside the point. The reality, however, is that this description is
not so far from the truth.
Each time I hear of an analysis of Mochizuki’s papers by an expert (off therecord) the report is disturbingly familiar: vast fields of trivialities followed by anenormous cliff of unjustified conclusions. The defense of Mochizuki usually restson the following point: The mathematics coming out of the Grothendieck school
followed a similar pattern, and that has proved to be a cornerstone of modern
mathematics. There is the following anecdote that goes as follows:
The author hears the following two stories: Once Grothendieck said
that there were two ways of cracking a nutshell. One way was tocrack it in one breath by using a nutcracker. Another way was toPERSIFLAGE: MATH BLOG POSTS

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soak it in a large amount of water, to soak, to soak, and to soak,
then it cracked by itself. Grothendieck’s mathematics is the latterone.
While rhetorically expedient, the comparison between Mochizuki and Grothendieckis a poor one. Yes, the Grothendieck revolution upended mathematics during the
1960’s “from the ground up.” But the ideas coming out of IHES immediately spreadaround the world, to the seminars of Paris, Princeton, Moscow, Harvard/MIT,
Bonn, the Netherlands, etc. Ultimately, the success of the Grothendieck school is
not measured in the theorems coming out of IHES in the ’60s but in how theideas completely changed how everyone in the subject (and surrounding subjects)
thought about algebraic geometry.
This is not a complaint about idiosyncrasy or about failing to play by the rulesof the “system.” Perelman more directly repudiated the conventions of academia bysimply posting his papers to the arXiv and then walking away. (Edit: Perelman
did go on an extensive lecture tour and made himself available to other experts,
although he never submitted his papers.) But in the end, in mathematics, ideas
always win. And people were able to read Perelman’s papers and find that theideas were all there (and multiple groups of people released complete accounts ofall the details which were also published within five years). Usually when there isa breakthrough in mathematics, there is an explosion of new activity when othermathematicians are able to exploit the new ideas to prove new theorems, usuallyin directions not anticipated by the original discoverer(s). This has manifestly not
been the case for ABC, and this fact alone is one of the most compelling reasons
why people are suspicious.
The fact that these papers have apparently now been accepted by the Publications of the RIMS (a journal where Mochizuki himself is the managing editor,
not necessary itself a red flag but poor optics none the less) really doesn’t change
the situation as far as giving anyone a reason to accept the proof. If anything, thevalue of the referee process is not merely in getting some reasonable confidence
in the correctness of a paper (not absolute certainty; errors do occur in publishedpapers, usually of a minor sort that can be either instantly fixed by any knowledgeable reader or sometimes with an erratum, and more rarely requiring a retraction).
Namely, just as importantly, it forces the author(s) to bring the clarity of the writing up to a reasonable standard for professionals to read it (so they don’t needto take the same time duration that was required for the referees, amongst otherthings). This latter aspect has been a complete failure, calling into question boththe quality of the referee work that was done and the judgement of the editorial
board at PRIMS to permit papers in such an unacceptable and widely recognizedstate of opaqueness to be published. We now have the ridiculous situation where
ABC is a theorem in Kyoto but a conjecture everywhere else. (edit: a Japanesereader has clarified to me that the newspaper articles do not definitively say thatthe papers have been accepted, but rather the wording is something along the lines
of “it is planned that PRIMS will accept the paper,” whatever that means. This
makes no change to the substance of this post, except that, while there is still achance the papers will not be accepted in their current form, I retract my criticism
of the PRIMS editorial board.)
So why has this state persisted so long? I think I can identify three basic reasons. The first is that mathematicians are often very careful (cue the joke about a

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sheep at least one side of which is black ). Mathematicians are very loath to claimthat there is a problem with Mochizuki’s argument because they can’t point to any
definitive error. So they tend to be very circumspect (reasonably enough) about
making any claims to the contrary. We are usually trained as mathematicians to
consider an inability to understand an argument as a failure on our part. Second,
whenever extraordinary claims are made in mathematics, the initial reaction takes
into account the past work of the author. In this case, Shinichi Mochizuki was someone who commanded significant respect and was considered by many who knew him
to be very smart. It’s true (as in the recent case of Yitang Zhang) that an unknown
person can claim to have proved an important result and be taken seriously, but if
a similarly obscure mathematician had released 1000 pages of mathematics written
in the style of Mochizuki’s papers, they would have been immediately dismissed.
Finally, in contrast to the first two points, there are people willing to come out
publicly and proclaim that all is well, and that the doubters just haven’t put in thenecessary work to understand the foundations of inter-universal geometry. I’m notinterested in speculating about the reasons they might be doing so. But the ideathat several hundred hours at least would be required even to scratch the beginnings of the theory is either utter rubbish, or so far beyond the usual experienceof how things work that it would be unique not only in mathematics, but in all ofscience itself.
So where to from here? There are a number of possibilities. One is that someone
who examines the papers in depth is able to grasp a key idea, come up with amajor simplification, and transform the subject by making it accessible. This was
the dream scenario after the release of the paper, but it becomes less and less likely
by the day (and year). But it is still possible that this could happen. The flip sideof this is that someone could find a serious error, which would also resolve thesituation in the opposite way. A third possibility is that we have (roughly) the
status quo: no coup de grâce is found to kill off the approach, but at the same time
the consensus remains that people can’t understand the key ideas. (I should saythat whether the papers are accepted or not in a journal is pretty much irrelevant
here; it’s not good enough for people to attest that they have read the argument
and it is fine, someone has to be able to explain it.) In this case, the mathematicalcommunity moves on and then, whether it be a year, a decade, or a century, whensomeone ultimately does prove ABC, one can go back and compare to see if (in theend) the ideas were really there after all.
Comment 96.1 (Jordan Ellenberg). Thanks for posting this, Frank.
Comment 96.2 (William Stein). Thanks for posting this Frank! A fourth possibility is that the fairly strong form of ABC that Mochizuki claims to have proved
turns out to not be true . . .
Comment 96.3 (Andrew Sutherland). I haven’t been following this closely —can one extract from Mochizuki’s arguments an explicit effective inequality (not an
asymptotic) that is falsifiable?
Comment 96.4 (Persiflage). I believe that this paper by Vesselin Dimitrov showsthat that one can formally extract such a quantity, although completely explicit
bounds are not found in that paper.

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Comment 96.5 (Matthew Emerton). Great post! It does indeed do an excellent
job of summarizing the situation from the perspective of professional number theorists.
Comment 96.6 (Toby Gee). Than you very much for posting this!
Comment 96.7 (Akshay Venkatesh). I couldn’t agree more.
Comment 96.8 (Terry Tao). Thanks for this. I do not have the expertise to havean informed first-hand opinion on Mochizuki’s work, but on comparing this story
with the work of Perelman and Yitang Zhang you mentioned that I am much morefamiliar with, one striking difference to me has been the presence of short “proof
of concept” statements in the latter but not in the former, by which I mean ways
in which the methods in the papers in question can be used relatively quickly toobtain new non-trivial results of interest (or even a new proof of an existing nontrivial result) in an existing field. In the case of Perelman’s work, already by thefifth page of the first paper Perelman had a novel interpretation of Ricci flow as a
gradient flow which looked very promising, and by the seventh page he had used
this interpretation to establish a “no breathers” theorem for the Ricci flow that,while being far short of what was needed to finish off the Poincare conjecture, wasalready a new and interesting result, and I think was one of the reasons why experts in the field were immediately convinced that there was lots of good stuff inthese papers. Yitang Zhang’s 54 page paper spends more time on material thatis standard to the experts (in particular following the tradition common in analytic number theory to put all the routine lemmas needed later in the paper in arather lengthy but straightforward early section), but about six pages after all thelemmas are presented, Yitang has made a non-trivial observation, which is thatbounded gaps between primes would follow if one could make any improvement to
the Bombieri–Vinogradov theorem for smooth moduli. (This particular observation
was also previously made independently by Motohashi and Pintz, though not quite
in a form that was amenable to Yitang’s arguments in the remaining 30 pages ofthe paper.) This is not the deepest part of Yitang’s paper, but it definitely reduces
the problem to a more tractable-looking one, in contrast to the countless papers
attacking some major problem such as the Riemann hypothesis in which one keeps
on transforming the problem to one that becomes more and more difficult looking,until a miracle (i.e. error) occurs to dramatically simplify the problem.
From what I have read and heard, I gather that currently, the shortest “proof ofconcept” of a non-trivial result in an existing (i.e. non-IUTT) field in Mochizuki’s
work is the 300+ page argument needed to establish the abc conjecture. It seems
to me that having a shorter proof of concept (e.g. ≤ 100 pages) would help dispel
skepticism about the argument. It seems bizarre to me that there would be an entire
self-contained theory whose only external application is to prove the abc conjecture
after 300+ pages of set up, with no smaller fragment of this setup having any
non-trivial external consequence whatsoever.
Comment 96.9 (Alon Amit). Thank you so much for weighing in so clearly andunambiguously on the situation. The mathematical community needs to speak up
more clearly about it.
Comment 96.10 (Dick Gross). This is an excellent post. Terry’s comment (fromthe outside of number theory) is particularly telling. For those of us inside of it, the

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situation is infuriating. Shortly after Faltings announced his proof of Tate’s isogeny
conjecture and the Mordell conjecture, he lectured on it at the Arbeitstagung,explaining the new tools he had introduced. Everyone in the audience who had
thought about the problem was immediately convinced. Instead of producing 300+
pages of manuscript, Mochizuki needs to give one or two lectures (in Bonn, or Paris,
or Boston, or . . . ) clearly explaining the new ideas in his argument and showing
how they lead to a proof of ABC. This shouldn’t be difficult — I have no idea why
he refuses to do so.
Comment 96.11 (Peter Scholze). Thanks for the wonderful post! I agree witheverything that was said.
One small thing I would like to add is that most accounts indicate that noexperts have been able to point to a place where the proof would fail. This is infact not the case; since shortly after the papers were out I am pointing out that
I am entirely unable to follow the logic after Figure 3.8 in the proof of Corollary
3.12 of Inter-universal Teichmüller theory part III: “If one interprets the above
discussion in terms of the notation introduced in the statement of Corollary 3.12,
one concludes [the main inequality].” Note that this proof is in fact the only proof
in parts II and III that is longer than a few lines which essentially say “This follows
from the definitions”. Those proofs, by the way, are completely sound, very little
seems to happen in those two papers (to me). Since then, I have kept asking otherexperts about this step, and so far did not get any helpful explanation. In fact, over
the years more people came to the same conclusion; from everybody outside theimmediate vicinity of Mochizuki, I heard that they did not understand that stepeither. The ones who do claim to understand the proof are unwilling to acknowledge
that more must be said there; in particular, no more details are given in any survey,
including Yamashita’s, or any lectures given on the subject (as far as they are
publicly documented). [I did hear that in fact all of parts II and III should be
regarded as an explanation of this step, and so if I am unable to follow it, I should
read this more carefully. . . For this reason I did wait for several years for someone
to give a better (or any) explanation before speaking out publicly.]
One final point: I get very annoyed by all references to computer-verification(that came up not on this blog, but elsewhere on the internet in discussions of
Mochizuki’s work). The computer will not be able to make sense of this step either.
The comparison to the Kepler conjecture, say, is entirely misguided: In that case,the general strategy was clear, but it was unclear whether every single case had
been taken care of. Here, there is no case at all, just the claim “And now the result
follows”.
Comment 96.12 (Brian Conrad). PS, thank you so much for writing in suchspecificity about your experience. In the spirit of stating things in public that havebeen known among some experts in arithmetic geometry for quite a while, I’d liketo now share something in public (I think for the first time) concerning Corollary
3.12 in IUT3 that I have been bringing to the attention of many mathematiciansin private during the past 2 years. Soon after I posted my essay on Cathy O’Neil’s
blog summarizing my impressions about the Oxford IUT workshop in December
2015, I received unsolicited emails from people whom I knew in quite distant partsof the world (one in Europe, one in Asia, and one in North America). Each of themtold me that they had worked through the IUT papers on their own and were able
to more-or-less understand things up to a specific proof where they had becomePERSIFLAGE: MATH BLOG POSTS

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rather stumped. For each of these people, the proof that had stumped them wasfor 3.12 in IUT3. It was striking to get three independent unsolicited emails in amatter of days which all zeroed in on that same proof as a point of confusion.
A focus of concern on the proof of 3.12 in IUT3 never came up in discussions
during the Oxford IUT workshop; my first awareness about it was from those three
unsolicited emails. Since that time, the number of people whom I know that haveinvested tremendous effort reading the IUT papers (some giving talks at IUT workshops) and became stumped by that proof has grown further. (I will not reveal theidentities of any of these people, since they communicated their concerns to me inprivate. It is also entirely unnecessary, since PS’s comment addresses the matterquite well.)
One reason that I have never before discussed this experience in “public” (=Internet posting) is that I assumed the referee process would ultimately lead toa revision that completely clarified the proof of 3.12 in IUT3 and thereby madethe matter disappear (so the earlier concerns would be rendered moot). I know
from much experience as an editor at various journals that it is very common
that papers submitted to math journals has errors that are caught during therefereeing process and then fixed by the author(s) before acceptance. Thus, there
is generally no purpose in publicizing such matters; we are all human, after all, and(as Frank notes) part of the referee’s job is to make a reasonable attempt to ferretout mistakes.
I was therefore very surprised when I heard recently (incorrectly, as it turnsout) that the IUT papers had been accepted, since the public version of IUT3
still did not have a revision to the proof of 3.12 that cleared up the matter (as I
immediately confirmed with several who have invested a lot of time on the IUT
papers). Of course, referees are human too and may sometimes overlook something;
this is why authors sometimes publish an erratum afterwards, and it is also why it
is imperative for papers to be written with a degree of clarity about the techniques
so that other can explore the ideas further. Clarity of communication is an essential
part of progress in mathematical research. I sincerely hope that wider awarenessof the genuine concern about the proof of 3.12 in IUT3 will finally lead to greatercollective understanding about what is going on there.
Comment 96.13 (Harald Helfgott). I am very glad that someone of note has put
in writing, and rather articulately at that, what many have long said or suspected.
Notes 96.14. The papers have been published [Moc21a, Moc21b, Moc21c, Moc21d],
but the community has certainly accepted that as things currently stand, to put it
generously, there is nothing there.

97. Abandonware
Mon, 25 Dec 2017
For a young mathematician, there is a lot of pressure to publish (or perish).
The role of for-profit academic publishing is to publish large amounts of crappy
mathematics papers, make a lot of money, but at least in return grant the authors acertain imprimatur, which can then be converted into reputation, and then into job
offers, and finally into pure cash, and then coffee, and then back into research. One
great advantage of being a tenured full professor (at an institution not run by bean

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counters) is that I don’t have to play that game, and I can very selective in what
papers I choose to submit. In these times — where it is easy to make unpublishedwork available online, either on the arXiv, a blog, or a webpage — there is no
reason for me to do otherwise. Akshay and I are just putting the finishing touches
on our manuscript on the torsion Jacquet–Langlands correspondence (a project
begun in 2007!), and approximately 100 pages of the original version has been
excised from the manuscript. It’s probably unlikely we will publish the rest, notbecause we don’t think it’s interesting, but because it can already be found online.
(Although we might collect the remains into a supplemental “apocrypha” to makereferencing easier.) Sarnak writes lots of great letters and simply posts them online.
I wrote a paper a few years ago called “Semistable modularity lifting over imaginary
quadratic fields.” (see [Cal]) It has (IMHO) a few interesting ideas, including one
strategy for overcoming the non-vanishing of cohomology in multiple degrees in an
l0 = 1 situation, a way of proving a non-minimal modularity lifting theorem inan (admittedly restricted) l0 = 1 situation without having to use Taylor’s Ihara
Avoidance or base change (instead using the congruence subgroup property), andan argument explaining why the existence of Nilpotent ideals in Scholze’s Galois
representation is no obstruction to the modularity lifting approach in my paper withDavid. But while I wrote up a detailed sketch of the argument, gave a seminar about
it, and put the preprint on my webpage, I never actually submitted it. One reason
was that David and I were (at the time, this was written in 2014-2015 or so) under
the cosh by an extremely persnickety referee (to give you some idea, the paper was
submitted in 2012 and was only just accepted), and I couldn’t stomach the idea ofbeing raked over the coals a second time merely to include tedious details. (A tiny
Bernard Woolley voice at the back of my head is now saying: excuse me minister,
you can’t be raked over by a cosh, it doesn’t have any teeth. Well done if you have
any idea what I am talking about.) But no matter, the paper is on my webpage
where anyone can read it. As it happens, the paper [ACC+ 23] has certainly madethe results of [Cal] entirely redundant, but there are still some ideas which might beuseful in the future someday. But I don’t see any purpose whatsoever in subjecting
an editor, a reviewer, and (especially) myself the extra work of publishing thispaper.
So I am all in favor of avoiding publishing all but a select number of papers if
you can help it, and blogging about math instead. So take a spoon, pass aroundthe brandy butter and plum pudding, and, for the rest of this post, let us tuck in
to something from the apocrypha.
97.1. Galois Extensions Unramified Away From One Place. I learned about
one version of this question in the tea room at Harvard from Dick Gross. Namely,
does there exist a non-solvable Galois extension K/Q unramified at all primes
except p? Modular forms (even just restricting to the two eigenforms of level one
and weights 12 and 16) provide a positive answer for p greater than 7. On the other
hand, Serre’s conjecture shows that this won’t work for the last three remaining
primes. Dick explained a natural approach for the remaining primes, namely toconsider instead Hilbert modular forms over a totally real cyclotomic extension
ramified at p (once you work out how to actually compute such beasts in practice).
And indeed, this idea was successfully used to find such representations by Lassina
Dembélé in [Dem09] and also this paper (with Greenberg and Voight [DGV11]). But
there is something a little unsatisfactory to me about this, namely, these extensions

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are all ramified at p and ∞. What if one instead asks Gross’ question for a single
place?
Minkowski showed there are no such extensions when v = {∞}, but I don’t see
any obstruction to there being a positive answer for a finite place. The first obvious
remark, however, is that Galois representations coming from Hilbert modular forms
are not going to be so useful in this case at least when the residual characteristic
is odd, for parity reasons.
On the other hand, conjecturally, the Langlands program still has something to
say about this question. One could ask, for example, for the smallest prime p forwhich there exists a Galois representation:
ρ : GQ → GL2 (Fp )whose image is big (say not only irreducible but also not projectively exceptional)
and is unramified at all places away from p including infinity. (This is related to § 2.
Here is how one might go about finding such a representation, assuming the usual
suite of conjectures. First, take an imaginary quadratic field F , and then look to
see if there is any extra mod-p cohomology of GL2 (OF ) in some automorphic local
system which is not coming from any of the “obvious” sources. If you find such aclass, you could then try to do the (computationally difficult) job of computing
Hecke eigenvalues, or alternatively you could do the same computation for a different such imaginary quadratic field E, and see if you find a weight for which there
is an “interesting” class simultaneously for both number fields. If there are no suchclasses for any of the (finitely many) irreducible local systems modulo p, then thereare (conjecturally) no Galois representations of the above form.
There are some heuristics (explained to me by Akshay) which predict that the
number of Galois representations of the shape we are looking for (ignoring twists)
is of the order of 1/p. On the other hand, no such extensions will exist for verysmall p by combining an argument of Tate together with the Odlyzko bounds. So
the number of primes up to X for which there exist such a representation might be
expected to be of the form
log log X − log log C
for some constant C to account for the lack of small primes (which won’t contributeby Tate + Odlyzko GRH discriminant bounds). This is unfortunately a function
well-known to be constant, and in this case, with the irritating correction term,
it looks pretty much like the zero constant. Even worse, the required computation
becomes harder and harder for larger p, since one needs to compute the cohomology
in the corresponding local system of weight (k, k) for k up to (roughly) p. Alas, as
it turns out, these things are quite slippery:
Lemma 97.2. Suppose ρ is absolutely irreducible with Serre level 1 and Serreweight k and is even. Assume all conjectures. Then:(1) The prime p is at least 79.
(2) The weight k is at least 33.
(3) If ρ exists with k ≤ 53, then p ≥ 1000.
(4) If ρ exists with k = 55, then p ≥ 200, or p = 163, and ρ is the uniquerepresentation with projective image A4 .
Of course the extension for p = 163 (which is well-known) does not have big image
in the sense described above. The most annoying thing about this computation

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F. CALEGARI(which is described in the apocrypha) is that it can only be done once! Namely,
someone who could actually program might be able to extend the computation
to (say) p ≤ 200, but the number of extensions which one would expect to see isroughly log log 200 − log log 79, which is smaller than a fifth. So maybe an extensionof this kind will never be found! (Apologies for ruining it by not getting it right thefirst time.)

98. The paramodular conjecture is false for trivial reasons
Mon, 15 Jan 2018
(This is part of a series of occasional posts discussing results and observations
in [BCGP21]).
Brumer and Kramer made a conjecture (see [BK14]) positing a bijection between isogeny classes of abelian surfaces A/Q over the rationals of conductor Nwith EndQ (A) = Z and paramodular Siegel newforms of level N with rational
eigenvalues (up to scalar) that are not Gritsenko lifts (Gritsenko lifts are thoseof Saito–Kurokawa type). This conjecture is closely related to more general conjectures of Langlands, Clozel, etc., but its formulation was made more specificallywith a view towards computability and falsifiability (particularly in relation to thestriking computations of Poor and Yuen), see [PY15].
The recognition that the “optimal level” of the corresponding automorphic forms
is paramodular is one that has proved very useful both computationally and theoretically. Moreover, it is almost certain that something very close to this conjectureis true. However, as literally stated, it turns out that the conjecture is false (thougheasily modifiable). There are a few possible ways in which things could go wrong.
The first is that there are a zoo of cuspidal Siegel forms for GSp(4); it so happensthat the forms of Yoshida, Soudry, and Howe–Piatetski-Shapiro type never haveparamodular eigenforms (as follows from a result of Schmidt), although this depends on the accident that the field Q has odd degree and no unramified quadratic
extensions (and so the conjecture would need to be modified for general totally real
fields). Instead, something else goes wrong. The point is to understand the relationship between motives with Q-coefficients and motives with Q-coefficients which are
invariant under the Galois group (i.e. Brauer obstructions and the motivic Galois
group.)
It might be worth recalling the (proven) Taniyama–Shimura conjecture whichsays there is a bijection between cuspidal eigenforms of weight two with rational
eigenvalues and elliptic curves over the rationals. Why might one expect this to betrue from general principles? Let us imagine we are in a world in which the Fontaine–
Mazur conjecture, the Hodge conjecture, and the standard conjectures are all true.
Now start with a modular eigenform with rational coefficients and level Γ0 (N ).
Certainly, one can attach to this a compatible family of Galois representations:
R = {ρp },ρp : Gal(Q/Q) → GL2 (Qp ).

with the property that the characteristic polynomials Pq (T ) = T 2 − aq T + q of
Frobenius at any prime q not dividing N p have integer coefficients, and the representations are all de Rham with Hodge–Tate weights [0,1]. But what next? Using
the available conjectures, one can show that there must exist a corresponding simple
abelian variety E/Q which gives rise to R. The key to pinning down this abelian

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variety is to consider its endomorphism algebra over the rationals. Because it issimple, it follows that the endomorphism algebra is a central simple algebra D/F
for some number field F. From the fact that the coefficients of the characteristic
polynomial are rational, one can then show that the number field F must be therationals. But the Albert classification puts some strong restrictions on endomorphism rings of abelian varieties, and the conclusion is the following:
Either:(1) E/Q is an elliptic curve.
(2) E/Q is a fake elliptic curve; that is, an abelian surface with endomorphisms
over Q by a quaternion algebra D/Q.
The point is now that the second case can never arise; the usual argument is to notethat there will be an induced action of the quaternion algebra on the homology of
the real points of A, which is impossible since the latter space has dimension two.
(This is related to the non-existence of a general cohomology theory with rationalcoefficients.) In particular, we do expect that such modular forms will give elliptic
curves, and the converse is also true by standard modularity conjectures (theorems
in this case!). A similar argument also works for all totally real fields. On the other
hand, this argument does not work over an imaginary quadratic field (more on
this later). In the same way, starting with a Siegel modular form with rational
eigenvalues whose transfer to GL(4) is cuspidal, one should obtain a compatible
family of irreducible 4-dimensional symplectic representations R with cyclotomic
similitude character. And now one deduces (modulo the standard conjectures andFontaine–Mazur conjecture and the Hodge conjecture) the existence of an abelianvariety A such that:
Either:(1) A/Q is an abelian surface.
(2) A/Q is a fake abelian surface; that is, an abelian fourfold with endomorphisms over Q by a quaternion algebra D/Q.
There is now no reason to suspect that fake abelian surfaces cannot exist. Taking
D to be indefinite, the corresponding Shimura varieties have dimension three, andthey have an abundance of points — at least over totally real fields. But it turns
out there is a very easy construction: take a fake elliptic curve over an imaginary
quadratic field, and then take the restriction of scalars!
You have to be slightly careful here: one natural source of fake elliptic curves
comes from the restriction of certain abelian surfaces of GL(2)-type over Q, andone wants to end up with fourfolds which are simple over Q. Hence one can do thefollowing:
Example 98.1. Let B/Q be an abelian surface of GL(2)-type which acquires
quaternion multiplication over an imaginary quadratic field K, but is not√ poten6) withtially CM.
For
example,thequotient
of
J
(243)
with
coefficient
field
Q(
0
√
K = Q( −3). Take the restriction to K, twist by a sufficiently generic quadratic
character χ, and then induce back to Q. Then the result will be a (provably) modular fake abelian surface whose corresponding Siegel modular form has rational
eigenvalues. Hence the paramodular conjecture is false.
Cremona (in his papers) has discussed a related conjectural correspondence between Bianchi modular forms with rational eigenvalues and elliptic curves over K.
His original formulation of the conjecture predicted the existence of a corresponding

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F. CALEGARI

elliptic curve over K, but one also has to allow for fake elliptic curves as well (as I
think was pointed out in this context by Gross). The original modification of Cremona’s conjecture was to only include (twists of) base changes of abelian surfaces ofGL(2)-type from Q which became fake elliptic curves over K, but there is no reason
to suppose that there do not exist fake elliptic curves which are autochthonous to K,
that is, do not arise after twist by base change. Indeed, autochthonous
fake elliptic√
curves do exist! We wrote down a family of such surfaces over Q( −6), for example.
(We hear through Cremona that Ciaran Schembri, a student of Haluk Sengun, hasalso found such curves) On the other hand, the examples coming from base change
forms from Q have been known in relation to this circle of problems for 30+ years,
and already give (by twisting and base change) immediate counter-examples to theparamodular conjecture, thus the title.
It would still be nice to find fake abelian surfaces over Q (rather than totally real
fields) which are geometrically simple. I’m guessing that (for D/Q ramified onlyat 2 and 3 and a nice choice of auxiliary structure) the corresponding 3-fold may
be rational (one could plausibly prove this via an automorphic form computation),
although that still leaves issues of fields of rationality versus fields of definition. But
let me leave this problem as a challenge for computational number theorists! (The
first place to look would be Jacobians of genus four curves [one might be lucky]
even though the Torelli map is far from surjective in this case.)
Let me finish with one fake counter example. Take any elliptic curve (say of conductor 11). Let L/Q be any Galois extension with Galois group Q, the quaternion
group of order 8. The group Q has an irreducible representation V of dimension 4over the rationals, which preserves a lattice Λ. If you take
A = E 4 = E ⊗Z Λ,
then A is a simple abelian fourfold with an action of an order in D, (now thedefinite Hamilton quaternions) and so gives rise to compatible families R of 4dimensional representations which are self-dual up to twisting by the cyclotomic
character. However, the four dimensional representations are only symplectic withrespect to a similitude character which is the product of the cyclotomic character
and a non-trivial quadratic character of Gal(L/Q), and instead they are orthogonalwith cyclotomic similitude character. So these do not give rise to counterexamples
to the paramodular conjecture. A cursory analysis suggests that the quaternion
algebra associated to a fake abelian surface which gives rise to a symplectic R withcyclotomic similitude character should be indefinite.
Comment 98.2. The title is a reference to this paper.
Notes 98.3. Ciaran Schembri’s paper is [Sch19].

99. The boundaries of Sato–Tate, part I
Mon, 09 Apr 2018
A caveat: the following questions are so obvious that they have surely been asked
elsewhere, and possibly given much more convincing answers. References welcome!
The Sato–Tate conjecture implies that the normalized trace of Frobenius bp ∈[−2, 2] for a non-CM elliptic curve is equidistributed with respect to the pushforwardof the Haar measure of SU(2) under the trace map. This gives a perfectly good

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√ √
account of the behavior of the unnormalized ap ∈ [−2 p, 2 p] over regions which√ √have positive measure, namely, intervals of the form [r p, s p] for distinct multiples√of p.
If one tries to make global conjectures on a finer scale, however, one quickly
runs into difficult conjectures of Lang–Trotter type. For example, given a non-CM
elliptic curve E over Q, if you want to count the number of primes p ≤ X such thatap = 1 (say), an extremely generous interpretation of Sato–Tate would suggest thatprobability that ap = 1 would be
1
√ ,
4π p
and hence the number of such primes ≤ X should be something like:X 1/2,
2π log(X)except one also has to account for the fact that there are congruence obstructions/issues, so one should multiply this factor by a (possibly zero) constant depending one adelic image of the Galois representation. So maybe this does givesomething like Lang–Trotter.
But what happens at the other extreme end of the scale? Around the boundariesof the interval [−2, 2], the Sato–Tate measure converges to zero with exponent one
half. There is a trivial bound ap ≤ t where t2 is the largest square less than 4p. How
often does one have an equality a2p = t2 ? Again, being very rough and ready, thegenerous conjecture would suggest that this happens with probability very roughly
equal to1,
6πp3/4and hence the number of such primes ≤ X should be something like:2X 1/4.
3π log(X)Is it at all reasonable to expect X 1/4±ϵ primes of this form? If one takes the elliptic
curve X0 (11), one finds a2p to be as big as possible for the following primes:√
a2 = −2 ≥ −2 2 = −2.828 . . . ,√a239 = −30 ≥ −2 239 = −30.919 . . . ,√a6127019 = 4950 ≤ 2 p = 4950.563 . . . ,but no more from the first 500000 primes. That’s not completely out of line for theformula above!
Possibly a more sensible thing to do is to simply ignore the Sato–Tate measurecompletely, and model E/Fp by simply choosing a randomly chosen elliptic curve
over Fp . Now one can ask in this setting for the probability that ap is as large as
possible. Very roughly, the number of elliptic curves modulo p up to isomorphism
is of order p, and√ the number with ap = t is going to be approximately the classnumber of Q( −D) where −D = t2 − 4p; perhaps it is even exactly equal to theclass number H(t2 − 4p) for some appropriate definition of the class number. Now
the behaviour of this quantity is going to depend on how close 4p is to a square.
If 4p is very slightly — say O(1) — more than a square, then H(t2 − 4p) is prettymuch a constant, and the expected probability going to be around 1 in p. On the

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other hand, for a generic value of p, the smallest value of t2 − 4p will have order
p1/2 , and then the class group will have approximate size p1/4±ϵ , and so one (moreor less) ends up with a heuristic fairly close to the prediction above (at least in thesense of the main term being around X 1/4±ϵ ).
But why stop there? Let’s push things even closer to the boundary. How small
can a2p − 4p get relative to p? For example, let us restrict to the set S(η) of prime
numbers p such that
S(η) := p p ∈ (n2 , n2 + n2η ) for some n ∈ Z .
√For such primes, the relative probability that ap = ⌊ 4p⌋ = 2n is approximatelynη /p ∼ n2η−1 . So the expected number of primes with this property will be infinite
providing thatX n3ηn2 log(n)is infinite, or, in other words, when η ≥ 1/3. So this leads to the following guess(don’t call it a conjecture!):Question 99.1. Let E/Q be an elliptic curve without CM. Is
lim inflog(a2p − 4p)1= ?
log(p)
3

Of course, one can go crazy with even more outrageous guesses, but let me stophere before saying anything more stupid.

100. Chicago seminar roundup
Sat, 28 Apr 2018
Here are two questions I had about the past two number theory seminars. I
haven’t had the opportunity to think about either of them seriously, so they may
be easy (or more likely stupid).
100.1. Anthony Várilly-Alvarado. Tony gave a talk on his joint work with Dan
Abramovich [AVA17] about the relation between Vojta’s conjecture and the problem of uniform bounds on torsion for abelian varieties. (Spoiler: one implies theother.) More specifically, assuming Vojta’s conjecture, there a universal bound onm (depending only on g and K) beyond which no abelian variety of dimension gover K can have full level structure.
If one wanted to prove this (say) for elliptic curves, and one was allowed touse any conjecture you pleased, you could do the following. Assume that E[m] =µm ⊕Z/mZ for some large integer m. One first observes (by Neron–Ogg–Shafarevich
plus epsilon) that E has to have semi-stable reduction at primes dividing NE . Thenthe discriminant ∆ must be an mth power, and then Szpiro’s Conjecture (which isthe same as the ABC conjecture) implies the desired result.
If you try to do the same thing in higher dimensions, you similarly deduce that A
must have semi-stable reduction at primes dividing NE .One then gets implications
on the structure of the Neron model at these bad primes, which one can hope toparlay in order to get information about local quantities associated to A analogous
to the discriminant being a perfect power. But I’m not sure what generalizations of

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209

Szpiro’s conjecture there are to abelian varieties. A quick search found one formulation attributed to Hindry in terms of Faltings height, but it was not immediately
apparent if one could directly deduce the desired result from this conjecture, nor
what the relationship was with these generalizations to either ABC or to Vojta’s
conjecture.
100.2. Ilya Khayutin. Ilya mentioned Linnik’s theorem that, if one ranges overimaginary quadratic fields in which a fixed small prime is split, the CM j-invariants
become equidistributed. The role of the one fixed prime is to allow one to use
ergodic methods relative to this prime. My naive question during the talk: given p
is split, let p be a prime above p. Now one can take the subgroup of the class groupcorresponding to the powers of p. Do these equidistribute? The speaker’s response
was along the lines that it would probably be quite easy to see this is false, but I
didn’t have time after the talk to follow up. It’s certainly the case that, most of thetime, the prime p will itself generate a subgroup of small index in the class group(the quotient will look like the random class group of a real quadratic field), butsometimes it will be quite large. For example, I guess one can take√
1 + −Dnn−2,
D = 2 − 1,p=2
and the subgroup generated by this prime has order log(D) compared to D1/2+ϵ .
So I decided (well, after writing this line in the blog I decided) to draw a picture forsome choice of Mersenne prime. And then, after thinking a little how to draw thepicture, realized it was unnecessary. The powers of p in this case are given explicitly
by
√


−D
m
m 1+
p = 2 ,,
2
It is transparent that for the first half of these classes, the first factor is much smaller
than the second, but since the second term also has small real part, the ratio alreadylies inside the (standard) fundamental domain. Hence the corresponding points will
lie far into the cusp. Similarly, the second half of the classes are just the inverses in
the class group of the first half, and so will consist of the reflections of those points
in x = 0 and so also be far into the cusp. So I guess the answer to my question is,indeed, a trivial no. So here is a second challenge: suppose that 2 AND 3 both split.
Then do the CM points generated by p for primes above 2 AND 3 equidistribute?
Actually, in this case, it’s not clear off the top of my head that one can easily write
down discriminants for which the index of this group is large. But even if you can,sometimes Z2 subgroups get you much closer to equidistribution than Z!
Comment 100.3 (Bisi Agboola). There’s a generalisation of Szpiro’s conjecture
to jacobians of hyperelliptic curves (due to Paul Lockhart) that might be morealong the lines of what you’re looking for [Loc94].

101. Update on Sato–Tate for abelian surfaces
Thu, 19 Jul 2018
Various people have asked me for an update on the status of the Sato–Tateconjecture for abelian surfaces in light of recent advances in modularity lifting

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theorems. My student Noah Taylor has exactly been undertaking this task, andthis post is a summary of his results. (Which have now appeared in [Tay20].)First, let me recall the previous status of this conjecture. An explicit form of
this conjecture (detailing all the 52 possible different Sato–Tate groups which could
occur for abelian surfaces over number fields — 34 of which occur over Q) was
given in a paper of Fité, Kedlaya, Rotger, and Sutherland (I recommend either
reading these slides or especially watching this video for the background and somefun animations, also see [FKRS12]). Christian Johansson [Joh17] gave proofs ofthis conjecture over totally real fields in many of the possible cases in which theabelian surface had various specific types of extra endomorphisms over the complex
numbers by exploiting modularity results that had been used in the proof of theSato–Tate conjecture for elliptic curves. Over totally real fields, this left essentiallyfour remaining cases:(1) The case when the Galois representations associated to A decomposes overa quadratic extension L/F into two representations which are Galois twistsof each other, and L/F is not totally real.
(2) The case when the Galois representations associated to A decomposes overa quadratic extension L/F into two representations which are not Galois
twists of each other, and L/F is CM.
(3) The case when the Galois representations associated to A decomposes overa quadratic extension L/F into two representations which are not Galois
twists of each other, and L/F is neither totally real nor CM.
(4) The case when the geometric endomorphism ring of A is Z.
Noah has something to say about each of these cases.
Case 1: Noah completed the proof of Sato–Tate in this case using only themethods from the paper [BLGGT14], by exploiting the fact that the corresponding
two-dimensional representations — while possibly only defined over a field L which
need not be totally real or CM — in fact give rise to projective representations which
extend to F . By a theorem of Tate, each of these representations can be extended
to F after twisting by a character, and so the original 4-dimensional representation
looks like the tensor product of a 2-dimensional representation over F (which ispotentially modular) and an Artin representation. At this point one is in good
shape.
Case 2: The Sato–Tate conjecture is proved in this case. This case required
the least amount work, because it is pretty much an immediate consequence of themodularity results proved in [ACC+ 23].
If the totally real field is Q this implies the Sato–Tate conjecture for all abeliansurfaces except those of type (4).
Cases 3 & 4: In these cases, one can apply the potentially modularity results proved in my (very close to being finished) paper with Boxer, Gee, and Pilloni [BCGP21]. It is too much to expect a full proof of Sato–Tate at this point.
However, knowing potential modularity allows one to obtain partial results, similar
to those of Serre and Kim–Shahidi for the case of elliptic curves (after Wiles but
before Clozel–Harris–Taylor). Here is a sample result:
Theorem 101.1 (Noah Taylor). Let C be a genus two curve over a totally real
field F . Then, for any ϵ ≥ 0, there exists a positive density of primes p (withPERSIFLAGE: MATH BLOG POSTS

211

N (p) = p), one has#C(O/p) − p − 1 ≥


2
√
p.
−ϵ
3

Compare this to the Hasse bounds, which imply that the quantity on the LHS√
has absolute value at most 4 p. Of course this theorem is much weaker than the
Sato–Tate conjecture. But even the weaker version of this theorem which says that
#C(Fp ) ≥ p + 1 for infinitely many primes was completely open before such curves
were known to be potentially modular. Similarly, I don’t think one can prove thecorresponding result for elliptic curves without either using something very close
to modularity (in the non-CM case) or the equidistribution theorems of Hecke in
the CM case. I think the following example is instructive: take the elliptic curvey 2 = x3 − x which admits CM by the Gaussian integers. One has a formula for thedifference ap = 1 + p − #E(Fp ) as follows: for a prime which is 1 mod 4, one may
write p = a2 + b2 uniquely in integers by imposing the additional congruence(a + bi) ≡ 1

mod (1 + i)3 .

Then one has the formula ap = 2a. The problem then becomes: do there exist infinitely many primes p ≡ 1 mod 4 such that one has a > 0? This seems suspiciously
like something that can be proven √using Cebotarev, but it is not. The problem isthat the infinite places of F = Q( −1) are all complex, so there is no choice of“conductor” which differentiates between complex numbers with positive or negative
real part at the infinite places in A×F.
Noah’s proof of the theorem above exploits the following idea. Potential modularity not only gives meromorphy of the L-function, but more importantly (in this case)holomorphy and non-vanishing in the (analytically normalized) halfplane ℜ(s) ≥ 1.
Moreover, again using functorialities, potential automorphy, and results of Shahidi,
one obtains similar results not only for the degree 4 L-function, but also the degree 5 L-function, and also crucially the Rankin-Selberg L-functions of degrees 16,
20, and 25. From this one can obtain various “prime number theorem” estimatesfor quantities involving the Frobenius eigenvalues, and then one has to massage
these into an inequality. A simple version of this argument is as follows: given some
infinite set of real numbers an ∈ [−2, 2] such thatn

1X
ai → 0,
n i=1

n

1X 2
a → 1,
n i=1 iOne can draw the conclusion that an ≥ 1/2 − ϵ infinitely often, by (for example)
considering the average of the quantity (2an − 1)(an + 2). Moreover, this is the best
possible bound given these constraints.
Note that since the Sato–Tate conjecture is known in all other cases, one only
has to consider cases (3) and (4), which behave slightly differently in this argument.
In fact, in case (3), one can do much better:
Theorem 101.2 (Noah Taylor). Let C be a curve over a totally real field F suchthat A = Jac(C) is of type (3). Then there exists a positive density of primes p(with N (p) = p), such that√
#C(O/p) − p − 1 ≥ 2.47 p.
(Note that once this result is known in case (3) it is known for all curves whose
Jacobian is not of type (4), that is, those whose Jacobians admit a non-trivial

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endomorphism over C.) The point is that, in this case, one knows not just thepotential automorphy of A, but also the potential automorphy of the corresponding two-dimensional representations over the quadratic extension L, and so one
can also exploit the automorphy of symmetric powers of the corresponding GL(2)automorphic representations (and further analyticity results for higher symmetricpowers) as well as a zoo of Rankin-Selberg L-functions coming from pairs of low
symmetric powers. (As for the constants involved in both of these theorems, they
are essentially optimal given the automorphic input.)
These results tie in to problems raised in various talks of Nick Katz (see for example this talk). Noah’s result above implies that, given an curve C over a totallyreal field, one can tell that it doesn’t have genus one from the distribution of the
traces of Frobenius except possibly in the case when its Jacobian has no non-trivial
geometric endomorphism (the “typical” case, of course). It’s a little sad that themodularity results are not sufficient to handle that last case as well — showing thatthe support of the normalized trace of Frobenius extends beyond [−2, 2] would require knowing something close to functoriality of the map Sym2 : GL(4) → GL(10),and this is currently out of reach, unfortunately. Oh well, that’s a shame: wow I
dearly would have loved to give a talk entitled Simple things that Nick Katz doesn’tknow (but I do).

102. Mazur’s program B on abelian surfaces
Fri, 07 Sep 2018
In the book “More mathematical people,” there is an interview with Robin Wilson
with the following quote:At the meal I found myself sitting next to Alistair Cooke who wasvery charming, and absolutely fascinating to listen to. The very
next Sunday when I was back in England I turned on his “Letterfrom America” on the radio — he started off by saying, “I went
to a very boring dinner at the White House. There was no one
interesting to talk to.” That amused me a lot.
So let me start off by saying that even though this post is about one or two things I
learnt at Oberwolfach, it is deliberately not about anything I learnt in the talks, lestmy choosing some talks over others leading to false inferences on what I thoughtinteresting. For example, the title of this post alludes to David Zureick-Brown’stalk, which I will not mention again.
Let g be a non-negative integer and p a prime. Suppose one starts with a representation
ρ : GQ → GSp2g (Fp )with cyclotomic similitude character. To avoid later circumlocutions, let me (mostof the time) assume it is absolutely irreducible. One can ask whether this representation arises infinitely often from the p-torsion on an abelian variety — perhaps
additionally assuming that it does arise from at least one such variety, or perhaps
not.
This problem is very well-studied in the case g = 1, where we know that theanswer is positive exactly for the primes p = 2, 3, and 5, where the corresponding
moduli space X(p) has genus zero, and the associated twists X(ρ) are Brauer–Severi

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213

varieties that also turn out to be rational over Q under the given hypothesis on
the similitude character. When p ≥ 5, the curves X(p) have genus at least 3, andso their twists always have at most finitely many points over any field, by Faltings.
So there is certainly a satisfactory answer in this case. (Of course, there are many
more subtle versions of this question — for example, replacing “infinitely” by “atleast twice” — and those variations are open in general.)If we move to genus g = 2, then the case of p = 2 is also straightforward —the 2-torsion of the Jacobian of y 2 = f (x) for a degree 6 polynomial with Galois
group G just comes from the isomorphism G ,→ S6 ≃ GSp4 (F2 ). (One needs tobe a little bit careful here because the outer automorphism of S6 means there are
two non-conjugate such maps and one has to choose the right one.) Given that one
can write down families of sextics where the etale Q-algebra Q[x]/f (x) is constant,it’s easy to see that the answer is positive in this case without any restrictions. For
example, given an S6 extension, there’s a six dimensional family of polynomials one
can write down whose splitting field generically gives this extension, and so after
accounting for the action of PGL2 on the roots, this still gives a three-dimensionalrational family of genus two curves whose two torsion comes from this extension.
In my paper [BCGP21] with Boxer, Gee, and Pilloni, we will also give a similarly
conclusive answer for p = 3, although there are some unexpected surprises, as well
as some complementary results recently proved by my student Shiva Chidambaram.
But more on this in a post coming up soon!
When p > 3, then the corresponding 3-folds obtained by taking full level pstructure of the corresponding Siegel 3-fold A2 are of general type. (Note that it isessentially known when Ag (n) is either geometrically rational of general type, see forexample Theorem II.2.1 and the surrounding comments in this paper, see [HS02].)
Of course, unlike the case of curves, varieties of dimension greater than one ofgeneral type can have many rational points. For example, it’s obvious that there aremany abelian surfaces over Q whose 5-torsion has the form (Z/5Z ⊕ µ5 )2 , because
one can take A to be E +E where E is an elliptic curve whose 5-torsion has the form
(Z/5Z ⊕ µ5 ), and there are infinitely many such E because the classical modular
curve of full level 5 is rational over Q. To put it a different way, the 3-fold A2 (5)corresponding to abelian surfaces with fixed 5-torsion will contain a number ofrational Shimura subvarieties coming both from Hilbert modular surfaces and frommodular curves, even though it itself is of general type. This can happen even if the
mod-p representation ρ is irreducible. For example, given an elliptic curve over aquadratic field K/Q, there will once more be a rational curve of elliptic curves withthe same mod-5 representation, and so the restriction of scalars will give a rational
curve on some twist A2 (ρ) of A2 (5). On the other hand, one might at least start
off by making the following naive minimal guess.
Question 102.1. Suppose that ρ is surjective for g = 2 and p ≥ 5. Then are there
only finitely many points on A2 (ρ)?
An even more extreme version of this question would be to ask if there is at
most one such point. This seems a little unlikely even by comparison with the caseof g = 1. I learnt the following nice example talking to John Cremona during the
hike through the Black Forest: for g = 1 and p = 7 and varying E, the twist X(E[7])has genus 3 (it is a twist of the Klein quartic). This twist is still geometrically aplane quartic. By considering the tangent to the point of X(E[7]) corresponding
to E, the line has two further intersections with the curve, and one obtains two

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F. CALEGARI

further points over X(E[7]) which now (in general) lie over a quadratic extension.
But one can parametrize the E for which these points are actually rational andthis turns out to be the rational cover of the j-line corresponding to asking thatthe invariant c4 is a square. So there are infinitely many elliptic curves A (evenwith A[7] surjective) for which there exist at least a pair of non-isogenous elliptic
curves A, B with A[7] = B[7] as symplectic Galois representations. So a better
question is the following:
Question 102.2. Can one find examples of non-isogenous abelian surfaces A and B
with A[5] = B[5] and such that the corresponding representation has a surjective
Galois representation?
This is the type of question where it is useful to have Andrew Sutherland nearby
with a laptop. Within an hour or two, he sent me the following examples (using
the [LMF24])
C1 : y 2 = − 120x6 − 264x5 + 186x4 + 276x3 − 201x2 + 24x
C2 : y 2 = 16x5 − 33x4 + 60x3 − 42x2 + 36x − 9both of conductor 210 · 37 with surjective and isomorphic mod-5 Galois representations which are not isogenous. Nice!
Naturally, the question turned to the existence of a pair with A[7] = B[7]. That
proved a tougher challenge, but not an insurmountable one, and here is such a pair(again found by Andrew the same day):C1 : y 2 + (x3 + x)y = − x6 + 2x4 + 2x3 + 16x2 + 4x + 16
C2 : y 2 + (x2 )y = 14x5 − 44x4 + 46x3 − 23x2 + 12x − 3
this time of conductor 27 · 32 · 74 . Any guesses as to whether there are any suchpairs for p = 11? I’m not sure I have any idea.
102.3. Other news from Oberwolfach. I do appreciate being invited to the
Oberwolfach conference on computational number theory — it pushes me outside
my usual range of interests. It’s also the conference I have attended most often,
now 8 times since 2003, although even that is far fewer than some of the regularparticipants. The conference is also chance to see a bunch of people I pretty muchnever get to see anywhere else. Even better, they are all nice enough to still invite
me after this post. On the other hand, every time I give a talk I think that this is thetime that I finally have something interesting to say to this audience, and it neverquite seems to work out that way. I was certainly convinced that this was going
to be the year, but then during my talk I managed to catch three people asleep inthe front row. To be fair, it was the third last talk of the conference. On the other
hand, Mike Bennett talked directly after me and completely failed to rise to mylevel of soporificness, despite his best efforts and his own predictions he would dootherwise.
There was a lunar eclipse on the final night of our stay. Most of us took to theroof to observe it, but the tall mountains of the Schwarzwald obscured our view
until the final moment. Mike Bennett took the following photo, which he describes
as the “best of a bad bunch”
Notes 102.4. it was an auspicious moon — I found out the very next day thatmy wife was pregnant with our daughter

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215

Figure 11. The distance of the Moon
Notes 102.5. Tom Fischer points out a reference for Cremona’s comment is [HK03,
Remarque 6.2].

103. More or less OPAQUE
Wed, 17 Oct 2018
I recently talked with Lynnelle Ye (a soon to be graduating student of Mark
Kisin) for a few hours about her thesis and related mathematics. In her thesis, she
generalizes (in part) the work Liu-Wan-Xiao [LWX17] on the boundary (halo) of theeigencurve to unitary groups. One of her main results gives a precise asymptotic
growth rate of the Newton Polygon of Up as one moves towards the boundary.
Turning this around, this leads to estimates for the function Nλ (X) which counts
the number of eigenvalues λ of Up (with multiplicity) of valuation at most X.
I have always had a soft spot for counting slopes, although I haven’t really done
anything in this business for many years. It is already interesting to estimate thisgrowth function for classical overconvergent modular forms in the centre of weightspace. Precise estimates were first obtained by Wan in his work on the Gouvea–
Mazur conjectures.
Suppose we fix a tame level Γ, and let X = X(Γ) denote the relevant modular
curve. Then it turns out that, conjecturally at least, that:
Vol(X0 (p))
X.
4π
But this is precisely the growth estimate in Weyl’s law for the Laplacian on X0 (p)!
This suggests an analogy between the spectrum of the compact operator Up in the padic case and the spectrum of the Laplacian operator in the complex case which
was first suggested to me by Don Blasius and which I always hoped but never quitemanaged to extract anything from (see section 5 of these notes, which also containmore precise details about Wan’s results and related results towards the conjecture
above, as well as many further speculations on Overconvergent p-adic Quantum
Unique Ergodicity, if you were wondering about the title).
What growth rate should one expect for the Unitary group U (n)? Lynnelle exploits the fact (as do Liu–Wan–Xiao) that one can work on a compact form of
Nλ (X) ∼?

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F. CALEGARI

the group which is zero dimensional. However, the eigenvariety is (or should be)
essentially the same as the eigenvariety for other forms of the group. Following theanalogy above, we can consider the growth rate of Weyl’s law for U (n−1, 1), which,since the Shimura variety for U (n − 1, 1) has complex dimension n − 1, grows likeX n−1 . However, the exponent in Lynnelle’s work turns out to be
X n(n−1)/2 .
If I understood correctly, this one can even predict (if not prove) by simply counting
the dimension of certain classical spaces of regular algebraic automorphic forms as
one ranges over local systems of appropriate weights (proving it requires more work,of course). However, this seems to spoil the very precise (up to the level of constants)
analogy for the complex dimension n = 1 case above. Is there something one cando to massage these results so they look more similar or was the n = 1 case simply
misleading?

104. Irregular lifts, Part I
Fri, 19 Oct 2018
This post motivated in part by the recent preprint of Fakhruddin, Khare, andPatrikis (see [FKP22]) and also by Matt’s number theory seminar at Chicago thisweek. (If you are interested in knowing what the calendar is for the Chicago number
theory seminar this quarter, then that makes two of us. Actually, if you are giving
a number theory seminar at Chicago this quarter, please leave a comment on thispost with the day you are visiting, because several readers of this blog would beinterested in finding out who is coming and what they are talking about.)Letρ : GQp → GLn (Fp )be a continuous representation. We now know, by the work of Emerton–Gee [EG23],
that this representation admits a lift to characteristic zero representation of regularweight which is de Rham (and is even potentially diagonalizable). On the other
hand, can it be the case that there do not exist any de Rham lifts in non-regularweight?
In the most extreme case, where we demand that all the Hodge–Tate weights arezero, then there are obstructions to lifting. In this case, the image of inertia on any
lift must have finite image, but the image of inertia of ρ may already be sufficiently
large to preclude this possibility. (This was exploited in the proof of Theorem 5.1
here, [Cal12].) So this answers the case when n = 2.
But what happens (for example) for n > 2 and Hodge–Tate weights = [0, . . . , 0, 1]?
Or even n = 2 and replacing Qp by a finite extension K? The first remark is thateven when the residual image lands inside the Borel, there will certainly be obstructions to finding lifts inside the Borel, which means that inductive arguments will
not be sufficient. On the other hand, this definitely smells like a tractable problem.
I offer an Aperol Spritz to an answer to this problem — let me do so even in
the constrained version in weight [0, 0, 1] and K = Qp . To recap:
Problem 104.1. Find a representation ρ : GQp → GL3 (Fp ) such that there isno lift ρ of ρ which is de Rham with Hodge–Tate weights [0, 0, 1] or prove that nosuch ρ exists.

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Comment 104.2 (Pierre Colmez). Some rule of thumb computation suggest thatde Rham liftings of representations of GQp with HT weights 0, 0, 1 come in (nonempty) varieties of dimension 2 (ignoring twists by an unramified character). Since
the mod p representation has a lifting in characteristic 0, the space of such liftings
V should be of dimension 10 (or just 9 if one ignores unramified twists). The theory+of (φ, Γ)-modules (or Fontaine’s extension of Sen’s theory) produces a module Ddifof rank 3 over Fn [[t]] for n big (and Fn is the n-th layer in the cyclotomic tower)with a connexion ∇ (with ∇t = t) and V is de Rham if and only if the connexion
is trivial (after inverting t). So, if all weights are 0, for example, this is equivalent+to ∇ = 0 on DSen = Ddif/t, which gives 9 conditions and explains why de Rham
representations with HT weights 0, 0, 0 are isolated. Now, in the case 0, 0, 1, youneed ∇ to be of trace and rank 1 on DSen which gives you 4 parameters (a line in
P 2 , 3 vectors in this line, and a relation implying that the trace is 1), and each such+∇ produces a plane W on which ∇ is trivial, and a line Kn [[t]]f with ∇f −f ∈ tDdif
2
and the connexion is trivial if and only if ∇f − f has no component in tW/t W ,which gives 2 extra conditions. So you are left with 2-dimensional families (if you
do the same thing with Hodge–Tate weights 0, 1, 2, you end up with 3-dimensionalfamilies). If one wants to turn the above into a rigorous argument, I am afraid thatsome work is needed . . . (One needs facts about the map sending a representation
to its Sen operator which are maybe not in the literature.)
Comment 104.3 (Persiflage). I certainly agree with the numerology here, having
recently done a similar calculation — at a weight corresponding to a dominantcocharacter λ, the stabilizer of this cocharacter inside the Weyl group will be of theform
Sa1 × Sa2 × . . . Sak ⊂ Sn
withai = n giving rise to a Levi subgroup GL(a1 ) × . . . GL(ak ), and then the“expected” dimension of the deformation ring with fixed determinant should bePdim(B(n)) −kX

dim(B(ai ))i=1kn(n + 1) X ai (ai + 1)
−
=
2
2
i=1

=

k
n(n − 1) X ai (ai − 1)
−
,
2
2
i=1

which (for 3 = 2 + 1 = 1 + 1 + 1) gives 0, 2, and 3 respectively. And this number
certainly positive unless one is in trivial weight. What I’m not sure of, however,
if you are arguing that this is a heuristic for the existence of lifts or a strategy
for proving so. The heuristic doesn’t entirely convince me — in part because theheuristic still somewhat suggests there should still be lifts of weight zero (because
0 ≥ −1). If you are saying this is a strategy, then the fact that these Sen maps
are only locally analytic certainly makes me nervous, combined with the fact thatp-adic balls are not projective spaces so if you want to impose A conditions in B
dimensions then you still have work to do when B ≥ A to show there are solutions. . . of course you know this better than me, so you might have a better feeling for
how worried one should be.

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Comment 104.4 (Pierre Colmez). That was more a heuristic than a strategy, but
I agree that your observation that 0 > −1 makes this heuristic somewhat shaky . . .

105. Irregular lifts, Part II
Sun, 28 Oct 2018
This is the global counterpart to § 104. I was going to write this post in a moregeneral setting, but the annoyances of general reductive groups got the better ofme.
Suppose we fix the following:(1) A number field F and a prime p ≥ 2.
(2) A conjugacy class of involutions (possibly trivial) cv of gln for all real placesof F .
Problem 105.1. Does every representation
ρ : GF → GLn (Fp ),with complex conjugation acting on the adjoint by cv for each real place of F
(1) Have a de Rham lift?
(2) Have a de Rham lift of non-regular weight?
I have basically come to the conclusion that the answer to this question is, almost
always, no. Namely, the only time the answer is yes is when F is totally real andall the complex conjugations are totally odd. (With one caveat that comes later.)
Most of the theoretical evidence — slim that it is — is in favour of this minimalist
conjecture. Namely:(1) When n = 2 and F = Q and c is non-scalar, there is a global obstruction
to lifting to a weight one modular form, since the image of such forms is a
finite subgroup of GL2 (C), and this can already be precluded from making
the image of ρ contain a large Borel subgroup.
(2) When n = 2 and p is totally split in F , there are also even local obstructions
to lifting to non-regular weight. (There may be local obstructions in othercases as well, although I’m not sure (see § 104).
(3) When F has a real place, the usual conjectures imply that, when cv is not
the “odd” involution, there are obstructions to lifting to regular weights.
In the extreme case when cv is trivial, all lifts should be of trivial weight,
and then one can prevent this happening by local (or conjecturally global)
reasons similar to those mentioned above.
One can also extend this conjecture to other settings, where one still might
conjecture the answer is always no unless one is in a context (regular weight) wherel0 = 0.
One caveat is that the case of GL(1) doesn’t quite work out. In this case, oddness
is automatic and regularity is automatic, but even when F is not totally real there
still exist lifts. I think this is too degenerate to really be so persuasive.
The first real case of this conjecture is when F is an imaginary CM field, andthen the claim is that there should be representations
ρ : GF → GL2 (Fp ),PERSIFLAGE: MATH BLOG POSTS

219

with no de Rham lifts. To be honest, I don’t have anything intelligent to say about
how to prove this, I merely wanted to put on the record that I think I used tobelieve that such lifts might always exist, and now I’m willing to go on the recordand conjecture that they don’t always exist. And, as I tell my group theory class,half the battle to answering a question in mathematics is determining what you
think the right answer should be!

106. A strange continuity
Sun, 25 Nov 2018
Returning to matters opaque (§ 103), , here is the following problem which may
well now be approachable by known methods. Let me phrase the conjecture in thecase when the prime p = 2 and the level N = 1.
As we know from Buzzard-Kilford (see [BK05]), in every classical weight κ “closeenough” to the boundary of weight space, the slopes of the space of overconvergent
forms are given by the arithmetic progression nt where t depends only on the 2-adic
valuation of κ(5) − 1. Now, for each of these overconvergent forms, one obtains a
Galois representation
ρn : GQ → GL2 (Q2 )for every positive integer n. This gives a map from the integers N considered asa discrete set to Spf(R) for a deformation ring R (there is only one residual representation in this setting. Yes, it is residually reducible, but ignore this for the
moment).
Problem 106.1. Show that this map from N extends to a continuous map fromZ2 .
I’ve never done any computations in these weights, but my spidey senses says
it should be true. Naturally, one should also try to work out the most precise
statement where N and p are now arbitrary.
I don’t have any sense about is whether, for a fixed weight κ, there is actually arepresentation
ρ : GQ → GL2 (O[[T ]])(for some O containing enough roots of unity) whose specialization to T = n for a
non-negative integer n is ρn , or whether the continuity is not so strong. That might
be interesting to check.
More natural questions:(1) Once one has the correct formulation in fixed weight κ, explain what happens over the entire boundary, and at the halo.
(2) I’m pretty sure that ρ0 will just be the Eisenstein series, or more accurately
the Galois representation 1 ⊕ χ, where χ is determined from κ in some easy
way involving normalizations which I don’t want to get wrong. But whatis ρ−1 ? I’m not sure if it is interesting or not. But is there any way ofparametrizing this family of Galois representations so that the potentiallycrystalline points transparently correspond to non-negative integers?
All of this is just to say that, even for N = 1 and p = 2, there’s a lot we don’tknow about the eigencurve over the boundary of weight space.

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Comment 106.2 (Summary). As pointed out in the comments, the paper thispaper in particular [Von18, §3.3] is certainly relevant. (Jan was in my AWS groupwhich touched on these questions)107. Local-global compatibility for imaginary quadratic fields
Thu, 17 Jan 2019
One of the key steps in [ACC 23] is to prove results on local-global compatibility
for Galois representations associated to torsion classes. The results proved in that
paper, unfortunately, fall well-short of the optimal desired local-global compatibility statement, because there are very restrictive conditions on how the relevant
primes interact with the corresponding CM field F/F + . This is not a difficultywhen it comes to modularity lifting providing one can replace F by a solvable CM
extension H/F where all the required hypotheses hold. However, there are certainly
other circumstances where one would like to work with a fixed F without making
such a base change. One particularly interesting case is the case when the maximal totally real subfield F + is the rational numbers, or equivalently when F is
an imaginary quadratic field. There are many reasons to be interested in this casein particular; it relates to classically studied objects (Bianchi groups) and it’s oneof the very few contexts in which we have optimal results about which homology
groups can have interesting torsion (in this case, you only have torsion in degreeone). So how restrictive are the local-global theorems in this case? The answer ispretty restrictive — that is, they never apply directly. If one is happy to restrict
to residual representations, however, then there are cheats in some cases.
For example:+

Lemma 107.1. Let F be an imaginary quadratic field in which p ≥ 3 splits, andsuppose that Γ is a congruence subgroup of GL2 (OF ) of level N prime to p. Letρ : GF → GL2 (Fp )be a semi-simple Galois representation associated to a Hecke eigenclass in H1 (Γ, Fp ).
Assume that the image of this representation contains SL2 (Fp ). Then ρ is finite flatat primes dividing p.
The point is as follows. One wants to apply Theorem 4.5.1 of the 10-author
paper, but not all the conditions are satisfied. First consider the decomposed genericcondition. This is guaranteed (a tedious lemma) by the big image assumption. (Infact, this hypothesis is no doubt much too strong, and possibly — in this setting
where F is an imaginary quadratic field — something close to irreducibility should
be enough, but I don’t really want to bother checking that now.) The more serious
hypothesis in 4.5.1 is that a certain inequality holds for the degrees of various
local extensions at primes dividing p in F . This inequality never holds unlessthere are at least three primes above p, not something that usually happens forimaginary quadratic fields. But it is possible to achieve this via a cyclic extension.
For characteristic zero forms, we can appeal to cyclic base change, but this doesn’t
apply for torsion classes. On the other hand, we see that we can achieve a transfer
of Galois representations in the case of a cyclic extension of degree p, by the main
result of this paper (see [TV16]) (I checked with at least one of the authors thispreserves the property of having level prime to p). We still have to assume that pPERSIFLAGE: MATH BLOG POSTS

221

splits in F because another condition of 4.5.1 is that F contains an imaginary field
in which p splits, and one can’t force this to happen after a cyclic extension H/Fof (odd) degree p unless it was true to begin with. So this hypothesis will always
be required if one wants to use the results of Venkatesh–Treumann in this way.
It’s an intriguing question to ask to what extent this argument could also be
applied to T/I valued representations, where T is the Hecke algebra acting on
mod-p classes and I is some nilpotent ideal with nilpotence of some fixed (absolute)order. This boils down to the corresponding question of how much of T one sees
after the cyclic degree p extension through the Venkatesh–Treumann argument.
I don’t know the answer to this, but possibly a reader will. (Having done that,
there are further tricks available in which one might hope to access the ring T
corresponding to all of H1 (Γ, Zp ) rather than just the p-torsion.)108. Jacquet–Langlands and a new R = T conjecture
Wed, 09 Jan 2019
It is somewhat mysterious how one should formulate the Jacquet–Langlands
correspondence integrally, particularly in the presence of torsion classes. Even theclassical case has many subtleties including for example some results in this paper(see [Rib90]) of Ribet.
In the case of imaginary quadratic fields, Akshay and I (see [CV19]) observed
a number of new pathologies that don’t occur in the classical case. One of theconfusing aspects was how to define a “space of newforms” which might match(in some vague sense) the cohomology of some inner form. I want to discuss herea new conjecture which is very speculative and for which I have absolutely nocomputational evidence. It started off as a troubling example in my mind where
things seemed to go wrong in the setting of my work with Akshay, and this is
the result of me trying to put down those concerns in written form. My guiding
principle is that R = T in every situation, so if this doesn’t seem to work, you haveto find the right definition of R (or T).
Let F be a fixed imaginary quadratic field, say of class number 1, and let P
and W be primes (of residue characteristic different from p). Suppose that
H1 (Γ0 (P ), Zp )m = Zp ,where localization is done with respect to a non-Eisenstein maximal ideal of the
Hecke algebra (assume all Hecke algebras are anemic for now). It can (and does)
totally happen that one might have
H1 (Γ0 (P Q), Zp )m = Z2p ,That is, at level P Q there are two old forms but nothing new either in characteristic
zero or at the torsion level. In this setting, there are apparently no “newforms” oflevel P Q, and so one might predict that, on the quaternionic side ramified at P Q,
there is no cohomology at this maximal ideal. This is certainly true in characteristic
zero by classical Jacquet–Langlands. But it is false integrally! In particular, supposethat the corresponding residual representation
ρ : GF → GL2 (Fp )has the property that the image of Frobenius at Q has eigenvalues with ratio N(Q).
Then one indeed expects a contribution on the non-split side. Akshay and I managed

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to find an interpretation of this result by giving a “better” definition of the spaceof newforms as the cokernel of a transfer map:Φ∨ : H1 (Γ0 (P ), Zp )2m → H1 (Γ0 (P Q), Zp )m ,and this can have interesting torsion even in the context above. In fact, by a versionof Ihara’s Lemma, one can (and we did) compute that the order of the cokernel inthis case will be exactly the order of
Zp /(a2Q − 1 − N (Q))Zp ,and (again in this precise setting) Akshay and I predicted that this should havethe same order as the corresponding localization at the same maximal ideal on the
non-split side. (In the Eisenstein case, this is not true, and one sees contributions
from various K2 groups). We even prove a few theorems which prove results of thisform taking a product over all maximal ideals of the Hecke algebra.
But even in this example, something a little strange can happen. In particular,
I want to argue in this post that there are two natural definitions of theappropriate global deformation ring, and in order to have a consistenttheory, one should consider both of them. To remind ourselves, we now have
two modules, one, defined in terms of the cokernel above, call it M , and then thecohomology localized at the appropriate maximal ideal on the non-split side, which
we call M ′ .
What should we predict about M ? The first prediction is that the image ofthe Hecke algebra should be precisely the universal deformation ring RQ which
records deformations that are Steinberg at Q (and what they should be at theother places). But what does Steinberg at Q even mean for torsion representations?
There are basically two types of guesses for the corresponding local deformation
ring, and correspondingly two guesses for the associated global deformation ring.
(1) A deformation ring defined in terms of characteristic polynomials. In particular, the maximal quotient of RQ which corresponds to classes unramified
at Q is the unramified deformation ring where the characteristic polynomialof FrobQ is (X − 1)(X − N (Q)).
(2) A more restrictive ring in which (on this same unramified quotient) theimage of FrobQ must actually fix a line.
These certainly will have the same points in characteristic zero, but they need nota priori coincide integrally. And this will save us below.
Returning to the corresponding global deformation rings (which should be framed,′
but now ignore the framing), call the corresponding rings RQ and RQ. There is a′surjection from RQ to RQ .
Now we make the following conjecture on the smell of an oily rag:
Conjecture 108.1. The Hecke action on M has image RQ while the Hecke action′
on M ′ has image RQ.
I base this conjecture entirely on the following thought experiment. Let’s suppose
for convenience that N (Q) is not −1 mod p. This implies that aQ is congruent toprecisely one of 1 + N (Q) or its negative — assume the former. Then the “space of
newforms” M as we define it (under all the hypotheses above) will be actually beisomorphic to
Zp /(a2Q − 1 − N (Q))Zp =: Zp /ηZp ,PERSIFLAGE: MATH BLOG POSTS

223

because one of the factors will be a direct summand. (The case when N (Q) =−1 mod p is no problem but one has to break things up more using the Hecke
operator at UQ which I am ignoring.) So the claim in this case is that RQ is′
isomorphic to this ring. What about RQ? Let us consider two possibilities. Note′that if N (Q) ̸= 1 mod p then RQ = RQ , so we are assuming that N (Q) ≡ 1 mod p
in the examples below.
Example 108.2. Suppose that aQ − 1 − N (Q) is exactly divisible by p2 , and that
1 1ρ(FrobQ ) =mod p.
0 1
In this case, the non-split property implies that the corresponding matrix modulo′
p2 will always have 1 as an eigenvalue, so the prediction is that RQ = RQ.
Example 108.3. Suppose that aQ − 1 − N (Q) is exactly divisible by p2 , and that
1 0ρ(FrobQ ) =mod p.
0 1
In this case, the split condition and the assumption that aQ − 1 − N (Q) is exactly
divisible by p2 force the lift to be of the form

1 + appb
ρ(F robQ ) =mod p2 .
0
1 + cpwhere a and c are non-zero. In particular, 1 will never be an eigenvalue. So in this′case, one predicts that RQ = Z/p2 Z but RQ= Z/pZ.
So how do we see this in terms of R = T and Jacquet–Langlands and ourConjecture above? First of all, my paper with Akshay suggests indeed that |M ′ | =|M | = p2 , and certainly M ′ should be an RQ -module. But now the following should
happen:(1) In Example 108.2, we should have multiplicity one, and so M ′ should be′.
free of rank 1 over RQ = RQ
(2) In Example 108.3, we should have multiplicity two, following Ribet (Helm,Cheng, Manning. . . [Hel07, Man21]), since multiplicities should be determined by local conditions, and in particular multiplicities should arise exactly when primes which ramify in the quaternion algebra are split and suchthat the image of the corresponding Frobenius is scalar. Hence M ′ should′be free of rank 2 over RQ
in this case.
′In particular, the Hecke action on M ′ should factor through RQ
in both cases,
and RQ does not act faithfully. Perhaps this conjecture is worth a computation!

Notes 108.4. This post (and the next!) are wrong; one should read Note 109.3
below to see what is going on.

109. Jacquet–Langlands and an old R = T conjecture
Sat, 12 Jan 2019

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Edit: this is still incorrect and there should have been a part 3, but I’ve been
distracted . . . in conversations with Boxer, Emerton, and Gee shortly after thispost, all issues were resolved. Jeff Manning also independently found the correct
formulation. (See Note 109.3 below.)
I feel that I should preface this post with the following psychological remark.
Occasionally you have the germ of an idea at the back of your mind that you sense
is in conflict with your world view. Perhaps you try subconsciously to banish itfrom your mind, or perhaps you are drawn towards it. But inevitably, the idea
breaks through your consciousness and demands to be addressed. The game is now
winner takes all — either you can defeat the challenge to your world view, or you
will be swallowed up by this new idea and emerge a new person. This is how I cameface to face with the non-trivial multiplicities in cohomology for non-split formsof GL(2) over an imaginary quadratic field. Part of me somehow, unconsciously,
worried about the conflict between extra multiplicities on the one hand and, on theother hand, the “numerical” equality between the space of “newforms” on the split
side with the corresponding space on the non-split side (this equality is not known
for each maximal ideal of the Hecke algebra, but rather the “averaged” version over
all maximal ideals is the topic of. Then, earlier this week, I turned my face directly
towards the problem and admitted its existence, which lead to the previous post.
But now . . . there may be a way to defeat the beast after all!
Here is the issue. I talked last time about two types of local framed Steinberg
deformation rings at l = 1 mod p. The first was defined by imposing conditions on
characteristic polynomials, but the second was a more restrictive quotient which
demanded the existence of an eigenvalue which was genuinely equal to 1. This modification seemed to pass some consistency checks, and more importantly resolved
the compatibility issue between having both the equality |M| = |M’| but also having
M be cyclic whilst M ′ was not. Then I went away for a few days and was distracted
by other math, until I flew back to Chicago this evening. While on the plane, I tried
to flesh out the argument a little more by writing down more carefully what thesetwo deformation rings R (and its smaller quotient R′ ) were like. And here’s the
problem. It started to seem as though this quotient R′ didn’t really exist — after
all, demanding the existence of an eigenvector without pinning it down in the residual representation is a dangerous business, and runs into exactly the same issues
one sees when trying to give an integral definition of the ordinary deformation ringfor l = p. Then I thought a little more about the ring R, and it turns out that, forall the natural integral framed deformation rings one writes down, the ring R is a
Cohen–Macaulay normal integral domain! In particular, since R′ has to be of thesame dimension of R, this pretty much forces R to equal R′ . So it seems that mylast post is completely bogus.
So what then is going on? When you have eliminated the impossible, whateverremains, however improbable, must be the truth. It is impossible that R does notequal T, so I can only conclude the improbable — that even when the representation
ρ is unramified at l and the image of Frobenius at l under rhobar is scalar, themultiplicity on the quaternionic side ramified at l will still have multiplicity one(See Note 109.3!). In other words, the local multiplicity behavior will be sensitive
to the archimedean places. This is not what I would (or did) guess, but I cannot see
another way around it. So, at the very least, we should investigate this assumption
more closely.

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225

Let’s talk about two situations where multiplicity two occurs. The first is in the
Jacobian J1 (N p) for mod-p representations which are ramified at p. In this case, thesource of multiplicities is coming from the fact that the local deformation ring R is
Cohen–Macaulay but not Gorenstein. On the other hand, the structure of the Tate
module is well understood to be of the form T⊕Hom(T, Zp ), and so the multiplicity
can (ultimately) be read off from the dialyzing module of R. This is what happens in
my paper with David Geraghty. The second, which is something I should have paid
more attention to last time, is in the work of Jeff Manning (I can’t find a working
link to either the paper or to Jeff!). The setting of Manning’s work is precisely as
above: one has l = 1 mod p and one is looking at the cohomology of an inner formof GL(2)/F . The only difference is that F is totally real and the geometric object isa Shimura curve. The corresponding local deformation ring R — which is basicallythe corresponding ring R above — is Cohen–Macaulay but not Gorenstein. On
the other hand, one doesn’t now know what the structure of the Jacobian is as amodule over the Hecke ring. Manning’s idea is to exploit the fact that, in his setting,the module M is reflexive (and generically of rank one), and then by studying theclass group of R, pin down M exactly. But here is the thing. The reflexivity of M iscoming, ultimately, from the fact that the cohomology group H 1 for Shimura curves
is self-dual. And this is fundamentally not the case for these inner forms for GL(2)over an imaginary quadratic field, where the cohomology is spread between H 1 andH 2 . So this is where the archimedean information can change the structure. At thispoint, I am pretty much obligated to make the following conjecture.
Conjecture 109.1. For inner forms of GL(2) over an imaginary quadratic field,
and for a minimal rhobar which is irreducible and finite flat at primes dividing p ≥2, the multiplicity of rhobar in cohomology is one. Moreover, the corresponding
module M ′ of this cohomology group localized at this maximal ideal is isomorphic(as R-modules and so as Zp -modules) to the space of newforms on the split side, as
defined in the last post.
To put it another way, in Example 108.3 of the previous post, I am now forced
to say that M ′ = Z/p2 Z rather than (Z/pZ)2 .
To reiterate from last time — perhaps this conjecture is worth a computation! I
guess we shall have to wait a few days to see whether there will be a part 3!
Comment 109.2. (There was some discussion between myself and Aurel Page but
in light of the note below there is no need to include it here.)Notes 109.3. WARNING! There is a fundamental confusion going on in these
posts. Imagine a situation where on the split side the space of newforms M was free
and of rank one over R, then M = R and |M | = |R| (imagine that R is finite). Now
let M ′ denote the space of forms on the quarternion algebra side. Let’s supposethat |R| < ∞. The conflict in the past two posts was coming from the following
desideratum:(1) We should have |M | = |M ′ |, following [CV19] (which proves a non-Hecke
equivariant version of this equality).
(2) We should often have, under suitable local conditions, that M ′ has multiplicity > 1 and so M ′ is not free over R.
(3) The Hecke algebra on the quaternion algebra is still R, so M ′ is a faithful Rmodule.

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But the point is that these are not in conflict — it’s certainly possible that a finite
ring R has a module M ′ which is faithful, has |R| = |M ′ |, and yet M ′ is not free.
For example, M ′ = Hom(R, Q/Z) is faithful with |M ′ | = |R| and M ′ is only free
if R is Gorenstein. In particular, the mod-p multiplicities should be determined by
the local behavior as I first believed (see [Man21]). In fact, M ′ will more or less be
self-dual (when it is finite) by the linking pairing.

110. Jean-Marc Fontaine, 1944-2019
Mon, 04 Mar 2019
The results which generate the most buzz in mathematics are usually those which
can be expressed in an elementary (or at least pithy) way to a general mathematicalaudience. It is certainly true that such results may be profound (see Wiles, Andrew),
but this is not always the case. An indirect consequence of this phenomenon is that
there are mathematicians who are considered absolute titans of their own field,
but who are less well-known by the broader mathematical community. Fontaine,
who died this year, might be considered one of these people. Fontaine will forever
be associated with p-adic Hodge theory, a subject which is absolutely central toalgebraic number theory today. While the initial seed of this subject came from
Tate’s paper on p-divisible groups, a huge part of its development was due to
Fontaine over a period of 30 years (both in his solo papers and in his joint work).
The usual audience for my posts is experts, but on the rare chance that someone
who knows less p-adic Hodge theory than me reads this post, let me give the briefest
hint of an introduction to the subject.
For a smooth manifold M, de Rham’s Theorem gives an isomorphismn
HdR(M ) → H n (M, R) = Hn (M, R)∨

which can more naturally be phrased as that the natural pairing between (classesof) closed forms [ω] and (classes of) paths [γ] given byZ⟨[ω], [γ]⟩ =ω
γinduces a perfect pairing on the corresponding (co-)homology groups. The classof paths in homology has a very natural integral basis coming from the paths
themselves. For a general M , the de Rham cohomology has no such basis. On theother hand, if M is (say) the complex points of an algebraic variety over the rational
numbers, then there are more algebraic ways to normalize the various flavours ofdifferential forms. To take an example which doesn’t quite fit into the world ofcompact manifolds, take X to be the projective line minus two points, so M is thecomplex plane minus the origin. There is a particularly nice closed form dz/z on
this space which generates the holomorphic differentials. But now if one pairs therational multiples of this class with the rational multiples of the loop γ around zero,
the pairing does not land in the rational numbers, sinceZdz= 2πi.
γ z
In particular, to compare de Rham cohomology over the rationals with the usual
Betti cohomology over the rationals, one first has to tensor with a bigger ring suchPERSIFLAGE: MATH BLOG POSTS

227

as C, or at least with a ring big enough to see all the integrals which arise in thisform. Such integrals are usually called periods, so in order to have a comparison
theorem between de Rham cohomology and Betti cohomology over Q, one first has
to tensor with a ring of periods.
It is too simplistic to say that p-adic Hodge theory (at least rationally) is a p-adic
version of this story, but that is not the worst cartoon picture to keep in your mind.
Returning to the example above, note that the period is a purely imaginary number.
This is a reflection of the fact that some arithmetic information is still retained,
namely, an action of complex conjugation on the complex points of a variety over
the rationals is compatible (with a suitable twist) with the de Rham pairing. A
fundamental point is that, in the local story, something similar occurs where now
the group Gal(C/R) = Z/2Z generated by complex conjugation is replaced by themuch bigger and more interesting group Gal(Qp /Qp ). Very (very) loosely, this isrelated to the fact that p-adic analysis behaves much better with respect to the
Galois group, for example, the conjugate of an infinite (convergent) sum of p-adic
numbers is the sum of the conjugates. In particular, there is a Galois action on
the ring of all p-adic periods. So now there is a much richer group of symmetries
acting on the entire picture. Moreover, the structure of the p-adic differentials can be
related to how the variety X looks like when reduced modulo-p, because smoothness
in algebraic geometry can naturally be interpreted in terms of differential forms.
So now if one wants to make a p-adic comparison conjecture between (algebraic)de Rham cohomology on the one side, and etale cohomology (the algebraic version of
Betti cohomology) on the other side, one (optimally) wants the comparison theorem
to respect (as much as possible) all the extra structures that exist in the p-adic
world, in particular, the action of the local Galois group on etale cohomology, andthe algebraic structures which exist on de Rham cohomology (the Hodge filtration
and a Frobenius operator), and secondly, involve tensoring with a ring of periods
B which is “as small as possible”.
Identifying the correct mechanisms to pass between de Rham cohomology andetale cohomology in a way that is compatible with all of this extra structure is very
subtle, and one of the fundamental achievements of Fontaine was really to identify
the correct framework in which to phrase the optimal comparison (both in this andalso in many related contexts such as crystalline cohomology). (Of course, his work
was also instrumental in proving many of these comparison theorems as well.) I
think it is fair to say that often the most profound contributions to mathematics
come from revealing the underlying structure of what is going on, even if only
conjecturally. (To take another random example, take Thurston’s insight into thegeometry of 3-manifolds.) Moreover, the reliance of modern arithmetic geometryon these tools can not be overestimated — studying global Galois representations
without p-adic Hodge theory would be like studying abelian extensions of Q without
using ramification groups.
Two further points I would be remiss in not mentioning: One sense in which
the ring BdR is “as small as possible” is the amazing conjecture of Fontaine–Mazur
which captures which global Galois representations should come from motives. Secondly, Fontaine’s work on all local Galois representations in terms of (φ, Γ) modules
which is crucial even in understanding Motivic Galois representations though p-adic
deformations, the fields of norms (with his student Wintenberger, who also sadly
died recently), the proof of weak admissibility implies admissibility (with Colmez,

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another former student, who surprisingly to me only wrote this one joint paper with
Fontaine), and the Fargues-Fontaine curve. (I guess this is more than two.)
Probably the first time I talked with Fontaine was at a conference in Brittany(Roscoff) in 2009. That was the first time I ever gave a talk on my work on even
Galois representations and the Fontaine–Mazur conjecture, about which Fontaine
had some very kind words to say. (One of the most rewarding parts of academia isgetting the respect of people you admire.) I never got to know him too well, due (inequal parts) to my ignorance of the French language and p-adic Hodge theory. But
he was always a regular presence at conferences at Luminy with his distinct senseof humour. Over a long career, his work continued to be original and deep. He will
be greatly missed.
Comment 110.1 (Persiflage). See also this much more substantial (at every level)
remembrance by Colmez and this note in the notices of the AMS.

111. The stable cohomology of SL(Fp )Wed, 19 Jun 2019
Today’s problem is the following: compute the cohomology of SL(Fp ) for a (modp) algebraic representation.
Step 0 is to say what this problem actually is. It makes sense to talk aboutcertain algebraic representations of SLn (Fp ) as n varies (for example, the standard
representation or the adjoint representation, etc.). For such representations, one
can prove stability phenomena for the corresponding cohomology groups. But myquestion is whether one can actually compute these groups concretely.
The simplest case is the representation L = Fp and here one has a complete
answer: these cohomology groups are all zero in higher degree, a computation firstdone by Quillen and which is closely related to the fact that the Kn (Fp ) ⊗ Fp = 0.
Most of the references I have found for cohomology computations of special linear
groups in their natural characteristic consider the case were p is very large compared
to n, but let me remind the reader that we are exactly the opposite situation. One of
the few references is a paper of Evens and Friedlander from the ’80s which computes
some very special cases in order to compute K3 (Z/p2 Z).
Note, however, that p should still be thought of as “large” compared to thepartition which defines the corresponding stable local system(s).
In order to get started, let us make the following assumptions:
ANSATZ 111.1. There exists a space X with a SL(Zp ) pro-cover such that:(1) The corresponding completed cohomology groups with Fp coefficients are
Fp for i = 0 and vanish otherwise.
(2) If L is the mod-p reduction of (an appropriately chosen) lattice in a (added
non-trivial irreducible) algebraic representation of SL(Zp ), then H i (X, L) =0 for i small enough compared to the weight of L.
Some version of this is provable in some situations and it may be generally
true, but let us ignore this√for now. (One explicit example is given by the locallysymmetric space for SL(Z[ −2]) and taking the cover corresponding to a prime pof norm p satisfying certain global conditions.) The point is, this anzatz allows us

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to start making computations. From the first assumption, one deduces by Lazardthat
H i (X(p), Fp ) = ∧i M,
where M is the adjoint representation. But now one has a Hochschild–Serre spectral
sequence:
H i (SL(Fp ), L ⊗ ∧j M ) ⇒ 0.
The point is now that one can now start to unwind this (even knowing nothing
about the differentials) and make some conclusions, for example:(1) H 1 (SL(Fp ), L) = 0.
(2) H 2 (SL(Fp ), L) = H 0 (SL(Fp ), L ⊗ M ).
In particular, the first cohomology always vanishes, and the second cohomology
is non-zero only for the adjoint representation where it is one dimensional. (Onecan see the non-trivial class in H 2 in this case coming from the failure of thetautological representation to lift mod p2 .) Note of course I am not claiming thatthe first cohomology vanishes for all representations, but only the “algebraic” ones,
and even then with p large enough (compared to the weight). Note also that one
has to be careful about the choice of lattices, but that is somehow built into thestability — for n fixed, the dual of M is given by trace zero matrices in Mn (Fp )and so (from the cohomology side) “L = M ” is the correct object to consider rather
than its dual since the dual is not stable even in degree zero. But I think you cansecretly imagine that p is big enough and the weight small enough so that you canchoose n so that all these representations are actually irreducible).
The first question is whether 1 & 2 are known results — I couldn’t find muchliterature on these sort of questions (they are certainly consistent with the very
special cases considered by Evens and Friedlander).
The second question is what about degrees bigger than 2? For H 3 things startgetting a little murkier, but it seems possible that H 3 always vanishes. Beyond that(well even before that) I am just guessing. But one might hope to even come up
with a guess the answer which is consistent with the spectral sequence above.
Notes 111.2. There were quite a few useful and relevant remarks in the comments
by Will Sawin. But I later gave a talk on this problem at the BIRS workshop
Cohomology of Arithmetic Groups: Duality, Stability, and Computations which canbe viewed here. This lead to some conversations with Oscar Randall-Williams, whothen resolved all of these questions in this paper (see [RW22]), see also § 137

112. I asked. . . and you responded!
Mon, 30 Sep 2019
I often ask mathematical questions on this blog that I don’t know how to answer.
Sometimes my smart readers are able to answer the questions I ask. Surely they
deserve some recognition for this? Here are two such occasions which come to mind
(one very recent):
In § 78, I asked whether there are infinitely many integers n such that all the odd
divisors of (n2 + 1) not of the form 1 mod 2m for large enough fixed m, and asked
whether that was an open problem. The answer: it was then, but no longer! It has

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now been answered by Soundararajan and Brüdern in Theorem 4 of this preprint(see [BS21]). Problem solved!
In § 34, I was looking at tables of George Schaeffer at non-liftable weight 1
modular forms of level Γ1 (N ) for various quadratic characters, and noting that
often there were forms with large prime factors. I said:
I said “However, something peculiar happens in the range of thetables, namely, there is not a single example with N prime. This
leads to the (incredibly) vague question: can this be predicted inadvance?”
But later George Boxer pointed out to me that when N is not prime, and thecorresponding quadratic character (in the tables) is not divisible by a prime q, thenthe Galois representation at the auxiliary prime q need not be unramified (it canbe of Steinberg type) and the corresponding Galois representations can have havesignificantly larger root discriminant — the ramification index at those primes is
eq = ℓ for the residue characteristic ℓ rather than eq = 2. And indeed, looking moreclosely at the tables, most of the big primes ℓ for which there exist non-liftable
forms of level (N, χ) occur when the conductor of χ strictly smaller than N .

113. Read my NSF proposal
Wed, 09 Oct 2019
Since this is NSF season, I took the opportunity to go back and look at some of
my old proposals. I am definitely too shy to put my most recent proposal online, but
I thought it might be interesting to share the very first proposal I ever submitted
back in 2006. You can find it here. Honestly, it’s not as bad as I might have imagined.
Here are some first impressions:• The first thing that strikes me is that there is no “results from prior support
section.” In particular, there is a pretty limited discussion of my previous
work. It looks like I don’t even try to name drop my paper with Matt in
Inventiones [CE05] which I been recently accepted before writing this grant;
how virtuous.
• I attribute a theorem to “Taylor” which is really a theorem of Taylor andHarris–Soudry–Taylor. Sorry Michael! (I do reference [HSBT10] later on in
the proposal.)• What is claimed in Theorem 3 is not entirely accurate — this was later
fixed by my student Vlad Serban in this paper (see [Ser22]), see also § 37.
• It’s less than the full 15 pages — Possibly this is an incomplete draft?
• Already in 2006, I had started thinking about the modularity of elliptic
curves over imaginary quadratic fields. Many ideas are missing. There is
at least one reasonable idea here, however, namely, that if one can provethat the “half” Hida families (taking limits for one prime above p = ππ ′but not the other) are flat over Zp ), then one is effectively in an ℓ0 = 0
situation. Of course, even today, nobody has any idea how to prove thisflatness. The problem is that one can sometimes show that it is pure ofco-dimension one over the Iwasawa ring, but then one has to deal with aµ-invariant type question proving that the support over Λ does not contain(p). George Boxer and I occasionally discussed whether it was reasonablePERSIFLAGE: MATH BLOG POSTS

231

even to conjecture this. I think I am more bullish that it should always beflat, but the question remains open.
• Using poles of (as yet unconstructed) p-adic L-functions to prove lifting criteria from smaller groups is a great idea! I’m sure I discussed this with Matt.
If you don’t want to find it in the PDF, here is the basic idea. Given an automorphic form π, Langlands explains how (morally) to determine whether
it arises via functoriality from a smaller group by considering L(π, ρ, s) for
every representation ρ and determining the order of vanishing (or the orderof poles) of this L-function at s = 1. This is the automorphic analog of thegroup theoretic fact that one can determine a representation V of a group Gby knowing not only the dimension of the invariant subspace of V but alsoof S(V ) for every Schur functor applied to V . Actually, it’s more than justan analogy, since both are just consequences of the Tannakian formalism(which only conjecturally applies to automorphic forms). For example, acompletely concrete example of this is that a cuspidal π for GSp(4) shouldarise as an induction from GL(2)/F for a quadratic extension F if and only
if L(π ⊗ χ, ρ, s) has a pole at s = 1 where ρ is the standard 5-dimensional
representation and χ is the quadratic character of F . I believe this is evena theorem in this case. The point made in the proposal is that this formalism should apply equally to ordinary Siegel modular forms of non-classicalweight, where the consequence of course is the weaker claim that π comesvia induction from a non-classical ordinary form ϖ for GL(2). Here is a niceexample which suggests that this picture is consistent. Start with a classicalordinary ϖ for GL(2) over an imaginary quadratic field (with some Galois
invariance condition on the central character). After inducing, we obtainan ordinary Siegel modular form π such that L(π ⊗ χ, ρ, s) has a pole. Thisshould also be true more or less for the p-adic L-function, defined correctly.
But now as we vary π over the ordinary family, the locus where the p-adic
L-function has a pole should have codimension one. Thus the philosophy
predicts a one-dimensional family of ordinary deformations of ϖ. And thisis indeed something that Hida proved. But everything we know strongly
suggests that this will be a non-classical family in general, so this lifting
criterion is something that is really completely different from the classicalanalog. It also suggests and even partially implies corresponding results forlifting torsion classes as well. I think that this project is definitely somethingworth pursuing, but I’ve never learnt enough about p-adic L-functions to doso. Whenever I have talked to someone who has constructed such functions,they are always working in some context where normalizations have been
made to ensure that the L-functions are Iwasawa functions and certainly
don’t have poles. Anyway, I think this remains the most attractive open
problem in this proposal.
• Question 2 has been answered (and much more) by Ian Agol [Ago13]. Agol(et. al.) pretty much put an end to the cottage industry of using number theory to answer various special cases of these Thurston conjectures.
Interesting problems still remain, of course.
• I haven’t had anything really interesting to say about the geometry of the
Eigencurve since writing this proposal. But Hansheng Diao and Ruochuan
Liu did end up proving that the Eigencurve is indeed proper in this paper.

232

F. CALEGARI• I redacted some stuff! There’s an idea in this proposal that I might want togive to a graduate student — so I blacked it out (no peeking using secrettechnologies)• The broader impact section suffers from the fact that this was my first year
as a tenure track assistant professor. But the panel understands that there
is only so much you can do at this point. The more senior you are, the more
you should be doing.
In the end, I think this was not a bad proposal from a young researcher. There
are some good ideas and some good problems. Probably the part on the geometry of
the eigencurve is the weakest bit, and that is not unrelated to the fact that I stopped
thinking about these types of questions. I think I accomplished less of what I set out
to do than for some of my more recent proposals. This is not entirely surprising fromlooking at the proposal — a (forgivable) weakness is that it’s somewhat speculativeand optimistic. What did I end up doing instead? Probably my most interesting
result in the next cycle was my result with Matt on bounds for spaces of tempered
automorphic forms using completed cohomology. This proposal was (in the end)
funded — I think I certainly must have benefited from the fact that panels look
generously on proposals from people within 5 years (or is it six?) from their PhD(“early career researchers”).
Comment 113.1 (Will Sawin). Thanks for posting this! Are you sure that the“fact that one can determine a representation V of a group G by knowing not only
the dimension of the invariant subspace of V but also of S(V ) for every Schurfunctor applied to V is, in fact, a fact?
My understanding that this is not true, for any reasonable interpretation of it
as a precise mathematical statement, and this poses a difficulty to Langlands’ program to develop a Tannakian theory of non-algebraic automorphic forms (though
not a difficulty anyone will have to deal with anytime soon, as there are many morepressing issues, like various basic cases of functoriality). For instance I believe that
there are two distinct groups G1 , G2 such that for every (!) irreducible representation of G1 , there is a corresponding irreducible representations of G2 , such thatthe invariants of all Schur functors on the two representations are equal.
I learned (maybe a slightly different version of) this non-fact from PS (who hasn’t
commented on this blog).
I said: Ha! I knew when writing these words I was being sloppy and that what I
was saying might not literally be true. (I should have made it more vague to coversuch a possibility.) But fortunately I have readers to keep me honest!
I do at least know that if V is a n-dimensional irreducible representation andV ⊗ V contains at least two one-dimensional summands, then V is induced. (Forexample, if V preserves a generalized symplectic form of dimension 4 and thecorresponding 5-dimensional representation inside ∧2 V contains an invariant onedimensional summand.) This is because the assumptions imply that Hom(V, V )contains (at least) two one-dimensional summands. By Schur’s Lemma, at mostone of these one-dimensional summands can be trivial, and thus V ≃ V ⊗ χ forsome non-trivial χ, which then implies V is induced.

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114. A homework exercise for Oaxaca
Sat, 12 Oct 2019
Here’s a homework problem for those coming to Oaxaca who have a facility forworking with Breuil-Kisin modules and finite flat group schemes. Let F be a finite
field of characteristic p, and consider a Galois representation:
ρ : GQp → GL2 (F).
which (one should imagine) is the local restriction of a global representation comingfrom a modular form. By a standard global argument, one can find a congruent
form in weight 2, and thus a lift to a representation which is de Rham with Hodge–
Tate weights [0, 1]. For almost all such representations one can ensure that lift ispotentially crystalline and hence comes from a representation which is potentially
Barsotti-Tate. An immediate consequence is that the representation ρ itself is –after restriction to some finite extension K – the generic fibre of a finite flat groupscheme. Without any other conditions this is obvious, since one can take K to be
the splitting field of ρ. However, the global argument gives a further restrictionthat one can take K/Qp Galois with the property that, for some 2-dimensional
representation VK lifting the restriction of ρ to GK , there is a representation:ϱ : Γ := Gal(K/Qp ) → GL(Dcris (VK )) → GL2 (Qp )which is faithful on the inertia subgroup. In particular, this forces Γ and K to be“small” in some sense. One can prove this result directly without recourse to any
global arguments. For example, consider the case when ρ is reducible, and, if theratio of characters is cyclotomic, then additionally assume the extension is not très
ramifée. In this case, I claim that one can take K to be the (unramified extension)of Qp (ζp ) which contains the fixed field of the characters on the diagonal. The
restriction of ρ to K is then the extension of the generic fibre of the trivial groupscheme by the multiplicative group scheme. But our assumptions imply that theKummer extension that arises will come from the pth power of a unit and hence
come from a finite flat group scheme over K. The (abelian) group Gal(K/Qp ) has noproblem admitting a representation of small dimension which is faithful on inertia.
When n = 3, the automorphic picture would suggest that one can find de Rham
lifts with Hodge–Tate weights [0, 1, 2], and this is the type of thing that I guess
one knows now in full generality by Emerton–Gee (but probably earlier in thiscase). But suppose we are still interested in whether there exist lifts of ρ which
are potentially Barsotti-Tate. We can ask the weaker question: does ρ come (after
restriction to K) from the generic fibre of a finite flat group scheme for a Galois
extension K/Qp which admits a representation:ϱ : Γ := Gal(K/Qp ) → GL3 (Qp )which is faithful on inertia? This seems like a question which one should be able toanswer. In particular, suppose that ρ is some representation with upper-triangularimage. It seems possible that if K/Qp is any extension such that ρ is the genericfibre of a finite flat group scheme over K then K might be “too big” to admit
such a ϱ. If that were true, this would give a direct proof that ρ does not admit
lift which are potentially crystalline with Hodge–Tate weights [0, 0, 1], which would(essentially) answer the final question in this post. (I say “essentially” because one
should also consider potentially semistable lifts as well. Certainly one should be234

F. CALEGARI

able to address this by similar methods, but for now, perhaps just assume that theratio of any two consecutive characters occurring in ρ is not cyclotomic.)
This seems to be an eminently answerable question to someone who knows what
they are doing, and there are certainly some experts in this sort of computation
who will be in Oaxaca in a few weeks time. So maybe one of you can work out theanswer (calling the Hawk!).
Comment 114.1 (Anonymous). “the automorphic picture would suggest” —could
you clarify how does it suggest that?
Comment 114.2 (Persiflage). via the standard argument: (Globalize to some cuspidal essentially self-dual situation; Generalizations of Serre’s conjecture then predicta corresponding eigenclass in H ∗ (X, V ) for some mod-p representation and somesuitable X, which, after increasing the level, gives a class in H ∗ (X, Fp ), which in
non-Eisenstein situations should be concentrated in a single degree and so give riseto an appropriate characteristic zero class. You can even take X to be zero dimensional. (I could have given more or less detail but when requests for clarification
are anonymous it is hard to respond appropriately.)Notes 114.3. The homework was too hard, as usual. These problems seem beyond
current technology.

115. Appropriate citations
Sun, 27 Oct 2019
Once I wrote a paper (two, in fact) on even Galois representations. The second
paper in particular (see [Cal12]) proved what I thought was a fairly definitive result
ruling out the existence of a wide class of even de Rham representations with distinct
Hodge–Tate weights. It turns out that almost nobody seems to cite these results,
probably because they aren’t particularly useful — at least in any obvious sense.
On the other hand, almost everyone who does cite the paper seems to cite it for a
specific proposition (3.2) which is an easy consequence of the results of Moret-Bailly.
The proposition, more or less, is a potential inverse Galois problem with (any finite
collection) of local conditions. The main application of such a proposition (both inmy paper and in papers which cite it) is that, given a local mod-p representation
which looks like it could come (say) from the localization of a global representation
associated to an automorphic form, the proposition often allows one to produce
such a form at the cost of making a finite totally real extension in which p splits
completely. This suffices for many purposes.
It turns out, however, that the lemma (pretty much in an equivalent form byan equivalent argument) was already proved by Moret-Bailly himself in this paper(see [MB90]). This means that if you cite my paper for this particular lemma, you
should definitely cite the paper of Moret-Bailly. Of course, if you are also applying
it in a context similar to my paper (say in order to construct automorphic forms
with certain local properties), you should certainly feel free to continue to cite mypaper as well.

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116. En Passant VI
Tue, 29 Oct 2019
I just learnt (from a comment on this blog) that Pierre Colmez hosts a wonderful
page on Fontaine and Wintenberger here. I particularly recommend reading boththe personal recollections of their friends and collaborators (sample quote from
Mark: These p-adic Hodge theorists seemed to me like an order of monks, who wereable to reveal the hidden design of a tapestry by examining it one thread at a time),
as well as this article by Colmez which gives a beautiful introduction to Fontaine’s
work (rather than my own somewhat superficial summary).
One can’t mention the early work of Fontaine in p-adic Hodge theory without
also mentioning the recent passing of John Tate (my mathematical grandfather).
Tate’s enormous contributions to mathematics are very well-known by readers of
this blog, many of whom certainly knew him personally much better than me. I
first met him at the 2000 Arizona Winter School, where there was an impromptu
celebration for his 75th birthday. We crossed paths a few times since then, chatting
about a number of things from p-adic modular forms to smoked trout (his wife
made a particularly tasty version of the latter for some Harvard holiday party). I
last saw him at the banquet for Barry’s 80th birthday when he called out my namein a friendly way to say hello, and I felt the flutter of satisfaction that comes when
one of your idols remembers who you are. Instead of trying to write a summary of
his work, however, let me instead recommend (again) that you purchase for yourself
a copy of the Serre–Tate correspondence ([Col15a, Col15b]), also § 83.

117. New results in modularity, Christmas update II
Mon, 30 Dec 2019
Just like last year, once again saint Nick has brought us a bounty of treasuresrelated to Galois representations and automorphic forms in the final week of theyear.
First there was this paper by Newton and Thorne [NT21a, NT21b], proving,
among other things, the modularity of symmetric powers for a large range of holomorphic modular forms, including ∆ and any newform associated to a semistable
elliptic curve. There is a lot to enjoy about this paper, not least of which is the niceapplication of an old computation of Buzzard and Kilford. But there are also somevery nice new results on Selmer groups and reducible modularity lifting provedin the substantial related papers by Newton–Thorne [NT23] and Allen-Newton–
Thorne respectively [ANT20]. (Also this paper by Thorne and Christos Anastassiades as well [AT22]) It’s often hard for the non-specialist to appreciate “technical”
improvements on previous theorems, but in this case, they are all wrapped up neatly
with a bow by such a clean application: Symn (∆) is modular!
Moving on, we have this paper (monograph?) by Liu, Tian, Xiao, Zhang, andZhu [LTX+ 22] on the Bloch-Kato conjecture for a very general class of motives
associated to Rankin-Selberg convolutions of forms on GLn and GLn−1 . I remembera few years ago talking to Yifeng during his interview at Northwestern (reader, we
hired him) about this beautiful paper, giving a totally new argument to study
questions of Selmer groups using cycles and level raising congruences. The current
paper seems to be not only a version of that on steroids but also with a nice hot cup

236

F. CALEGARIof tea with 3 lumps of potassium. It’s an amazing achievement which pulls together
a lot of wonderful ideas, including Xiao–Zhu’s work on the Tate conjecture, not tomention all the previous work on the Gan–Gross–Prasad conjecture.
Well done to both groups of authors!
(In different times I would have given more details as to what these papers
actually do, but as my free time nowadays consists of brief moments like this at
5:00AM in the morning you will have to forgive me, and anyway, these papers allseem to be very well written with nice introductions. That said, there will be somemore technical mathematics posts coming up, not least of which relates to work of
my own students. Stay tuned, Persiflage intends to keep posting!)
Comment 117.1 (David Loeffler). Happy new year from me as well. Just a remark
on the paper by Liu et al: the “level-raising congruences” approach to bounding
Selmer groups didn’t emerge entirely from nowhere in Liu’s 2015 paper – it isrecognisably a generalisation of earlier work of Bertolini and Darmon [BD05], which
does a version of this for Heegner points on modular curves.
Liu was the first to make this idea work in a higher-dimensional case, and now
he and his coauthors have pushed this far beyond anything that one could havedreamed possible in 2005 or even in 2015 – it’s a magnificent piece of work.

118. The last seven words of Kedlaya–Medvedovsky
Tue, 14 Jan 2020
New paper by my student Noah Taylor! (see [Tay22]) It addresses some conjectures raised by Kedlaya and Medvedovsky in this paper [KM19]. Let T denote the
Hecke algebra acting on modular forms of weight two and prime level N generated
by Hecke operators Tp for p prime to N and 2 (the so-called “anemic” Hecke algebra). If m is a maximal ideal of T of residue characteristic two, and T/m = k, there
exists a corresponding Galois representation:
ρ : GQ → GL2 (T/m) = GL2 (k).
If S denotes the space of modular forms modulo 2, then certainly dimk (S[m]) ≥ 1.
Since there can exist congruences between modular forms, it is certainly possiblethat the generalized m-eigenspace of S has dimension greater than one. Kedlaya andMedvedovsky observe that if one assumes that ρ has (projectively) dihedral image,then one can systematically predict lower bounds for this generalized eigenspacecontingent on various properties of ρ. They prove a number of such results, butthey finish the paper with what amounts to six more conjectures. Actually, one of
the conjectures splits into two completely different cases, and so I like to think of
it as seven conjectures.
Before stating the conjectures, first√ note that ρ (when projectively dihedral) is
necessarily induced from the field Q( ±N ). The corresponding representation may
or may not be ordinary
at the prime 2. Also, let h(N ) denote the even part of the√class number of Q( N ). Now we can state the conjectures, which are now all provedby Noah:
Theorem 118.1 (Noah Taylor).
√(1) Suppose that N ≡ 1 (mod 8). If m is Q( N )-dihedral, then the generalized
m-eigenspace has dimension at least 4.

PERSIFLAGE: MATH BLOG POSTS

237√(2) Suppose that N ≡ 1 (mod 8). If m is Q( −N )-dihedral, then the generalized m-eigenspace has dimension at least h(−N ).
(3) Suppose that m is Eisenstein. Then the generalized m-eigenspace has dimension at least (h(−N ) − 2)/2.
√(4) Suppose that N ≡ 5 (mod 8). If m is ordinary-Q( N )-dihedral, then thegeneralized m-eigenspace has dimension at√ least 4.
(5) Suppose that N ≡ 5 (mod 8). If m is Q( −N )-dihedral, then the generalized m-eigenspace has dimension at least√2.
(6) Suppose that N ≡ 3 (mod 4). If m is Q( N )-dihedral, then the generalized
m-eigenspace has dimension at least 2.
√(7) Suppose that N ≡ 3 (mod 4). If m is ordinary-Q( −N )-dihedral, then thegeneralized m-eigenspace has dimension at least 2.
Noah uses quite a number of different arguments to prove this theorem. One
basic idea is that the extra dimensions are related to deformations of ρ, but only
in some of the proofs is this connection transparent. More directly, Noah exploits
the following:(1) The existence of weight one dihedral representations. When ρ is unramified
at 2 it is natural to look to such forms. However, even when ρ is ramified
at two, the weight one forms, after giving rise via congruences to weight
two forms, can often be level-lowered to level N using an argument similar
to that employed by me and Matt in our paper on the modular degree of
elliptic curves.
(2) Known properties of the real points of the Jacobian J0 (N ), in particular
the connectedness of J0 (N )(R) for prime N as proved by Merel. This canbe used to give a lower bound of 2 when ρ is totally real. In order to get abetter bound in the even case (if necessary) one has to combine this withother arguments.
(3) The difference between the Hecke algebra T and the Hecke algebra where
the operator T2 is also included. If this Hecke algebra is strictly larger than
T after localization at m, then one can show that the m-torsion of S has
to be at least two, and moreover one can make this argument work nicely
with some of the other methods for producing non-trivial lower bounds.
Concerning the third point: the difference between the Hecke algebra T and thefull Hecke algebra is the addition of the operators T2 and TN . Noah’s arguments
crucially use this in the case of T2 but not of TN . But this is also explained in
the paper: once you add the Hecke operator T2 , it turns out that you have the full
Hecke algebra! The fact that the Hecke algebra is integrally generated by Tp for p
prime to the level is not true for general levels N but just happens to be true for N
prime. It suffices to prove the result after localizing at any maximal ideal m. Mazur
proved it in the Eisenstein case by a somewhat subtle argument (it’s false in generalfor Eisenstein primes at non-prime level). Second, in the non-Eisenstein case, theargument uses the result that all irreducible representations modulo 2 are ramified
at N . If there were such a representation, it would be an absolutely irreducible andunramified away from 2, and Tate prove that no such representations exist!
Of course, apropos of the title, this post must finish with the following:
The last seven words of Christ

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119. Vesselin Dimitrov on Schinzel–ZassenhausMon, 10 Feb 2020
Suppose that P (x) ∈ Z[x] is a monic polynomial. A well-known argument of
Kronecker proves that if every complex root of P (x) has absolute value at most 1,
then P (x) is cyclotomic. It trivially follows that, for a non-cyclotomic polynomial,
the largest root α in absolute value satisfies |α| > 1. Elementary considerations
imply that this can be improved to|α| > 1 + cn
for some real constant cn > 0 that only depends on the degree. What is the true
rate of decay of this parameter as the degree increases? By considering the example
xn − 2, the best one can hope for is that cn can be taken to have the form c/n
for some constant c. This is exactly what is predicted by the Schinzel–Zassenhaus
conjecture:
Conjecture 119.1 (Schinzel–Zassenhaus). there is an absolute constant c and aboundc|α| > 1 +n
for the largest root of all non-cyclotomic polynomials.
In fact, Schinzel–Zassenhaus don’t actually make this conjecture. Rather, theyfirst prove a bound where cn has the form 2−n up to a constant, and then go on
to say that they “cannot disprove” the claim. And of course, this then gets turned
into a conjecture named after them! The best bounds were rapidly improved fromexponential to something much better, but the original conjecture remained open.
That is, until Vesselin Dimitrov in this paper [Dim19] proved the following:
Theorem 119.2 (Vesselin Dimitrov). The Schinzel-Zassenhaus conjecture is true.
Vesselin’s result is completely explicit, and gives the effective bound |α| ≥ 21/4n ,orlog(2).
4n
The actual proof is very short. Step 0 is to assume the polynomial is reciprocal,
which is a quite reasonable assumption because the conjecture (and much more,
including Lehmer’s conjecture) was already known by work of Smyth the nonreciprocal case (see [Smy71]). I’m not sure this step is even needed, since the conjecture is certainly true for polynomials whose constant term is not plus or minus
one, and so one can simply replace the polynomial by the reciprocal polynomial inwhat comes below. Step 1 is to show the inclusionqY(1 − αi2 /X)(1 − αi4 /X) ∈ Z[[1/X]].
cn = 21/4n − 1 ∼The argument here is elementary (the only prime to worry about is p = 2). If theoriginal polynomial is cyclotomic, then this squareroot is actually a polynomial,
but otherwise it is a power series which is not rational. But now one has a power
series which has an analytical continuation outside a very specific region in theplane, namely the “hedgehog” (I would have called it a spider) consisting of raysfrom 0 to α4 and α2 in C. These rays may overlap, but that only improves the final
bound. The complement of the Hedgehog is a simply connected region, and know
one wants to say that any power series with integer coefficients that has an analytic

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continuation to such a region with sufficiently large transfinite diameter has to berational. Step 2 is then to note that such theorems exist! The transfinite diameterof the region in question can be computed from results already computed in theliterature, and the consequent bounds are enough to prove the main theorem, all in
no more than a couple of pages! It is a very nice argument indeed. As a comparison,
to orient the reader not familiar with Bertrandias’ theorem (which is used to deduce
rationality of the power series in question), it might be useful to give the following
elementary variation. Suppose that instead of the hedgehog, one instead took thecomplement of the entire disc or radius r for some r < 1. (Importantly, this does
not contain the hedgehog above which has spikes outside the unit circle.) Replacing
X by 1/X, one ends up with a power series on the complement which is a disc ofradius greater than one. Now one can apply the following
Theorem 119.3 (Trivial Theorem). A power series in Z[X] with radius of convergence bigger than one is a polynomial.
Plugging this into Dimitrov’s setup, one deduces a new proof of Kronecker’s
theorem! So the main technical point is that the “trivial theorem” above can be
replaced by a more sophisticated version (to due Bertrandias and many others)
where the region of analytic continuation can be taken to be something other thana disc. (For an exposition of some of these rationalization/algebraization theorems,
a good point to start is this post of Matt Baker.)

120. Counting solutions to ap = λ, Part II
Tue, 03 Mar 2020
This is a sequel to § 72, where the problem of counting eigenforms with ap = λand λ ̸= 0 was considered. Here we report on recent progress in the case λ = 0.
It is a somewhat notorious conjecture attributed to Lehmer (who merely asked
the question, naturally) that the coefficients of
∆=q

∞
Y

(1 − q n )24 =Xτ (n)q n = q − 24q 2 + 252q 3 + . . .

n=1

never vanish. One problem with this conjecture is that there really isn’t any compelling reason it should be true except (basically) on probability grounds given thegrowth of the coefficients. As with a number of problems concerning “horizontal”
questions about modular eigenforms (fixing the form and varying the prime p), it
is often easier to consider the analogous “vertical” question where one fixes p andvaries the weight. Namely: fix a tame level, say Γ = Γ1 (N ), fix a p prime to thelevel, and then consider the eigenforms of level Γ and varying weight with ap = 0.
Unlike in the case of Lehmer’s conjecture, is certainly can happen that ap = 0, forexample:(1) If f is associated to a modular elliptic curve E with supersingular reduction
at p ≥ 5.
(2) If f has CM by an imaginary quadratic field K in which p is inert.
Let S(X) denote the number of cuspforms of level Γ and weight ≤ X such thatap = 0. Consider bounds on this function. The trivial bound, given by counting
all cuspforms, is S(X) ≪ X 2 . If you try to improve this bound using analytic

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techniques, namely via the trace formula, you can only do very slightly better, say
S(X) ≪ X 2 / log(X). The problem is that the condition ap = 0 within the spaceof all spherical representations πp which could possibly occur has measure zero, so
any trace formula approach will have to use a test function where ap has support insome non-trivial interval [−ε, ε] depending on the weight. This is the same problem(more or less) which prevents the analytic approach from giving optimal bounds tothe number of weight one modular forms (where now the measure zero condition
is being imposed at the infinite prime instead of at the prime p). One approach to
improving these bounds is to use non-commutative Iwasawa theoretic methods as
employed in my paper with Matt and then used by Simon Marshall to give the first
non-trivial bounds for spaces of modular forms for GL(2) over imaginary quadratic
fields of fixed level and varying weight. This approach should lead, in principle, toa power saving over the trivial bound.
On the other hand. the best possible bound on S(X) will have the shape S(X) ≪X, because the number of CM forms of each weight will be bounded independentlyof the weight, and there is no reason to imagine that the other exceptions willcontribute anything of this order. Indeed, in § 72, I conjectured that there should
only be finitely many such forms of fixed level which are not CM as the weightvaries.
When I last visited Madison in 2018, Naser Sardari was working on this problem, and in a preprint from late 2018, he proved exactly a bound of the optimal
shape S(X) ≪ X, with the slight caveat that one should restrict to even weights.
Quomodocumque blogged about it here.
Just a few weeks ago, Naser was in town in Chicago, and we got to talking
about this problem again. Happily, we were able to come up with one more extra
ingredient to push the original result to a (close to) optimal conclusion, and prove
the aforementioned conjecture:
Theorem 120.1 (C, Sardari, [CTS21]). Fix a prime p > 2 and a tame level Γ1 (N ).
Then there are only finitely many eigenforms of level Γ and even weight with ap = 0
which are not CM.
This establishes a vertical version of Lehmer’s conjecture, up to a congruence on
the weight, which arises for a technical reason discussed more below.
The first main idea of the proof is as follows. The p-adic Galois representation
associated to ρf for a modular form can be very complicated viewed as a representation of the Galois group of Qp . However, if ap = 0, then the local representation
has a very special form: it is induced from an unramified extension K/Qp . Breuil
gave a precise formula for the representation, but a fairly soft argument shows that
it is induced – 2-dimensional irreducible crystalline representations over Qp are determined by ap , and twisting by an unramified character fixes both the determinant
and the condition ap = 0, hence V = V ⊗ ηK is induced. That means that one cancapture the locus of such representations by a local deformation condition. It is
not the case that locally induced implies globally induced, as can be seen from theexample of supersingular elliptic curves. This is related to the fact that the map
π : Rloc → Rglobof (unrestricted at p) local to global deformation rings is not a surjection. On theother hand, we know in some generality that this is a finite map. This was exploredin §22, and then more properly taken up in [AC14]. The argument to this pointPERSIFLAGE: MATH BLOG POSTS

241

is now enough to prove the original result of Sardari. Let Rloc,ind denote the local
deformation ring of induced representations. If R = Rglob ⊗Rloc Rloc,ind denotes theglobal deformation ring of locally induced representations, we know that the forms
with ap = 0 and a fixed weight are the points of this deformation ring which lie in
the fibre over some fixed point in local deformation space. Hence the finiteness of π
gives a uniform bound on the number of points in this fibre, and hence a uniform
bound over the number of such modular forms in any fixed weight. BTW, for thosewondering why there is a restriction on the parity of the weight, it is only really
there to prevent the residual representation from being globally reducible, a settingin which one doesn’t quite yet know the finiteness of π. (When the optimal R = T
theorems become available in the reducible case, our methods should apply without
any restrictions.)
Now comes the second ingredient. In order to explain it, let me describe thering Rloc,ind in more detail, or at least the part coming from inertia. This local
deformation ring is basically equal to the deformation ring of the trivial characterof GK , and in particular the ring has the form
×Zp [[OK(p)]]where A(p) denotes the maximal pro-p subgroup of A. This ring is isomorphic (atleast for odd p) to the Iwasawa algebra Zp [[X, Y ]] after (via the p-adic logarithm)
×
fixing a choice of multiplicative basis for OK(p). Imagine that some component of
the global deformation ring (with a locally induced condition) has infinitely many
points which correspond to classical non-CM modular forms of level prime to p.
The points in weight space correspond to the algebraic characters of the followingform:
OK → K × ,z 7→ z n
We now exploit the following fact which might (at first) be surprising: any infinite
collection of these weights are Zariski dense! To make things a little more concrete,×η
suppose we choose a basis of OK(p) of the form
√ 1 + p and (1 + p) , for a suitableη ∈ OK which will not be in Zp , for example, u for some non-quadratic-residue.
The corresponding points with respect to the usual Iwasawa parameters have theshape:
X 7→ (1 + p)n − 1,
Y 7→ (1 + p)ηn − 1.
Instead of proving here why these are Zariski dense, it might be more useful toexplain a very close analogy that Naser brought up with Lang’s Conjecture: if you
take an infinite set of pairs of points of the form (exp(x), exp(ηx)) ⊂ (C× )2 , thenthey will be Zariski dense whenever η ∈/ Q. In other words, the group subvarietiesof the formal torus going through (X, Y ) = (0, 0) basically all have to be of theform (1 + X)η = (1 + Y ) for η ∈ Zp . (Coincidentally, the arithmetic applicationsof Lang’s conjecture was the subject of the recent Ahlfors lecture by Peter Sarnak
which you can watch here. Our result is yet another application!)Once your non-CM points are Zariski dense, you are home and hosed: using anidea due to Ghate–Vatsal [GV04], you now specialize at lots of points which are
inductions of finite order characters. The corresponding Galois representations havefinite image on inertia and so are classical by known results. But then (apart fromfinitely many exceptions) they have to all be CM, because they are classical weight
one forms, and the image of inertia is sufficiently large to rule out them having
exceptional image.

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One might ask whether the results are effective. I’m not so sure because of thefollowing issue. Suppose you take p = 79 and level one (I’m not sure this case will
exhibit the required behavior but it might.) Then you might be able to prove thatthe global locally-induced deformation ring is (now over all weights) Zp = Λ/P. But
it might be very hard to tell if that weight P corresponds to a classical weight or arandom weight, simply because Z is dense in Zp . This is not unlike the problem ofshowing that the zeros of the Kubota–Leopoldt zeta function are not in arithmetic
weights.

121. NSF proposal, graduate fellowship edition
Tue, 17 Mar 2020
Note: I feel as a service to the number theory entertainment complex that I
should blog more often in these times, even if it means being less coherent thanusual. I might even try to get a few guest posts since I won’t be going to any
conferences any time soon.
I recently linked to my first NSF proposal in § 113, but just today I stumbled upon
my graduate NSF fellowship application from 1998. There is really only one page
which involves any proposal (rather than a list of courses I took or references), andI include the mathematical portion here in full (the only changes from the original
are one or two spelling errors and getting the latex to compile. Here we go:
My research interests center mainly around the study of two dimensional Galois
representations, the connection of such representations to Modular forms, and application of these connections to the arithmetic of Elliptic curves. Here are several
possible questions which are of interest to me.
(1) Serre’s conjectures predict that for any odd, absolutely irreducible Galois
representation ρ into GL2 (Fq ), there exists a modular form f which gives
rise to ρ, in the sense of Deligne/Shimura/Deligne-Serre. The characteristic zero representation ρf , however, need not be defined into GL2 (W (Fq )),(W (Fq ) = Witt–vectors of Fq ), but perhaps in GL2 (O), for some ramified
extension O of W (Fq ). Is there any sense in which one can quantify theramification of O in advance? Is there perhaps a clear cohomological obstruction to a modular lift over W (Fq )? Can one quantify this in terms ofsome H 2 (GQ , ∗), or perhaps in terms of Runiv , where Runiv is the universal
deformation ring of ρ? Perhaps if this is too difficult, some qualitative resultin this direction?
(2) Applications of the above ideas to rational cuspidal eigenforms of Level
1. Such forms are only known to exist if dimS2k (1) = 1. Can one use
ideas from representations to show that no other cuspidal eigenform can be
defined over Q?
My first thought is “I guess I haven’t changed that much as a mathematician
over 22 years” followed by “not a bad problem but too optimistic.” The funny thingis that I do think of myself as a number theorist with a certain amount of breadth(despite protestations to the opposite from my most dyspeptic collaborator), so I
guess I have to claim that I work on a large circle of ideas and sometimes returnto very similar points on the circle. There are also echos in the first proposal of

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243

future work with Matt where we studied the ramification of O for the reducible
representation occurring in weight 2 and prime level N , as studied by Mazur. The
most definitive result in that paper was computing the exact ramification degreeh−1when p = 2, where in the case that the ring was a DVR one−1, where√ had e = 2h2 was the order of the 2-part of the class group of Q( −N ). Other progress on
this problem more in the spirit of the formulation above has been done by Lundell
and Ramakrishna (see [LR11]), although I still think there are many open questions
around this problem which are of interest.
On the other hand, the second problem is too optimistic. One reason is related
to Buzzard’s observation that, in high weight with p fixed, the representations all
seemed to be defined over rings with very little ramification. (He goes on to make aconjecture along these lines for which nobody has made any progress.) So it seems
unlikely to rule out forms of large weight with coefficients in Q by showing that
there are no such forms over Qp because the latter seems to be false in a strongway. The problem of showing there are no more eigenforms over Q when the weightis at least 24, which is very close to Maeda’s Conjecture, is something on which
virtually no progress has been made since my proposal, so I guess I don’t have tofeel bad for not making any progress myself. On the other hand, I don’t actually
think it is an impossible problem. Maybe I should work on it!

122. More on Lehmer’s conjecture
Sat, 21 Mar 2020
Lehmer said it was a “natural question” whether there existed an integer such that
τ (n) = 0 or not. I’ve wondered a little bit recently about how reasonable this is. The
historical context is presumably related to the fact that, by the multiplicativity ofcoefficients, the vanishing of τ (p) for one prime guarantees that a positive proportionof other coefficients vanish. From the perspective of Galois representations, however,I’m a little confused as to whether we expect any sort of “automorphic” Lehmer’s
conjecture to hold. To recall, we have
∆=q

∞
Y

(1 − q n )24 =

n=1

∞
X

τ (n)q n .

n=1
11/2

Deligne’s bound says that |τ (p)| ≤ p, so a probabilistic argument suggests that
there should only be finitely many primes for which τ (p) = 0. Since there aren’t any
such primes in the first few thousand primes, that’s a fairly convincing heuristic
for why it might be true. But it’s basically impossible to prove anything this way,
so one might hope to formulate a more general conjecture which is true for a more
systematic reason.
A first attempt might be to ask that ap (f ) ̸= 0 for any eigenform f of weightk ≥ 3 and level prime to p which is not CM. (When k = 2, of course, there areplenty of modular elliptic curves without CM, and (thanks to Noam) there areplenty of primes p with ap (f ) = 0). At first thought this seems a little strong; after
all, if we just work in weight 12 (say) then we know that |a2 (f )| < 211/2 < 46, so
surely if you take enough such forms you should find one with a2 (f ) = 0. However,this secretly assumes that there are many weight 12 forms with coefficients in Z,
and it seems that there are only finitely many such forms. So, for most forms, the

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F. CALEGARI

coefficients would lie in (presumably) larger and larger number fields, and there
would be more possibilities for a2 (f ).
For those who did the computation and might be worried, note that the probabilistic heuristic above only applies when the weight k ≥ 4. On the other hand,an easy exercise shows that when the weight is odd and the coefficients are integral then the form has CM. The conjecture that there only finitely many non-CM
forms with rational coefficients in large even weight is certainly made in this paper,
although Dave seems to be hesitant on numerical grounds to make the conjecture
for k = 4. There seem to be enough forms of weight k = 4 and integer coefficientsthat perhaps there exists a form of odd level with a2 (f ) = 0. In fact, it shouldbe easy to search for such forms if you can search the LMFDB with a fixed Hecke
eigenvalue, which I remember John Voight demonstrating at the Simons Institutegeneral meeting, but I couldn’t work out just know when writing this post. Ah, but
I guess one can just search for forms with coefficients in Q and just look at themby hand. It appears that there is a form
f = q + 4q 3 − 8q 4 − 5q 5 − 22q 7 − 11q 9 . . . ∈ S4 (Γ0 (95), Q)
with a2 (f ) = 0. Are there any examples in higher > 4 weight? (Yes, see the notes).
All of this becomes similarly confusing on the level of Galois representations. The
modular forms with ap (f ) = 0 have the special property that the local p-adic Galois
representation ρf is induced from the unique unramified quadratic extension of Qp .
From this perspective, the Lehmer conjecture looks a little bit like Greenberg’s
conjecture that an ordinary modular form is split if and only if it has complex
multiplication. But whereas that conjecture (or at least a stronger version where
one assumes such a splitting at all primes of the coefficient field) does follow fromplausible conjectures about motives as explained by Matt. I wonder if Matt has any
more opinions on what happens if one makes the assumption for only one primeof the coefficient field. Note that if you read Matt’s paper, you might be confused
why you can’t also use Serre–Tate theory to prove that elliptic curves with ap = 0
have CM. I think Florian Herzig gives a nice explanation here of what is going on.
This is also related to the question raised in this post. While that conjecture is
not unreasonable, it does skirt the border of conjectures which are actually false,for example, the conjecture that any exceptional splitting in a local Galois representation is caused by (more or less) some global splitting. After all, taken to its
logical conclusion it would imply not only Lehmer’s conjecture but also (combinedwith Elkies’ theorem) that all elliptic curves are CM. Greenberg’s conjecture excludes the case of weight one forms, since certainly any form with finite image has
many primes for which the local Galois representation is split but the global representation is not if the image is of exceptional type. One can still argue, however,that these forms are potentially CM. On the other hand, non-CM Hilbert modular
forms of partial weight one, induced to Q, also admit some exceptional splitting on
inertia. (Note that non-CM Hilbert modular forms actually exist, as follows fromthe computation of Moy and Specter [MS15]. While these induced forms are notof regular weight (the HT weights are, up to twist [0, 2, 2, 4], the splitting of thelocal Galois representation is not explained by any correspondences over any finite
extension.
I guess the summary is that all of this discussion points to the fact that Lehmer’s
conjecture is not true for any good reason beyond random probability grounds andso is kind of rubbish. Actually this reminds me of an experience one occasionallyPERSIFLAGE: MATH BLOG POSTS

245

has after giving a seminar in which you feel like you proved a snazzy result and thenthe questions from the audience are somewhat deflating. Rest assured, this happens
to the best of us — I remember watching a talk online where Richard was talking
about his (joint) proof of the Sato–Tate conjecture for ∆, and the only question
from the audience was does this have any implications for Lehmer’s conjecture?
Comment 122.1 (Emmanuel Kowalski). Here’s an old blog post of mine that ispartially about whether Lehmer’s conjecture is reasonable or not:It contains the bolder statement that maybe the tau function is injective . . .
Comment 122.2 (Aurel Page). Here is how to search for a value of ap in the
LMFDB: on the classical modular forms page, first fill in the coefficient field to be
1.1.1.1, then click “Traces Table”; you will get extra search options including “Trace
constraints” where you can input a2 = 0,
Notes 122.3. At the time, it wasn’t possible to also search for forms which had
level prime to 2, since a2 = 0 a lot when 4|N . But now it is (thanks in part to aticket request by Aurel), and there are 53 cuspidal eigenforms f of weight ≥ 4 withcoefficients in Z, level prime to 2, and a2 = 0 in [LMF24], and all have weight 0.
See this search. For p = 3, however, there is this form
f = q + 14q 5 − 170q 7 + . . . ∈ S6 (Γ0 (832), Z)
with a3 = 0 and weight 6. There is also a form in S6 (Γ0 (66), Z) with a5 = 0.
Another very amusing example is f ∈ S6 (Γ0 (390), Z), this time with a7 = 0. But
look at the coefficients:
f = q + 4q 2 − 9q 3 + 16q 4 + 25q 5 − 36q 6 + 64q 8 + 81q 9+100q 10 − 36q 11 − 144q 12 + 169q 13 − 225q 15 + 256q 16 + . . .
They are all squares! Alas, the coefficient of q 17 is 866. Part of the mystery is
revealed when considering the level: 2 · 3 · 5 · 13, by newform theory this forces theidentity ap = ±p2 for these values, so it’s just a7 = 0 and a11 = −36 which “luckily”
turn out to be squares (the other squares come from multiplicativity).

123. Chidambaram on genus two curves, I
Wed, 15 Apr 2020Those who study elliptic curves certainly know that if you start with an elliptic
curve E/Q, the p-torsion gives rise to a Galois representation:
ρ : GQ → GL2 (Fp )with cyclotomic determinant. Conversely, if p = 2, 3, 5 then the converse is true,that is, any such Galois representation comes from an elliptic curve. Moreover, anysuch representation comes from an infinite number of curves which are parametrizedby P1Q . This is intimately related to the fact that the curves X(p) have genus zerofor these p.
What is also true is that, given any E, one can write down explicit parameterizations of these families. This was done by Rubin and Silverberg [RS95] forp = 3, 5 around the time Fermat’s last theorem was proved. Indeed, the idea of
passing between elliptic curves with the same mod-3 Galois representation features
prominently in Wiles’ argument.

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F. CALEGARI

One might ask what happens for higher genus. First of all, there is a geometricproblem over the complex numbers: when is the moduli space Ag (p) of PPAV ofdimension g with full p-level structure a rational variety? It turns out the only
possibilities when g > 1 are p = 2, 3 when g = 2 and p = 2 when g = 3. The case(g, p) = (2, 3) arose in my work with Boxer, Gee, and Pilloni [BCGP21]. In that
paper, we proved a weaker version of the result above, namely the following:
Proposition 123.1 ([BCGP21]). If ρ : GQ → GSp4 (F3 ) is any continuous representation with cyclotomic similitude character, then the corresponding twist A2 (ρ)
is unirational over Q via a map of degree at most six. In particular, it has many
rational points.
The degree six cover is not so mysterious. When g = 2, PPAV are (more or less,the less being the Humbert surface) are Jacobians of genus two curves. So certainly
birationally one can replace A2 (ρ) by M2 (ρ), the moduli of genus two curves whose
Jacobian has the given 3-torsion. The degree six cover is then the moduli of genus
two curves with a fixed Weierstrass point, or, more prosaically, the genus two curvesof the form:
y 2 = x5 + ax3 + bx2 + cx + d
whose Jacobian has the given 3-torsion. (Any fixed Weierstrass point can be moved
to ∞, and then the x4 term can be suppressed by an obvious linear transformation.)
This moduli space will be discussed in more detail in the next post. But for now,
this leaves open the question of whether A2 (ρ) itself is rational.
Over the complex numbers, things are well understood. The space A2 (3) has anumber of compactifications, including the (singular) Satake compactification, andthe various smooth toroidal compactifications. When g = 2, things work out extra
nicely: there is a somewhat canonical compactification A∗2 (3) due to Igusa. It turns
out that A2 (ρ) is birational to a very nice 3-fold known as the Burkhardt quartic.
The Burkhardt quartic is given explicitly in P5 by the equations:
σ1 = x0 + x1 + x2 + x3 + x4 + x5 = 0,σ4 = x0 x1 x2 x3 + . . . + x2 x3 x4 x5 = 0.
Eliminating any variable using the first equation leads to a quartic in P4 , but thisis the most symmetric presentation. This variety B is singular and has 45-nodes— a maximal number, as it turns out [dJSBVdV90]. Not surprisingly, it also hasan action by automorphisms of the simple group G = PSp4 (F3 ). Blowing up B atthese nodes gives the smooth variety A∗2 (3).
Things are more subtle over Q. It turns out that for the trivial level 3 structure
corresponding to the representation (Z/3Z)2 ⊕ (µ3 )2 with the obvious symplectic
structure, the corresponding variety A∗2 (3) is still rational (e.g. see here [BN18]). But
it is no longer so obvious whether the twists we are considering should be rational
over Q or not. (There are actually some twists of a different flavor which don’t
have points, but all the ones we are considering do.) Note there is a big difference
between what happens in higher dimensions and what happens in dimension one: In
dimension one the only unirational smooth projective curve with a rational point is
projective space itself, but this is completely false in higher dimensions (for example,
take products of projective spaces).
We left the question of the rationality of A2 (ρ) open in [BCGP]. But my student
Shiva Chidambaram took up the question. The first question is how can you provePERSIFLAGE: MATH BLOG POSTS

247

a smooth projective variety X is not rational over Q assuming that it is rational
over C and has rational points. One obstruction was found by Manin [Man86]. If X
is projective space, then the geometric Picard group of X is Z. The Picard groupdoes not always have to be Z for a smooth rational variety, but Manin showed
that, still assuming that X is smooth and projective, if it is birational to projective
space then its (geometric) Picard group is similar to the trivial representation ina technical sense we now explain. Here we say that two Z[GQ ]-modules A and B(which are free finitely generated abelian groups) are similar if there are integral
permutation representations P and Q of GQ such thatA ⊕ P ≃ B ⊕ Q.
(I think one should imagine a sequence of birational maps where one introduces (orremoves) the class of some cycle and all of its conjugates.)
Now we can hope to apply this in practice if we can compute the GQ action on
M = PicQ (A∗2 (ρ)).
How might one go about computing M ? First of all, consider the non-twisted
space A∗2 (ρ). Using the explicit geometry of this space, one can hope to go about
computing the Neron–Severi group completely explicitly, together with the actionof G = PSp4 (F3 ). And this was indeed done by Hoffman and Weintraub (amongst
other things) in this paper (see [HW01]). In particular, they show that the cohomology of this variety is all torsion free, trivial in odd degrees, and satisfies
H 2 (X, Z) ≃ H 4 (X, Z) = Z61 .
Moreover, the cohomology
is entirely generated by cycles, and these cycles are all√
defined over E = Q( −3) and can be written down explicitly, together with thecorresponding intersection pairing, and the action of the group G = PSp4 (F3 ) on
these cycles is self-evident because of their geometric nature. Clearly the NeronSeveri group of any twist will also be Z61 , because the geometric object is thesame — the only thing that will
√ change is the Galois action. For this, it is moreconvenient to work over E = Q( −3). In this case, the GE action will be as follows:
the projective image of ρ when restricted to GE factors through PSp4 (F3 ) given
the assumption on the similitude character. Thus ρ gives a canonical map
GE → G,and the action of GE on M is simply the restriction of the action of G. Manin’s
obstruction says that for the variety to be rational over E, the action of GE has
to factor through a representation similar to the trivial representation. But that
depends only on the image H ⊂ G. Thus the problem (at least in terms of when
we can apply Manin’s criterion) is “reduced” to group theory.
Some more caveats: It turns out to be pretty hard to tell if a representation issimilar to the trivial representation. There is one obstruction coming from cohomology: using Shapiro’s lemma, if H is acting on M by a permutation representation,then
H 1 (P, M ) = H 1 (P, M ∨ ) = 0, all P ⊂ H.
But then it follows that the same is true of M is similar to a permutation representation. This gives a way to explicitly verify in some cases that M is not similar toa permutation representation by finding a subgroup P for which the group above
is non-trivial. Moreover, computing cohomology is something that magma can do!
So it remains to:248

F. CALEGARI(1) Explicitly translate the description of Hoffman=-Weintraub into a presentation of Z61 as a G = PSp4 (F3 )-representation.
(2) Determine for what subgroups P of G one has H 1 (P, M ) = H 1 (P, M ∨ ) = 0.
(3) Deduce that A∗2 (ρ) is not rational whenever the projective image contains
such a P as above.
The conclusion (see [CC22]):
Theorem 123.2 (C–Chidambaram, [CC22]). For all but 27 of the 116 √conjugacyclasses of G, the corresponding twist A∗2 (ρ) is not rational over E = Q( −3) andhence not rational over Q either. In particular, if the projective image over E has
order greater than 20, the twist is not rational.
You actually get something stronger from Manin’s criterion — if the variety
becomes rational over some map of degree d, then the cohomology of the modules
must be annihilated by d. From our computations we find, for example:
Theorem 123.3 (C–Chidambaram, [CC22]). Suppose that ρ : GQ → GSp4 (F3 )is surjective with cyclotomic similitude character. Then the minimal degree of any
dominant rational map P3Q → A∗2 (ρ) is six.
Note that from the construction of [BCGP21] we know that there is such a coverof degree six, so the six in this theorem is best possible! (It was good that thecomputation was consistent with the existence of this cover!)
It turns out that (in the surjective case) one can give a softer argument that
only depends on the rational representation. The point is that there can still be
an obstruction to a rational representation to being a difference of permutation
representations. This is easy enough to compute using the character table; you takethe group of all virtual representations over Z and compute the subgroup of all
induced representations. For G = PSp4 (F3 ), this quotient, sometimes called theBurnside cokernel (at least this is what it is called in the magma documentation),turns out to be Z/2Z (magma computes it). It’s also not so hard to see that there
exist subgroups G40 and G45 of the obvious index such that[H 2 (X, Q)] = [G/G40 ] + [G/G45 ] − [χ24 ],where χ24 is the unique representation of G of dimension 24 which also happens
to be defined over Q and also generates the Burnside cokernel. On the other hand,
this method this gives weaker results for subgroups of G = PSp4 (F3 ) and even in
the surjective case only shows the minimal cover has degree two rather than six.
123.4. A word on the actual computation. Shiva went off and did the taskof converting the description in Hoffman–Weintraub into a form which could be
used by magma. I also went off and tried to do this independently. We then bothproduced codes (mine much messier) which computed the cohomology of all thesubgroups and arrived at completely different answers, which was a bit troubling.
But then Shiva pointed out to me that magma automatically does something withmatrices that converts right actions to left actions or something like that [could it
really be that Magma treats matrices as acting from the right? that sounds crazy],and so his computation of H 1 (P, M ) was correct, but I was computing H 1 (P, M ∨ ).
But fortunately both are useful! (Of course, one could easily also extract that datafrom Shiva’s code which was much cleaner than mine.)PERSIFLAGE: MATH BLOG POSTS

249

124. Chidambaram on genus two curves, II
Fri, 24 Apr 2020
We now continue a series of posts on the work of my student Shiva Chidambaram
following on from § 123. Today I would like to discuss another project [CCR20] with
Shiva that was also joint with David Roberts (no, not David Roberts).
We saw last time that the moduli spaces A2 (ρ) and M2 (ρ) are not in general
rational over Q. On the other hand, the degree six cover Mw
2 (ρ) is always rational.
So the next question is: what is an explicit parametrization? Slightly differently,
start with a genus two curve with a Weierstrass pointy 2 = x5 + ax3 + bx2 + cx + d
Problem 124.1. Parametrize all other genus two curves with a Weierstrass point
which have the same 3-torsion representation.
It might be worth briefly revisiting the argument from [BCGP21] that such aparameterization exists. The key point is that there is an birational map
3Mw
2 (3) → P

which is PSp4 (F3 )-equivariant. This allows one to show that the corresponding
twists are Brauer–Severi varieties, and then deduce they are rational by the samegroup theoretic trick which appears in this paper of Shepherd-Barron and Richard
Taylor. More explicitly, there are maps×

H 1 (Q, GL4 (Q)) → H 1 (Q, PGL4 (Q)) → H 2 (Q, Q )Here the LHS is trivial by Hilbert 90. One shows, using the fact that the Darstellungsgruppe of PSp4 (F3 ) is Sp4 (F3 ), that the cocycle corresponding to any Galois
representation ρ : GQ → GSp4 (F3 ) with cyclotomic determinant lifts in an explicit
way to a cocycle in the LHS and hence is trivial. The problem is to bridge thegap between a theoretical argument that a cocycle is trivial and a way to produce
an equation of the corresponding twist. That amounts to the problem of taking acocycleZ 1 (Q, GL4 (Q))
and writing it as a coboundary. Before going further, it’s worth pointing out thatthe case of (g, p) = (2, 3) is very similar (but more complicated) than the case of(g, p) = (1, 3). In the latter case, one has that
dimH 0 (X(3), ω) = 2,
and this simple equality leads to an identification of X(3) with ProjH 0 (X(3), ω).
So, let’s talk about the problem of parametrizing elliptic curves as a warm-up case.
If you start with a curve
E : y 2 = x3 + ax + b
for which (for convenience of exposition) you assume ρE,3 surjective,√ then the splitting field L/Q is a GL2 (F3 ) extension L which contains F = Q( −3). There is an
isomorphism of H = SL2 (F3 )-modules
L = F [H].
The group H admits a certain specific 2-dimensional representation V , and therepresentation ρ can be interpreted as giving an explicit map
V → L.

250

F. CALEGARIOTOH, the identification above means there is an inclusion V 2 → L becausedim(V ) = 2. The problem of solving Hilbert 90 (ignoring a certain descent from Fto Q) then becomes the question of finding the “other” extension. Now if you canwrite L down you can do this by linear algebra. But even in any specific example,
L has degree 48, and computations with fields of that size can be pretty formidable
and are at the limit of what one can do with explicit number fields in (say) pari/gp
or magma.
Of course, one wants to do this over the field Q(a, b) with general parametersin order to have the general formula. The extension L, for example, is the Galois
closure of the equation
27y 8 + 216by 6 − 18∆y 4 − ∆2 = 0,
but you probably don’t want to even write down a polynomial of degree 48 in thesegeneral variables which gives L, let alone try to compute the Galois action. We
did succeed in solving this problem by a certain amount of trickery — working inspecial cases and making the right ansatz for the general case. There were many intermediate formulas which involved polynomials with (say) 100 terms, but the final
answer turns out to be perhaps surprisingly simple, namely, the general equation
is given by
y 2 = x3 + A(a, b, s, t)x + B(a, b, s, t),
where
3A(a, b, s, t) =3as4 + 18bs3 t − 6a2 s2 t2 − 6abst3 − (a3 + 9b2 )t4 ,9B(a, b, s, t) =9bs6 − 12a2 s5 t − 45abs4 t2 − 90b2 s3 t3 + 15a2 bs2 t4− 2a(2a3 + 9b2 )st5 − 3b(a3 + 6b2 )t6 .
Here [s, t] is the P1 parameter. Curiously enough this exact formula was also found
before in the literature. That reflects something a little surprising about this equation. The moduli space we are looking for is a P1 , and this has many automorphisms.
On the other hand, we are starting with an E so we have a fixed point normalized
here to be [1, 0]. But the projective line with one fixed point still has many automorphisms! However, it turns out that there is some extra hidden structure which
gives rise to a second canonical point normalized here as [0, 1], which is why different people would possibly end up with the same equation independently. (P1 with
two fixed points still has a Gm ’s worth of automorphisms, but an informal consideration of the integral structure can be used to pin this down further.) The map
which takes one E and spits out the other point therefore ends up giving a canonical rational map on P1j which has the property that it preserves the (projective)
3-torsion representation. Explicitly it is given by:
j 7→(6912 − j)3
27j 2

I wonder if this has interesting dynamical properties?
The computation above was not so easy, even though the answer turned out to be
simple enough. But for (g, p) = (2, 3) things are looking pretty bad. First of all, theextension L now has degree 103680, which one is not going to write down explicitly.
Even the analogue of the degree 8-polynomial above is a degree 40 polynomial inx6 with 1673 terms.
Despite that, we found the answer:

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251

Theorem 124.2 (C–Chidambaram–Roberts [CCR20]). There exist (and we compute) explicit polynomials A, B, C, D in Q[a, b, c, d, s, t, u, v] which specialize to a, b, c, d
at [s, t, u, v] = [1, 0, 0, 0] such thaty 2 = x5 + Ax3 + Bx2 + Cx + D
is the general genus two curve with a rational Weierstrass point and fixed 3-torsion
representation.
Even though we find the simplest form of these polynomials, they turn out tobe quite big. As in, the number of monomial terms they contain are 14671, 112933,
515454, and 1727921 respectively. The text files were so big that I ran into space
problems on my university account! (OK, so it’s only 200MB or so, but that’s a big
text file!)
The reason such a computation is ultimately possible relates to an accidental
fact that is common between the two cases, namely, that the groups SL2 (F3 ) andSp4 (F3 ) × Z/3Z are two of the 37 exceptional complex reflection groups as determined by Shephard and Todd. The story is explained in our paper [CCR20] so I
won’t discuss it here, but it might be worth mentioning two further facts:
The first is that these methods can also deal (in principle) with an analogue of
this problem for g = 3 and p = 2. Just as with g = 2, the moduli space which admits
an equivariant birational map to P6 is not M3 (2) but once more a finite cover, andthis cover does not correspond to any level structure but rather some cover coming
genuinely from the mapping class group. This picture relates to the isomorphism
Z/2Z × Sp6 (F2 ) ≃ W (E7 ), another exceptional complex reflection group. There iseven a less analogous version for g = 4 and p = 2 related to the fact that the largest
complex reflection group W (E8 ) admits a description W (E8 ) ≃ 2.O+8 (F2 ) : 2, andthe projective version of this group O+
8 (F2 ) : 2 is a subgroup of Sp8 (F2 ), althoughof genuine index (136) rather than as an isomorphism, which is the main reason why
this is a little different to the other cases. We estimated that an explicit version of
the last moduli problem would involve polynomials with approximately 100 trillion
terms, so needless to say we did not try to compute it.
Second, there is an interesting story concerning the auxiliary copy of Z/3Z thatturns up in the g = 2 setting. The formulas that we write down actually correspond
not only to projective spaces P1 and P3 but actually to affine spaces A2 and A4
which represent moduli problems related to the complex reflection groups. In theseaffine families, not only is the representation corresponding to ker(ρ) fixed, but the
splitting field of X 3 −∆ also remains unchanged. When g = 1, this is not surprising,f4 → S4 → S3 . On
because the S3 extension comes from the map GL2 (F3 ) = S
the other hand, that’s obviously not happening in the genus two case where thegroup is almost simple. This is a little peculiar! However, it related to the factthat the splitting field of X 3 − ∆ for genus two curves depends on the Weierstrassequation. If you scale the Weierstrass equation by (x, y) 7→ (t2 x, t5 y), this sends∆ → t40 ∆. So the affine equation represents a moduli space for some larger groupwhich disappears when considering the equation projectively, and you can always
normalize your Weierstrass equation so that ∆ is a perfect cube.

125. Picard groups of moduli stacks
Thu, 30 Apr 2020

252

F. CALEGARI

Here are some algebraic geometry musings related to § 124, most of which is
hopefully correct. Everything below is secretly over Z[1/6] but I think one may as
well think about what is happening over C. Warning: I don’t know any algebraic
geometry, please correct me if you see any nonsense.
As mentioned in the last post, if you fix a 3-torsion representation with cyclotomic determinant and look at the corresponding moduli space of elliptic curves
with this 3-torsion, you get a P1 (at least accounting for cusps). A natural followup
question is: what geometric object do you get over the stack A1 = M1,1 ? Thinking
about stacks in the most naive way, we just consider
y 2 = x3 + ax + b
for (a, b) in P(4, 6) minus ∆ = 0 in the stacky sense. But just thinking about thisas an elliptic curve over Q(a, b), you can write down:y 2 = x3 + Ax + B
where
3A(a, b, s, t) =3as4 + 18bs3 t − 6a2 s2 t2 − 6abst3 − (a3 + 9b2 )t4 ,9B(a, b, s, t) =9bs6 − 12a2 s5 t − 45abs4 t2 − 90b2 s3 t3 + 15a2 bs2 t4− 2a(2a3 + 9b2 )st5 − 3b(a3 + 6b2 )t6 ,Now one thing you notice straight away about these equations is that they change
when one replaces a, b by aλ4 , bλ6 , namely:A(λ4 a, λ6 b, s, t) = A(a, b, λs, λ3 t)
and the same equation holds for B. That is, the parametrization of P1 changes,
and so the family is not literally projective space over this stack. Of course, if∆(a, b) = 16(−4a3 − 27b2 ),then∆(aλ4 , bλ6 ) = λ12 ∆(a, b),
where ∆ trivializes ω 12 . In order to remove the ambiguity, one can then definetsA∗ (a, b, s, t) = A a, b, 1/12 , 3/12∆∆
and similarly with B ∗ , then the equation is well defined, at least after addressing
the issue of taking 12th roots correctly. This suggests that after pulling back to thespace where you adjoin ∆1/12 you get projective space, but that the original space
is not projective space at all but maybe something like the projective bundle
Proj(OX ⊕ ω 2 ) = Proj(ω ⊕ ω 3 )where ω is the usual line bundle which has order 12 in the Picard group of A1 .
Something very similar happens for the equations for families of fixed three
torsion over Mw
2 , the moduli stack of genus two curves with a fixed Weierstrass
point. In this case, the base looks likey 2 = x5 + ax3 + bx2 + cx + dor P(4, 6, 8, 10) minus ∆ = 0. (You need to be a little bit more careful at the prime5.) Here the corresponding identity for A, B, C, D isA(λ4 a, λ6 b, λ8 c, λ10 d, s, t, u, v) = A(a, b, c, d, λs, λ7 t, λ13 u, λ19 v)

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253

and
∆(λ4 a, λ6 b, λ8 c, λ10 d) = λ40 ∆(a, b, c, d).
So now one wants to trivialize the family by taking the cover with various roots of∆, including ∆1/20 . Except now I don’t really know what the Picard group of Mw
2
is. Somehow I first assumed that the Picard group would be the same as that of thecorresponding moduli space of abelian surfaces Aw
2 , and since ∆ seems to give a
trivialization of some power of the determinant bundle it should be related to torsionin H1 (Γ, Z) for the corresponding congruence subgroup Γ of Sp4 (Z). But because of
the congruence subgroup property, presumably H1 (Sp4 (Z), Z) is equal to Z/2Z, andthat’s not going to change by taking the map to S6 = PSp4 (F2 ) and taking the prew
image of S5 . But it is pure folly to imagine the Picard group of Mw
2 and A2 coincide.
The latter contains an extra divisor, the Humbert divisor, consisting of direct sumsof elliptic curves. Moreover, (I guess) the Siegel modular form corresponding to ∆is probably very close to the Igusa form, which vanishes not only at the cusp but
also along the Humbert divisor. So the line bundle ω on A2 has infinite order eventhough its pullback to M2 does not because ∆ itself is giving a trivialization ofsome power of ω. So it is indeed plausible that abelianization of the corresponding(index five subgroup of) the g = 2 Torelli group has 20-torsion. One way to try
to compute this is to explicitly compute the abelianization of the corresponding
cover of the mapping class group (I guess there are explicit presentations?). So thefirst question is can someone confirm that Pic(Mw2 ) does indeed have 20-torsion? If
only there was someone in my department who could prime me on the properties ofmapping class groups . . . Actually, Andrew Putman is probably the obvious person
to ask. The second problem is confirm that the family explicitly computed in thelast post does indeed coincide with Proj(OX ⊕ ω 6 ⊕ ω 12 ⊕ ω 18 ).
I confess my efforts to do a literature search in this case have not been very
thorough. In my mind I somehow thought that the Picard group of the stack Mg(for g ≥ 2) was Z, but that is transparently false, at least for g = 2. I got as far as
doing a google search for Picard groups of moduli stacks and found a few pages ofnotes written by Daniel Litt. So I naturally zoomed in to Daniel Litt’s office hours
once after he advertised them on twitter . . . but I soon realized that it would take
too long to explain and he had better things to do like explaining modular forms
to his students . . . so here it is now in blog form!
Comment 125.1 ( Najmuddin Fakhruddin). g = 2 is special, for g > 2 the Picard
group of Mg is indeed Z (at least over C) (see, e.g., Albarello–Cornalba Picard
groups of the moduli spaces of curves (see [AC87]). The paper of Arsie–Vistoli
Stacks of cyclic covers of projective spaces (see [AV04]) contains a computation of
the Picard group of the moduli stack of hyperelliptic curves from which it follows
that Pic(M2 ) = Z/10Z.
My response: Excellent! So that surely means that ω is the generator 10 andthat ∆ as a Siegel modular form has weight 10. So now I guess I am implicitly
suggesting that ω admits a square root over Mw
2 . Naïvely I would have even guessthat there is an O(1) on P(4, 6, 8, 10) ∖ (∆ = 0) which had order 40 and may evengenerate Pic(Mw2 ) but stacky weighted projective spaces confuse me.
Najmuddin Fakhruddin replies: I don’t see any conceptual way of showing thata square root of ω (which is indeed a generator of Pic(M2 )) exists on Mw
2 , but itshould be easy to compute the Picard group of P(4, 6, 8, 10)\(∆ = 0). In general,

254

F. CALEGARI

if X is say a smooth variety and G is an algebraic group acting on X, then the
Picard group of the quotient stack [X/G] can be computed as follows:
Let V be any linear representation of G on which the action is sufficiently free–tocompute the Picard group it suffices to assume that the codimension of the nonfree locus is at least two–and let U be the locus on which the action is free. Then
Pic([X/G]) = Pic((X × U )/G) where the action is the diagonal action. (Of course,
the point here is that (X × U )/G is a smooth variety.)
If G = Gm , then one can just take V = A2 with the action λ · (x, y) = (λx, λy).
In the case of weighted projective space one may even replace X = An − 0 by An
when computing Pic(X × U/Gm ) so the quotient is an affine bundle over P1 . In
your example, I think for any irreducible ∆ (with the given homogeneity properties)
one can then see that the Picard group is indeed Z/(40).

126. Picard groups of moduli stacks update
Wed, 27 May 2020
A tiny update on § 125. I was chatting with Benson and realized that I may
as well ask him directly for a presentation of the mapping class group of a genus
two surface. Perhaps unsurprisingly, it can be found in his book with Dan Margalit(see page 122 of their book [FM12] which might be downloadable from a Russian
website) and is given as follows:G ≃ ⟨a1 , a2 , a3 , a4 , a5 | [ai , aj ] for |i − j| > 1, ai ai+1 ai = ai+1 ai ai+1 ,(a1 a2 a3 )4 = a25 , [(a5 a4 a3 a2 a1 a1 a2 a3 a4 a5 ), a1 ], (a5 a4 a3 a2 a1 a1 a2 a3 a4 a5 )2 , ⟩.
The next task is to find the representation
G → Sp4 (Z) → Sp4 (F2 ) ≃ S6
and then take the index 6 preimage Γ ⊂ G of the S5 ⊂ S6 corresponding to fixing
a Weierstrass point. Note there are two conjugacy classes of S5 , the correct one isthe one whose restriction to A5 still acts absolutely irreducibly on (F2 )4 . Then one
can use Reidemeister–Schreier to compute a presentation of Γ and then compute
H1 (Γ, Z). This is all good in theory, and Farb-Margalit does have a chapter on thesymplectic representation, but actually having to read the book in detail to extract
the precise symplectic representation sounded like too much work, especially since
all of this is ultimately just for a two sentence comment in a paper that might beremoved for space reasons anyway. So instead I just fired up magma with the representation G and asked it to find all index six subgroups. It turns out that there are
only two of them (up to conjugation), which must come exactly from the two subgroups of S5 ⊂ S6 . The abelianization of one is Z/10Z ≃ Gab , but the other groupis Γ = ⟨a1 , a2 , a3 , a4 ⟩, and one finds that H1 (Γ, Z) ≃ Z/40Z. Hence this is (in lightof the previous discussion) the correct subgroup, and this (unsurprisingly although
not entirely independently) confirms the analysis of Najmuddin Fakhruddin in thecomments. Now I suspect that if you think a little harder than I am prepared todo (or if you just know a little bit more than me), you might be able to see directlyfrom the definition of the ai that a1 , a2 , a3 , a4 fix a Weierstrass point; if you aresuch a person please make a comment!

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127. Families of Hilbert modular forms of partial weight one.
Wed, 03 Jun 2020
Today I would like to talk about a beautiful new theorem of my student Eric
Stubley. The first version of Eric’s result assumed (unknown) cases of the general
Ramanujan conjecture for Hilbert modular forms, and relied on a beautiful idea
due to Hida. The final argument, however, is unconditional, and goes beyond Hida’s
ideas in a way (I hope) that he would be delighted to see.
Suppose that F is a real quadratic field in which p = vw splits. If f is a Hilbert
modular form of (paritious) weight (1, 2k + 1) and level prime to p, then the corresponding Galois representation (really only defined up to twist):ρf : GF → GL2 (Qp )has the property that, for exactly one of the places v|p, the restriction ρf |Gv isunramified. Forms of partial weight one are slippery objects — one can construct
such forms which are CM, but the existence of any such form which is not CM was
open until an example was found by my students Richard Moy and Joel Specter
see § 8, 73. They behave in many ways like tempered cohomological automorphic
forms for groups without discrete series, more specifically Bianchi modular forms or
cohomological forms for GL(3)/Q. In each of these cases, the invariant l0 as considered in Calegari–Geraghty (see for example [Cal20, §2.8]) is equal to 1. Following
work of Ash–Stevens and Calegari–Mazur, one might consider whether or not f
deforms into a family of classical forms. For example, the form f will be ordinary
at v, and so it lives in a Hida family H over Λ = Zp [[Ov× (p)]] ≃ Zp [[T ]] where wekeep the weight and level at w fixed and consider (nearly) ordinary forms at v. The
specialization of this family to regular paritious weights will give a space of classicalHilbert modular forms. What can one say about the other specializations in partial
weight one?
Theorem 127.1 (Stubley). Only finitely many partial weight one specializationsof the one variable v-adic Hida family H associated to f are both classical and not
CM.
This gives a completely general rigidity result for all partial weight one Hilbert
modular forms in the split case. Over the past decade or so, the prevailing philosophy is that the only algebraic automorphic forms which are not exceedingly rare are
either those coming from automorphic forms which are discrete series at infinity,
or come from such forms on lower rank groups by functoriality. In this setting, thispredicts that non-CM forms of partial weight one should be rare. It might even beplausible to conjecture that, up to twisting, there are only finitely many such formsof fixed tame level. However, such conjectures are completely open, and Stubley’s
result is one of the first general theorems which points in that direction. (Stronger
results for very specific F and p and tame level were obtained by Richard Moy andare discussed in some of the links above.)
One way to think about this theorem is in terms of the Galois representation
associated to H. Assume for convenience of exposition that the family is free ofrank one over Λ. The Galois representation ρf extends to a family:ρ : GF → GL2 (Zp [[T ]])where Λ = Zp [[T ]] represents weight space, so T = 0 corresponds to the original specialization, and T = ζ − 1 for a p-power root of unity ζ corresponds to256

F. CALEGARI

a specialization to partial weight one with non-trivial level structure at v. These
representations are all nearly ordinary at v. Is it possible that they could be split
locally at v for infinitely many specializations to partial weight one? Since a nonzero Iwasawa function has only finitely many zeros, this would actually force thelocal representation to split for all T. Moreover, it should imply (and does in manycases) that the specializations T = ζ − 1 are all classical by modularity lifting
theorems. Thus, by Stubley’s theorem, this can only happen when the family ρ is
CM. In particular, Stubley’s result implies a theorem (assuming some Taylor–Wiles
hypothesis) that a family of Galois representations which is (say) nearly ordinary
at w of fixed weight and level and nearly ordinary at v is locally split at v if andonly if it is CM.
Experts should recognize the similarity between the Galois theoretic versionof Stubley’s theorem and the work of Ghate–Vatsal [CV19], who prove that an
ordinary family over Q cannot be locally split unless it is CM. The main ingredientin their proof is the fact that there are only finitely many weight one forms of
fixed tame level (up to twist) which are not CM, since these correspond either to
A4 , S4 , A5 extensions of Q unramified outside a fixed set of primes, which are clearly
finite, or real multiplication forms, whose finiteness comes down to the finiteness of
the ray class group of conductor N p∞ for a split prime p in a real quadratic field.
However, the analogous statement for partial weight one forms is completely open
as mentioned above, so Stubley’s theorem requires a quite different argument.
Before discussing the proof, we first need to discuss a result of Hida (see thispaper) (see [Hid16]) about fields of definition of ordinary forms in families. Consider an ordinary family over Q, and consider specializations in some fixed weight,
amounting (with some normalization) to specializing T to ζ − 1 for a pth powerroot of unity. The coefficient field will automatically contain Q(ζ). Suppose thatfor any prime q, the degrees [Q(aq , ζ) : Q(ζ)] are bounded for infinitely many specializations. Then Hida proves the family has to be a CM family. Let αq be one of
the corresponding Frobenius eigenvalues. Hida’s key insight is to note that αq is a
Weil number, and that Weil numbers over extensions of Q(ζ) of uniformly bounded
degree are extremely restricted, and in particular given an infinite collection of such
numbers then infinitely many of them have to be of the form αζ for a fixed α. Usinga rigidity lemma fashioned for this very purpose, he then deduces that αq in the
Iwasawa algebra more or less has to equal α(1 + T )s for some s ∈ Zp , and this puts
enough restrictions on aq for him to be able to deduce the family is CM.
Stubley’s first idea is to use Hida’s result in the context of partial weight one
forms. The key fact that is different in partial weight one is that when av ̸= 0, theform f is automatically ordinary at v, and hence the Gal(Q/Q(ζ)) conjugates of fwill still be ordinary at v! This is completely false in regular weights. However, inpartial weight one, the only possible (finite) slope of any form at a split prime is 0.
As a consequence, the boundedness assumption of Hida’s theorem is always going
to be satisfied, because all of the conjugates have to lie on one of the finitely many
Hida families which all have bounded rank over Λ.
There is, however, a problem. Hida’s argument crucially uses the fact that αq isa Weil number, which uses the Ramanujan conjecture for forms of regular weight.
The Ramanujan conjecture is completely open for partial weight one forms, since
we have no idea how to prove they occur motivically (nor prove modularity of theirsymmetric powers). This is where Stubley’s second idea comes in. Instead of the

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257

Ramanujan conjecture, one does have standard bounds on the coefficients aq . This
is not enough to deduce that αq has the form αζ for some fixed α. Instead, Stubley
shows that it does allow one to show that the trace of aq (together with the traceof any if its powers) to Q(ζ) (which has uniformly bounded degree) can be written
as a finite sum of roots of unity where the number of terms does not depend on ζ.
Again for convenience of exposition and to avoid circumlocutions with traces, let
us suppose that the rank of the Hida algebra is one and so Q(ζ, f ) = Q(ζ). Then
Eric shows that infinitely many of the aq satisfy:
aq = α1 ζ1 + α2 ζ2 + . . . + αN ζN
for varying p-power roots of unity ζi , but where αi and N are fixed. Then Stubley
proves a new rigidity theorem in this context (not unrelated to results of Serban)
showing that one must have an equality
aq = α1 (1 + T )s1 + α2 (1 + T )s2 + . . . + αN (1 + T )snover the Iwasawa algebra. This is probably enough to show the family has to be CM
using ideas similar to Hida, but even that is not necessary — by using this formulafor specializations in regular weight one deduces that the αi are in Q, and thenapplying Hida’s theorem in this fixed regular weight one deduces that the family is
CM.
Stubley’s theorem is the first result that gives general theoretical evidence towards the conjecture (if one is so bold to make such a conjecture) that there are
only finitely many non-CM partial weight one forms of fixed tame level up to twist.
It also shows that certain v-ordinary deformations of a non-CM partial weight one
form f will not be classical. But there is also a second possible way to deform f ,namely, to vary the weight at w|p instead (or as well). For example, if the form
f was also ordinary at w|p, one could look at the ordinary at w Hida family. One
might also conjecture that this family only contains finitely many non-CM points,
but this is still open. (Boxer has raised this question.) I think this is an interesting
but very hard question!
Notes 127.2. One version of Stubley’s results can be found here, [Stu21], although
unfortunately I am not sure he plans to submit this to a journal having left mathematics.

128. Chidambaram on Galois representations (not) associated toabelian varieties
Tue, 13 Oct 2020
Today’s post is about a new paper [Chi24] by my student Shiva Chidambaram.
Suppose that A/Q is a principally polarized abelian variety of dimension g and p
is a prime. The Galois representation on the p-torsion points A[p] gives rise to a
Galois representation:
ρ : GQ → GSp2g (Fp )with the property that the similitude character coincides with the mod-p cyclotomic
character. A natural question to ask is whether the converse holds. Namely, givensuch a representation as above with the constraint on the similitude character, doesit necessarily come from an abelian variety (principally polarized or not)?

258

F. CALEGARI

When g = 1, the answer is that all such representations come from elliptic curves
when p ≤ 5, but that for p ≥ 7 there exist representations for any p which do not.
For p ≤ 5, more is true: the twisted modular curves X(ρ) all are isomorphic to
P1 . When p ≥ 7, the curves X(ρ) are of general type, so one might expect a“random” such example to have no rational points. Dieulefait was the first person
to find explicit representations (for any such p) which do not come from elliptic
curves (and there is a similar result in my paper here) (see [Cal06]). Both of thesearguments exploit the Hasse bound. Namely, if
ρ : GQ → GL2 (Fp )is unramified at l ̸= p ≥ 5 and ρ comes from E/Q, then E must have either goodor multiplicative reduction at l. But this puts a constraint on the possible trace of
Frobenius at the prime l. For l = 2, for example, this leads to explicit examples ofnon-elliptic mod-p representations for p ≥ 11. The case p = 7, however, requires
a different argument. More generally, while the Hasse argument does generalize tolarger g, it only works when p is large compared to g. On the other hand, the Siegel
modular varieties Ag (p) of principal level p are rational over C for only very few
values of g and p. Indeed, they are rational only for(g, p) = (1, 2), (1, 3), (1, 5), (2, 2), (2, 3), (3, 2)
whereas Ag (p) turns out to be of general type for all other such pairs. When (g, p)
is on this list, then, as discussed in §123, 124, the twists Ag (ρ) can all be shown
to be unirational over Q and so any such representation ρ does indeed come frominfinitely many (principally polarized) abelian varieties.
Thus one is left to consider all the remaining pairs. This is exactly the question
resolved by Shiva:
Theorem 128.1 (Chidambaram [Chi24] ). Suppose that (g, p) is not one of the six
pairs above such that Ag (p)/C is rational. Then there exists a representation:
ρ : GQ → GSp2g (Fp )with cyclotomic similitude character which does not come from an abelian variety
over Q.
Shiva’s argument does not use the Weil bound. Instead, the starting point for
his argument is based on the following idea. Start by assuming that ρ comes froman abelian variety A. Suppose also that ρ is ramified at v ̸= p and the image of theinertia group at v contains an element of order n for some (n, p) = 1. Using this,one deduces (using independence of p arguments) that|Sp2g (Fl )| = l

g2

g
Y

l2m − 1m=1

is divisible by n for all large enough primes l, and hence divides the greatest common
divisor Kg of all these orders. This is actually a very restrictive condition on n. For
example, using Dirichlet’s theorem, the number Kg is only divisible by primes atmost 2g + 1. But now if the order of the group Sp2g (Fp ) for any particular pis divisible by a prime power n with n not dividing Kg , then one can hope toconstruct a mod-p Galois representation whose inertial image at some prime v hasorder divisible by this n, and this representation cannot come from an abelianvariety over Q.

PERSIFLAGE: MATH BLOG POSTS

259

The good news is that one can show that (most) symplectic groups have orders
divisible by large primes using Zsigmondy’s theorem. Combined with a few extra
tricks and calculations for some boundary cases, the groups Sp2g (Fp ) contain elements of “forbidden” orders exactly when one is not in the case of the six exceptional
pairs (g, p). Note that Zsigmondy’s theorem already arises in the literature in thiscontext in order to understand prime factors of the (corresponding) simple groups.
So now one would be “done” if one could (for example) solve the inverse Galois
problem for GSp2g (Fp ) with local conditions. The inverse Galois problem is solved
for these groups, but only because there is an obvious source of such representations
coming from abelian varieties. Of course, these are precisely the representations
Shiva wants to avoid.
Instead Shiva looks for solvable groups inside GSp2g (Fp ) containing elements oforder n for suitable large prime powers n. Note that the obvious thing would simply
be to take the cyclic group generated by the element of the corresponding order.
The problem is that there is no way to turn the corresponding representation intoa Galois representation whose similitude character is cyclotomic. The groups Shiva
actually uses are constructed as follows. Start by finding prime powers n|pm + 1
for some m ≤ g, then embed the non-split Cartan subgroup of SL2 (Fpm ) into
GSp2g (Fp ), and then consider the normalizer of this image. One finds a particularly
nice metabelian subgroup whose similitude character surjects onto F×
p . Shiva thenhas to prove the existence of a number field whose Galois group is this metabelian
extension with the desired ramification properties at some auxiliary prime v but also
crucially satisfying the cyclotomic similitude character condition. This translates
into a (typically) non-split embedding problem — such problems can be quite subtle!
Shiva solves it by a nice trick where he relates the obstruction to a similar one which
can be shown to vanish using methods related to the proof of the Grunwald-Wang
Theorem. Very nice! In retrospect, the case of g = 1 and p = 7 in my original
paper is a special example of Shiva’s argument, except it falls into one of the “easy”
cases where the relevant metabelian extension actually is a split extension over the
cyclotomic field. In general, this only happens when the the maximum power of 2
dividing g is strictly smaller than the maximum power of 2 dividing p − 1 which is
automatic when g is odd. (The case when p = 2 is easier because the cyclotomic
similitude character condition disappears!)

129. Hire my students!
Wed, 18 Nov 2020
(Excised: some exhortations to hire my graduating students.) Here’s a result
from Noah’s thesis which I haven’t discussed before:
Let N be prime, and let T denote the Z2 -Hecke algebra generated by Tl for le denote the Hecke algebra where T2 is also included. These
prime to 2, and let T
Hecke algebras are famously not the same in general. For example, when N = 23,
the space of cusp forms is 2-dimensional and has a pair of conjugate cusp forms asfollows:√√√
5+1 2 √ 3
5−1 4
q−q + 5q +q − (1 + 5)q 5 + . . .
2
2

260

F. CALEGARI√
e =Z
So T = Z[ 5] whereas T

"√

#
5+1
. Noah gives a formula for the index:
2

e : T]
Theorem 129.1 (Noah Taylor [Tay21]). Let N be prime. Then the index [Tis given by the order of the space
S1 (Γ0 (N ), F2 )of Katz modular forms of weight one and level Γ0 (N ).
In particular, the index at level 23 is coming from the fact that there is a classicalweight one form of this level. From this one sees that the index is non-trivial forall primes N ≡ 3 mod 4 except for N = 3, 7, 11, 19, 43, 67 and 163. For primes
N ≡ 1 mod 4, on the other hand, I might guess that there would be a positive
density of primes for which either the index was trivial or non-trivial. The question
more or less hinges on the expected number of SL2 (F2n ) representationsof Q (with√n ≥ 2) which become unramified at all finite places over Q( N ).

130. Ramanujan machine redux
Thu, 11 Feb 2021
I had no intention to discuss the Ramanujan Machine again, but over the pastfew days there has been a flurry of (attempted) trollish comments on that post,
so after taking a brief look at the latest version, I thought I would offer you myupdates. (I promise for the last time.)
Probably the nicest thing I have to say about the updated paper is that it isbetter than the original. My complaints about the tone of the paper remain thesame, but I don’t think it is necessary for me to revisit them here.
Concerning the intellectual merit, I think it is worth making the following remarks. First, I am only address the contributions to mathematics, Second, whatcounts as a new conjecture is not really as obvious as it sounds. Since continued
fractions are somewhat recherché, it might be more helpful to give an analogy with
infinite series. Suppose I claimed it was a new result that
∞
X
1
5
5
269
1219
2G =
an = 1 + +
+
+
−
+ ...
2
36
72
3600
705600n=0where for n ≥ 4 one has
2n2 an = n2 an−1 − 2(n − 2)2 an−2 + (n − 2)2 an−3 .
How can you evaluate this claim? Quite probably this is the first time this result
has been written down, and you will not find it anywhere in the literature. But itturns out that
!
!
∞
∞
∞
X
X
X
xn
(−1)n x2n+1
×
=
an xn
n
2
2
(2n
+
1)
n=0
n=0
n=0
and letting x = 1 recovers the identity above and immediately explains how toprove it. To a mathematician, it is clear that the proof explains not only why theoriginally identity is true, but also why it is not at all interesting. It arises as moreor less a formal manipulation of a definition, with a few minor things thrown inlike the sum of a geometric series and facts about which functions satisfy certainPERSIFLAGE: MATH BLOG POSTS

261

types of ordinary differential equations. The point is that the identities produced by
the Ramanujan Machine have all been of this type. That is, upon further scrutiny,
they have not yet revealed any new mathematical insights, even if any particular
example, depending on what you know, may be more or less tricky to compute.
What then about the improved irrationality measures for the Catalan constant?
I think that is a polite way of describing a failed attempt to prove that Catalan’sconstant was irrational. It’s something that would be only marginally publishable ina mathematics journal even with a proof. Results about the irrationality measure in
the range where they are irrational have genuine implications about the arithmeticof the relevant numbers, but these results do not.
What then about the new continued fractions developed over the last year —maybe these are now deeper? Here you have to remember that continued fractions,especially of the kind considered in this paper, are more or less equivalent to questions about certain types of ordinary differential equations and their related periods.
(But importantly, not conversely: most of these interesting ODEs have nothing to
do with continued fractions since they are associated with recurrences of length
greater than two.) For your sake, dear reader, I voluntarily chose to give up anhour or two of my life and took a closer look at one of their “new conjectures.” I
deliberately chose one that they specifically highlighted in their paper, namely:
6 × 132=3+0×7−
8×23−1 + 2G
3 + 1 × 10 −10×33
3+2×13−...

Where G here is Catalan’s constant L(2, χ4 ). As you might find unsurprising, once
you start to unravel what is going on you find that, just as in the example above,the mystery of these numbers goes away. This example can be generalized in a
number of ways without much change to the argument. Let p0 = 1 and q0 = 0, andotherwise letpn
3 33 765 30105 1790775
,
,
,...
= , ,
qn
1 13 313 12453 743403
denote the (non-reduced) partial fraction convergents. If
P (z) =

X 4n pn z n
n!2= 1 + 12z + 132z 2 + . . .

Q(z) =

X 4n qn z nn!2= 4z + 52z 2 + . . .

Then, completely formally, DP (z) = 0 whereD = z(8z − 1)(4z − 1)d2d+ (160z 2 − 40z + 1) + 12(8z − 1)2dz
dz

and DQ(z) = 4. If K and E denote the standard elliptic functions, one observes
that P (z) is nothing but the hypergeometric function
3/2 1/2
2E(16z(1 − 4z))
; 16z(1 − 4z) =2 F11π(1 − 8z)2
But now one is more or less done! The argument is easily finished with a little help
from mathematica. Another solution to DF (z) = 0 is of course
R(z) =

2E((1 − 8z)2 ) − 2K((1 − 8z)2 )= log(z) + 2 + . . .
(1 − 8z)2

262

F. CALEGARI

and knowing both homogenous solutions allows one to write Q(z) = u(z)P (z) +v(z)R(z) and then easily compute that
pn
P (z)
2
= lim
=
.
n→∞ qn−1 + 2Gz→1/8 Q(z)lim

as desired. For those playing at home, note that a convenient choice of u(z) andv(z) can be given by
Z
E(16z(1 − 4z))v(z) == 4z − 8z 2 + . . .
π





−1 + 2G π(1 − 8z)−1/2 1/2 1/2
1/2 1/2 1/2
2
2u(z) =
+
F
;
(1
−
8z)
−
F
;
(1
−
8z)
3 2
3 2
2
2
1 3/2
1 3/2
= −4z log(z) − 4z + . . .
Comment 130.1 (Pupshaw). “Here you have to remember that continued fractions, especially of the kind considered in this paper, are more or less equivalent
to questions about certain types of ordinary differential equations and their related
periods.” — I’d love to understand this statement, where might I start?
Comment 130.2 (Persiflage). Suppose you have a Picard–Fuchs equation overP1 ∖ {0, 1, ∞} with rational coefficients (so an ODE with coefficients in Q(z) .
You can expand a basis of solutions around 0 as power series (with perhaps somelogarithm terms) but with rational coefficients. The coefficients of the holomorphic
solutions will satisfy recurrence relations of the form
A0 (n)un − A1 (n)un−1 − . . . − Ak (n)Un−k = 0
for certain polynomials Ai (n). Given two solutions P (z) and Q(z), say, then since 1
is a singular point, typically what will happen is that P (z) − αQ(z) will have better
convergence properties for some α which will imply that the ratio of the coefficientsof P and Q converge to α. So how to determine α? Well, you can also find a basis ofsolutions with rational coefficients in power series expanded around the point z = 1.
To determine α, you want to write the rational basis around z = 0 in terms of therational basis around z = 1, and for a Picard–Fuchs equation you will exactly see
the periods arising in this matrix (in particular periods of the degenerate motives
in the Picard–Fuchs equations above the singular points).
What is the relation with continued fractions? Well, given any recurrence relationof length 2, i.e. un = R(n)un−1 + S(n)un−2 , then for any two such sequences pn
and qn you can write down a generalized continued fraction out of R(n) and S(n)with partial convergents pn /qn . Of course, you can do this even if the original ODE
is not of geometric type, but then it is a little less obvious what the change of basis
matrix will be, although if you know enough about the solutions to the ODE thenthis might not be a problem.
Comment 130.3 (Will Sawin). And if I understand correctly, you can reverse thisprocess — i.e. given a polynomial continued fraction, find the recurrence satisfied
by its convergents, and write down the associated ODE? Then the next step inconstructing a proof like the one you have is relating that ODE to ODE’s satisfied
by classical special functions (e..g hypergeometrics) and calculating some special
values of these functions?

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263

Comment 130.4 (Persiflage). There are of course many wrinkles! One particularly fascinating one is that for Picard–Fuchs equations you can have non-singularpoints where the Mumford–Tate group of the corresponding Motive changes, andthis has implications for the special values of the corresponding solutions to the
ODE. The classical manifestation of this is of course the hypergeometric function
associated to the Legrendre curve and then specialized at CM points. (This coincidentally comes full circle back to some hypergeometric identities observed by
Ramanujan.) For generalized hypergeometric functions this degeneration also happens but more sporadically, because the codimension of the special locus is > 1.
But it still happens! See, for example, the paper of Dembélé, Panchiskin, Voight,
and Zudilin here. (see [DPVZ22]).

131. Test Your Intuition: p-adic local Langlands edition
Tue, 09 Mar 2021
Taking a page from Gil Kalai, here is a question to test your intuition about
2-dimensional crystalline deformation rings.
Fix a representation:
ρ : GQp → GL2 (Fp )after twisting, let me assume that this representation has a crystalline lift of weight[0, k] for some 1 ≤ k ≤ p. Let R denote the universal framed local deformation
ring with fixed determinant. Now consider positive integers n ≡ k mod p − 1, and
let Rn denote the Kisin crystalline deformation ring also with fixed determinant.
Global considerations suggest that for n ≡ m ≡ k mod p − 1 and n ≥ m, there
should be a surjection Rn /p → Rm /p, and quite possibly one even knows this tobe true. Global considerations also suggest that any representation can be seen inhigh enough weight, which leads to the following problem:
Question 131.1. How large does n have to be to see the entire tangent space of
the unrestricted local deformation ring R? That is, how large does n have to be forthe map R/(p, m2 ) → Rn /(p, m2 ) to be an isomorphism? Naturally, one can also
ask the same question with m2 replaced by mk for any k ≥ 2.
The first question came up in a discussion with my student Chengyang. I madea guess, and then we proceeded (during our meeting) to do a test computation on
magma, where my prediction utterly failed, but in retrospect my computation itself
may have been dodgy so now I’m doubly confused.
Matt remarked that this question is not entirely unrelated in spirit to the Breuil–Mezard conjecture. Instead of counting multiplicities of geometric cycles, one ismeasuring the Hilbert-Samuel function and its “convergence” to that of the freemodule. Also, if you know everything about GL2 (Qp ) and 2-dimensional Galois
representations then you should be able to answer this question too.
Of course I could have re-done the initial computation for this blog post, but I
think at least some readers are happier when I ask questions for which I don’t know
the answer . . .
Notes 131.2. This question was taken up by Chengyang in her thesis, see § 152.
The (still conjectural) answer is surprising to me — I would have guessed that

264

F. CALEGARI

with mk the answer would be of order O(pk ), but it turns out (conjecturally, andasyptotically) to have order O(k 2 ), according to Chengyang’s conjectures.
There was also some interaction between me and Vytas Paškūnas (somewhat
at cross purposes) in the comments about this question (related to whether one
took local deformation rings to include Tp or not. One point Vytas highlighted
was (with the standard way of defining the local Kisin deformation rings) whether
the map Rn+p−1 /p → Rn was surjective. This is still open, although Chengyang’s
conjectures certainly seem to strongly suggest this is true.

132. Potential automorphy for GL(n)Thu, 29 Apr 2021
Fresh on the arXiv, a nice new paper (see [Qia23]) by Lie Qian proving potential
automorphy results for ordinary Galois representations
ρ : GF → GLn (Qp )of regular weight [0, 1, . . . , n − 1] for arbitrary CM fields F . The key step in light of
the 10-author paper is to construct suitable auxiliary compatible families of Galois
representations for which:(1) The mod-p representation coincides with the one coming from ρ,(2) The compatible family can itself be shown to be potentially automorphic.
The main result then follows by an application of the p-q switch. Somethingsimilar was done by Harris–Shepherd-Barron–Taylor [HSBT10] in the self-dual case.
They ultimately found the motives inside the Dwork family. Perhaps surprisingly,
Qian also finds his motives in the same Dwork family, except now taken from a partof the cohomology which is not self-dual!
This result doesn’t quite have immediate implications for the potential modularity of compatible families: If you take a (generically irreducible) compatible familywith Hodge–Tate weights [0, 1, . . . , n − 1] then one certainly expects (with someassumption on the monodromy group) that the representations are generically ordinary, but this is a notorious open problem even in the analogous case of modular
forms of high weight. One way to try to avoid this would be by proving analogous
results for non-ordinary representations. But then you run into genuine difficulties
trying to find such arbitrary residual representations inside the Dwork family overextensions unramified at p. This difficulty also arises in the self-dual situation, andthe ultimate fix in [BLGGT14] was to bypass such questions by applying Khare–
Wintenberger lifting style results. However, such lifting results can’t immediately
be adapted to the l0 > 0 situation under discussion here.
On the other hand, I guess one should be OK for very small n: If M is (say) arank three motive over Q with HT weights [0, 1, 2], determinant ε3 , and coefficientsin some CM quadratic field E (you have to allow coefficients since otherwise themotive is automatically self-dual, (see § 4), then one is probably in good shape. For
example, the characteristic polynomials of Frobenius are Weil numbers α, β, γ ofabsolute value p and will have (as noted in the blog post linked to in the previous
sentence) the shape
X 3 − ap X 2 + ap pX + p3 ,

PERSIFLAGE: MATH BLOG POSTS

265

and now for primes p which split in E, the corresponding v-adic representation will
be ordinary for at least one of the v|p unless ap is divisible by p, which by purity
forcesap ∈ {−3p, −2p, −p, 0, p, 2p, 3p}.
From the usual arguments, one sees that there is at least one ordinary v for almost all
split primes p. The rest of the Taylor–Wiles hypotheses should also be genericallysatisfied assuming the monodromy of M is GL(3), potential modularity in any
other case surely being more or less easy to handle directly. Hence Qian thus proves
such motives are potentially automorphic. A funny thing about this game is that
actually finding examples of non-self dual motives is very difficult, but in this case,van Geemen and Top studied (see [vGT94]) a family of such motives St occurringinside H 2 of the surfacez 2 = xy(x2 − 1)(y 2 − 1)(x2 − y 2 + txy)for varying t (they note that this family was first considered by Ash and Grayson.
Also apologies for changing the notation slightly from the paper, but I prefer todenote the parameter of the base by t). They then compare their particular motive
when t = 2 to an explicit non-self dual form for GL(3)/Q of level 128. I’m sure bythis time (after [HLTT16] and [Sch15b]) someone has verified using the Faltings–
Serre method that S2 is automorphic, but now by Qian’s result we know that the
St are potentially automorphic for all t.
Notes 132.1. The details are carried out in this paper [Mia24]. In particular,[Mia24, Thm 1.1] says that St is potentially automorphic.

133. Divisors near√nTue, 08 Jun 2021
Analytic Number Theory Alert! An even more idle question than normal(that’s because it comes from twitter). Alex Kontorovich noted with pleasure thefollowing pictorial representation of the integers from a Veritasium youtube video,
where prime numbers are represented by 1 × n rectangles and all other numbers
represented as a × b rectangles (of area n) for some a > 1.
This leads to the natural followup questions.
Question 133.1. How much horizontal space does it take to graph the first X
integers this way if one either:(1) Plots the integers n as a × b with a ≤ b as big as possible?
(2) Plot the integers n as a × b with a = 1 if n is prime, and otherwise with aas small as possible, that is, the smallest divisor of n greater than 1?
(From the graph, it actually appears that the second algorithm is actually used.)In both cases, there is a trivial upper bound ≪ X 3/2 . On the other hand, simply
by considering products of primes in the interval [X 1/2 /C, X 1/2 ] for some constantC > 1 you get at least a constant times (X 1/2 / log X)2 integers less than X witha ≫ X 1/2 , and hence a lower bound (in both cases) of ≫ X 3/2 /(log X)2 . But
neither of these bounds are presumably best possible. What then are the precise
asymptotics? This seems like the type of question Kevin Ford might be able toanswer. Actually, this might be a question that Kevin Ford already knows how to266

F. CALEGARI

Figure 12. Number boxes
answer. I summon his spirit from the whispers of the internet to come and answer
this for me. But if that doesn’t work, anyone else should feel free to give an answer
or make a guess.
Friend of the blog Boaty McBoatface emails me to say: I think the second one
is quite easy. I think you just want to computeX
pF (X/p, p)
p<X 1/2

where F (y, p) is the number of integers ≤ y with all prime factors ≥ p. (This has a
name, the Buchstab function). Here the X/p should be ⌊X/p⌋ but this is of little
consequence. Using the trivial bound F (y, p) ≤ y shows that essentially all thecontribution is from p > X 1/2−ε , and in this range a number ≤ X/p has all its
prime factors ≥ p if and only if it is in fact a prime ≥ p . So in fact you want tocomputeXpπ(X/p).
p≤X 1/2

There are various ways to do this more or less carefully, but by splitting into ranges
cX 1/2 < p < (c + 1/N )X 1/2 , summing over c = 0, 1, 2, . . . , N − 1 and then letting
N → ∞ I think one gets
Z 1
4X 3/2
8
X 3/2
2
(1
−
c
)dc
∼
·
.
(log X)2 0
3 (log X)2
The first question is definitely much harder and, as you guess, feels pretty close to
the kind of stuff Ford and Tenenbaum do in their work.
Comment 133.2 (John Voight). Carl Pomerance points to this paper.
Comment 133.3 (Persiflage). Indeed the Erdös’ multiplication problem (howmany distinct integers can you form by multiplying two integers less than N ) seems
closely related, and the work of Tenenbaum also seemed relevant (though I didn’tknow he had worked on the exact problem, giving upper bounds
O(X 3/2 /(log X)c (log log X)1/2 )PERSIFLAGE: MATH BLOG POSTS

267

and lower bounds
O(X 3/2 /(log X)c+ε ).
But since then Kevin Ford made significant advances on this circle of problems (seehere, [For08]), and now perhaps by these methods one can give the precise growthrate, or at least up to some constant terms. For example, maybe the upper boundabove is actually correct up to a constant here. (Note I am lazy and haven’t tried
to work out if it follows easily from the linked paper above . . . )134. 59281
Thu, 19 Aug 2021
The target audience of this blog (especially the mathematics) is usually professional mathematicians in the Langlands program. I do sometimes, however, haveposts suitable for a broader mathematical audience. Very rarely though do I haveanything (possibly) interesting to say to a popular audience. In my recent talk in
the Number Theory Web Seminar, I gave a talk about some math that I’ve discussed with Soundararajan (and which will possibly be written up at some day)about the “average” digit of 1/p in its decimal expansion, in particular, discussing
the distribution of primes for which the average digit of 1/p is less than, equal to,or greater than 4.5 respectively. An easy argument using Cebotarev shows that thedensity of primes for which the average is exactly 4.5 is 2/3. More subtle, however,is that there are more primes for which the average is less than 4.5 than greaterthan 4.5, but still the (upper and lower) density of primes for which the average isgreater than 4.5 is still positive, assuming GRH (the actual percentages of primes
with digit average less than, equal to, and greater than 4.5 are approximately 28%,
67%, and 5% respectively).
I think the talk went well, and one reason I suspect is that it was self-contained.
Moreover, quite a lot of the setup was completely elementary, although certainly
it did move towards deeper topics (Kummer’s Conjecture and work of Patterson and Heath-Brown on equidistribution of Gauss sums, and work of Granville–Soundararajan on the distribution of L-values), it was a result that could more orless be appreciated by an undergraduate.
I decided that this was the time — if ever — that I should make a video post.
I decided to make a “numberphile” style video — complete with brown paper anda title consisting of a single number — by taking my talk and significantly scaling
back the mathematical content. My first attempt was, to put it mildly, a bit ofa disaster. First of all, the aspects of making a video that I know nothing about(lighting, audio, glare, video, editing) were unsurprisingly a complete mess and a
distraction from the actual mathematics. Second, my resident expert felt that it
was still a bit too long, a bit too much like a recording of some lecture, and lacking
a hook. So I cut down the script and made a second even more elementary version.
This version (unfortunately) no longer has me writing on physical brown paper,but it might at least reach a bare minimum audio/video quality.
What’s Special about 59,281?
Just in case you want to skip the video and skip straight to the challenge problem,
here it is:

268

F. CALEGARI

Conjecture 134.1. Let p ̸= 2, 5 be prime, and let C(p) denote the average of thedigits of 1/p in its decimal expansion. (Since the digits repeat this makes sense.)
Then the maximum of C(p) for all primes is achieved by p = 59281, with:
C(59281) =486= 5.11 . . .
951= 0.000016868811254870869249843963495892444459438943337662994
59281
88875018977412661729727906074458932879
A rough heuristic why this should be true: if the period of p is sufficiently large,
then, if the digits are sufficiently random, the probability that the average deviatesthat much from 4.5 becomes exponentially small. Since there are not that many
primes with small period, this leads to the heuristic that all but finitely many
primes should have C(p) very close to 4.5. Moreover, it suggests finding them asfactors of 10n − 1 for small values of n. (59281 is a factor of 1095 − 1.) Making theabove idea more precise suggests that it is highly unlikely to find a counterexample
with period more than 400 or so. Now pari/gp can’t factor most numbers of thisform even for small n, but there is a second competing heuristic. If p is too large
and still has small period, then because 1/p starts out with a bunch of zeros, thissuppresses the digit average. So any big prime factors of 10n − 1 that pari/gp
doesn’t find probably won’t be counterexamples anyway. Note this secondary effectalso explains why C(p) can be significantly less than 4.5 — if p = (10q − 1)/9 isprime, for example, then C(p) = 9/q. Since one expects infinitely many primesof this form (q = 2, 19, 23, 317, 1031, . . .) one expects that C(p) can be arbitrarily
small.
That said, I certainly have not done any significant computation on this question— possibly pari/gp is not finding 10 digit factors of 10n − 1 for odd n < 400 — it
was just an idle question I added to the end of my talk for fun. Hence:(1) I offer a beer to the person who finds the first counterexample.
(2) I offer a bottle of fine Australian wine to the first person who proves theresult. Proofs assuming GRH, for example, are certainly acceptable.
Probably the first thing to try (in order to look for a counter-example) would beto test all primes p < 1010 (say) which are factors of 10n − 1 for some odd n < 1000
or so.
Comment 134.2 (Persiflage). In binary, the corresponding prime could well be
p = 4721 with period 295, where the string of length 295 has 160 ones and 135
zeros for a corresponding average
C2 (4721) =

160
32
=
= 0.542 . . .
295
59

Notes 134.3. Update from the youtube link: Matthew Bolan has carried out thecomputation I suggested above, using in addition information about the factorization of 10n − 1 for small n given at the Cunningham Project (Jonathan Webster
told me about this link). The current records for the primes p with the six highest
values of C(p) = A(1/p) are given in the following table. (I had already found thefour smallest of these primes in my initial search.) After this computation, it looks
like my beer is pretty safe!

PERSIFLAGE: MATH BLOG POSTS

Prime
59281
3076279269802917
53
173
561470969

269

C(p) = A(1/p) Period
5.115789474
4.898734177
4.866028708
4.846153846
4.813953488
4.803108808

95
79
209
13
43
193

135. Polymath proposal: 4-folds of Mumford’s typeTue, 24 Aug 2021
Let A/K be an abelian variety of dimension g over a number field. If g ̸≡ 0 mod 4
and End(A/C) = Z, then Serre proved that the Galois representations associatedto A have open image in GSp2g (Zp ). The result is not true, however, when g = 4,
as first noted by Mumford (in this paper) (see [Mum69]).
The goal of this polymath project is to find an “explicit” example of such a
Mumford 4-fold over Q. There are a number of things I have in mind for what
“explicit” might mean (this is, after all, supposed to be a polymath project so I’m
not supposed to know how to do everything). But here is one way: associated to A
is a compatible family of Galois representations
ρp : GQ → GSp8 (Zp )such that, for some integer N , the Galois representations ρp are unramified outside
N p, and for all other primes q the characteristic polynomial of ρp (Frobq ) is equalto
Qq (T ) ∈ Z[T ]for some polynomial which does not depend on p. Then for example one could hope
to give a list of the polynomials Qq (T ) for a collection of primes q.
Here is the strategy to find such Galois representations. We start by choosing
a totally real cubic field, which for reasons to possibly be explained later should
perhaps be F = Q(ζ7 )+ . (One reason: it is the Galois cubic field of smallest possible
discriminant.)
Step I: Find a Hilbert modular form over F of weight (1, 1, 2) with coefficientsin F .
The idea here will be to follow the strategy employed by [MS15] (following Schaeffer,√ [Sch15a]) to compute a Hilbert modular form of weight (1, 3) over the fieldQ( 5). Namely, Let W denote the space of Hilbert modular forms of weight (2, 2, 3)of some fixed level. Now divide by some suitable Eisenstein series of weight (1, 1, 1)
to get a space V of meromorphic forms of weight (1, 1, 2). This will contain the(possibly zero) space U of holomorphic forms of weight (1, 1, 2). The holomorphic
forms will be preserved under the action of Hecke operators whereas V in general
will not be. Hence one can start computing the intersection of V with its Hecketranslates, which will also contain U . Either you eventually get zero, or you (mostlikely) end up with an eigenform which you can hope to prove is holomorphic byproving its square is holomorphic.
Some Issues: The way that Moy–Specter compute the (analogue) of W is touse Dembélé’s programs to compute the Hecke eigensystems of that weight, and270

F. CALEGARI

then use the fact that q-expansions are determined by the Hecke eigenvalues for
Hilbert modular forms (suitably interpreted, one has to compute spaces of oldforms of lower level etc.). The same idea should certainly work, but note that weare working here in non-paritious weight (that is, not all weights are congruent
modulo 2). My memory is that the current programs on the contrary assume thatthe weight is paritious. This would have to be fixed! Perhaps this is an opportunity
for someone to code up Dembélé’s algorithms in sage?
Step II: Suppose one finds such a form π. Note that I am also insisting thatthe coefficient field be as small as possible, namely the field F itself. Even though
π is of non-paritious weight, there are still associated Galois representations (Somerelevant references are this paper of Patrikis (see [Pat19]) and also this paper of
Dembélé, Loeffler, and Pacetti) [DLP19]. More precisely, there are nice projective
Galois representations, and these lift to actual representations, but they will not
be Hodge–Tate; rather, up to twist (making the determinant have finite order, forexample), they will have Hodge–Tate weights [0, 0], [0, 0], and [−1/2, 1/2]. But now
consider the tensor induction (twisted by a half!) of this representation from GF
to GQ , that is, for σ ∈ Gal(F/Q), the representations2ϱ := ρ(π) ⊗ ρσ (π) ⊗ ρσ (π)(1/2)
Now these representations will be crystalline with Hodge–Tate weights[0, 0, 0, 0, 1, 1, 1, 1].
Moreover, they will be symplectic, have cyclotomic similitude character, and (thisis where the assumption on the coefficients of π comes in) will also have Frobenius
traces in Q. OK, I literally have not checked any of those statements at all, but it
kind of feels like it has to be true so that’s what I’m going with. The point of insistingthat the coefficients of π was just F is to make the coefficients of this new system
in Q. But this means (at least conjecturally) that these Galois representations haveto come exactly from an abelian variety of Mumford’s type, because the Galois
representations tell you that the Mumford–Tate group has Lie algebra (sl2 )3 .
Step III: Find this family in a different way. One issue with the construction
above is that the Galois representations are not obviously motivic (or even satisfy
purity!), so they certainly don’t provably come from an abelian variety. But it might
be easier to find the actual variety once one knows its exact level. I’m not quite surewhat I mean by “find” here — it’s an open question as to whether these Mumford
4-folds are Jacobians so I’m not entirely sure what one should be looking for.
Step IV: Bonus: prove that these 4-folds have L-functions with meromorphic
continuations (at least for H 1 but it’s worth checking the other degrees as well)using triple product L-functions.
Some Further Remarks: There are a number of relevant papers by Rutger
Noot that one should be aware of (An particularly relevant example: this one)(see [Noo06]). There are restrictions on the possible level structures that can arisefor Hilbert modular forms of this weight (in particular, they can’t be Steinberg
at some place), so make sure not to waste time computing at those levels. This isrelated to the fact that the corresponding Shimura variety is compact. The actual
associated Shimura variety is isomorphic to P1 over the complex numbers; there’s
some discussion in section 5.4 of Elkies’ paper (see [Elk08]). These Shimura curves

PERSIFLAGE: MATH BLOG POSTS

271

naturally have models over the reflex field, which is F in this case, but actually
they can sometimes be defined over even smaller fields, such as Q. Now I confess I
am confused by a number of points, in increasing order:(1) What is the exact relationship between the model of this Shimura curve
over Q and the moduli problem? This is an issue both with understanding
the moduli problem but also (because of the stackiness issues) differences
between fields of moduli and fields of definition.
(2) Does this Shimura curve have points over R? I think so. If I understandShimura’s paper here, I think the answer is yes. (see [Shi75]).
(3) Does this Shimura curve have points over Q? I think so! Assuming it haspoints over R you only need to check all other finite primes, and the onethat is most worrying is p = 7 but you don’t really even need to check that
one either if the others all work.
(4) Assuming it is P1Q , does that help at all? At the very least it provides succor that lots of A should exist over Q, but it’s not so clear how to go froma point to an equation. (Consider the easier case of Shimura curves corresponding to fake elliptic curves, for example.) Given a complex point, can
one at least reconstruct some complex invariants of A such as its periods?
Probably understanding this Shimura curve and its relationship with themoduli problem (over different fields) as concretely as possible would be a“second track” in this problem. (Presumably an advantage of a polymath
project is that you can attack it from several angles at once.)
Notes 135.1. Some of these steps are active work in progress of my student Abhijit
Mudigonda — stay tuned and ask me more before working on this!

136. Schur–Siegel–Smyth–Serre–Smith
Thu, 25 Nov 2021
If α is an algebraic number, the normalized trace of α is defined to be
T (α) :=Tr(α)[Q(α) : Q].

If α is an algebraic integer that is totally positive, then the normalized trace is atleast one. This follows from the AM-GM inequality, since the normalized trace is
at least the nth root of the norm, and the norm of a non-zero integer is at least
one. But it turns out that one can do better, as long as one excludes the special
case α = 1. One reason you might suspect this to be true is as follows. The AMGM inequality is strict only when all the terms are equal. Hence the normalized
trace will be close to one only when many of the conjugates of α are themselvesclose together. But the conjugates of algebraic integers have a tendency to repel
one another since the product of their differences (the discriminant is also a nonzero integer.) In an Annals paper from 1945 [Sie45], Siegel (building on a previous
inequality of Schur) proved the following:
Theorem 136.1 (Siegel). There are only finitely many algebraic integers withT (α) < λ for λ = 1.7336105 . . .

272

F. CALEGARI

Siegel was also able to √find that the only such integers with normalized trace at
most 3/2 are 1 and (3 ± 5)/2 = ϕ±2 for the golden ratio ϕ (We will also prove
this below). On the other hand (generalizing these examples), one has



1
,
T ( (ζp + ζp−1 )2 = 2 1 −p−1
and hence the optimal value of λ is at most 2. Sometime later, Smyth had a very
nice idea to extend the result of Siegel. (An early paper with these ideas can befound here.) Consider a collection of polynomials Pi (x) with integral coefficients,and suppose that
X
Q(x) = −λ + x −ai log |Pi (x)| ≥ 0
for all real positive x where Q(x) is well-defined, and where the coefficients ai arealso real and non-negative. Now take the sum of Q(x) as x ranges over all conjugatesof α. The key point is that the sum of log |Pi (σα)| is log of the absolute value ofthe norm of Pi (α). Assuming that α is not a root of this polynomial, it follows thatthe norm is at least one, and so the log of the norm is non-negative, and so thecontribution to the sum (since −ai is negative) is zero or negative. On the other
hand, after we divide by the degree, the sum of λ is just λ and the sum of σα is thenormalized trace. Hence one deduces that T (α) ≥ λ unless α is actually a root of
the polynomial Pi (x). So the strategy is to first find a bunch of polynomials withsmall normalized traces, and then to see if one can construct for a constant λ as
close to 2 as possible some function Q(x) which is always positive. One can makethis very explicit. Suppose that
Q(x) = −λ + x −43187· log |x| −· log |x − 1| −· log |x − 2|,
50
25
50

Problem 136.2 (Calculus Exercise). Show that, with λ = 1.488753 . . ., that
Q(x) ≥ 0 for all x where it is defined. Deduce that the only totally real algebraic
integer with T (α) ≤ λ is α = 1.
The graph is as follows:

Figure 13. The function Q(x) is positive
One can improve this by increasing λ and modifying the coefficients slightly,
but note that we can’t possibly modify this with the given polynomials to get λ >PERSIFLAGE: MATH BLOG POSTS

273

3/2, because T (ϕ2 ) = 3/2. Somewhat surprisingly, we can massage the coefficients
reprove the theorem of Siegel and push this bound to 3/2. Namely, take3
Q(x) = − + x − a log |x| − (2a − 1) log |x − 1| − (1 − a) log |x − 2|,2
and note that the derivative satisfies
Q′ (x)x(x − 1)(x − 2) = (x2 − 3x + 1)(x − 2a),Hence the minimum occurs at either x = 2a or at the conjugates of ϕ2 where ϕ is
the golden ratio. Since ϕ2 − 1 = ϕ and ϕ2 − 2 = ϕ−1 , one finds that3
Q(ϕ2 ) = − + ϕ2 + (2 − 5a) log ϕ,2and so choosing a so that this vanishes when, we get21
a= + √= 0.864674 . . .
5 2 5 log ϕand then we find that Q(x) ≥ 0 for all x where it is defined with equality at ϕ2
and ϕ−2 . So this reproves Siegel’s theorem by elementary calculus. Of course we
can strictly improve upon this result by including the polynomial x2 − 3x + 1, forexample, replacing Q(x) by
1
3
P (x) = Q(x) −· log |x2 − 3x + 1| +−λ15
2
where λ = 1.5444 . . . is now strictly greater than 3/2. By choosing enough polynomials and optimizing the coefficients by hook or crook, Smyth beat Siegel’s valueof λ (even with an explicit list of exceptions), although he did not push λ all theway to 2. This left open the following problem: is 2 the first limit point? That is,does Siegel’s theorem hold for any λ < 2? This was already asked by Siegel and it
became known as the Schur–Siegel–Smyth problem.
Some point later, Serre made a very interesting observation about Smyth’s argument. (Serre’s original remarks were in some letter which was hard to track down,
but a more recent exposition of these ideas is contained in this Bourbaki seminar,see [Ser19].) He more or less proved that Smyth’s method could never prove that
2 was the first limit point. Serre basically observed that there existed a measure µ
on the positive real line (compactly supported) such thatZ
log |P (x)|dµ ≥ 0
for every polynomial P (x) with integer coefficients, and yet withZ
xdµ = λ < 2for some λ ∼ 1.89 . . .. Since Smyth’s method only used the positivity of these
integrals as an ingredient, this means the optimal inequality one could obtain bythese methods is bounded above by Serre’s λ. On the other hand, Serre’s result
certainly doesn’t imply that the first limit point of normalized traces of totally
positive algebraic integers is less than 2. A polynomial with roots chosen uniformly
from µ will have normalized trace close to λ, but it is not at all clear that one candeform the polynomial to have integral coefficients and still have roots that are all
positive and real.

274

F. CALEGARI

I for one felt that Serre’s construction pointed to a limitation of Smyth’s method.
Take the example of Q(x) we considered above. We were able to prove the resultfor λ = 3/2 by virtue of the fact that Q(x) = 0 at these points. But that required
the fact that the three quantities:
ϕ2 , ϕ2 − 1 = ϕ, ϕ2 − 2 = ϕ−1
were all units and so of norm one. The more and more polynomials one inputs
into Smyth’s method, the inequalities are optimal only when Pi (α) is a unit for all
the polynomials Pi . But maybe there are arithmetic reasons why non-Chebychev
polynomials (suitably shifted and normalized) must be far from being a unit whenevaluated at (ζ + ζ −1 )2 for a root of unity ζ.
However, it turns out my intuition was completely wrong! Alex Smith has justproved (see [Smi24]) that, for a measure µ on (say) a compact subset of R withcountably many components and capacitance greater than one, that if Serre’s (necessary) inequality
Z
log|Q(x)|dµ ≥ 0
holds for every integer polynomial Q(x), then you can indeed find a sequence ofpolynomials with integer coefficients whose associated atomic measure is weakly
converging to µ. In particular, this shows that Serre’s example actually proves the
maximal λ in the Schur-Siegel-Smyth problem is strictly less than 2, and indeed isprobably equal to something around 1.81 or so. Remarkable! I generally feel thatmy number theory intuition is pretty good, so I am always really excited when I
am proved wrong, and this result is no exception.
Problem 136.3 (Exercise for the reader). One minor consequence of Smith’s argument is that for any constant ε > 0, there exist non-Chebyshev polynomials
P (x) ∈ Z[x] such that, for primes p say and primitive roots of unity ζ, one haslog NQ(ζ)/Q P (ζ + ζ −1 ) < ε[Q(ζp ) : Q]
for all sufficiently large primes p. Here by non-Chebyshev I mean to rule out “trivial”
examples that one should think of as coming from circular units, for example withP (ζ + ζ −1 ) = ζ k + ζ −k for some fixed k. Is there any other immediate constructionof such polynomials?
For that matter, what are the best known bounds for the (normalized) norm of
an element in Z(ζ) which is not equal to 1, and ruling out bounds of elements in
the group generated by units and Galois conjugates of 1 − ζ? I guess one expects
the class number h+ of the totally real subfield field to be quite small, perhaps
even 1 infinitely often. Then, assuming GRH, there should exist primes which split
completely of order some bounded power of log |∆K |, which gives an element of verysmall norm (bounded by some power of [Q(ζ) : Q]). However, this both uses many
conjectures and doesn’t come from a fixed polynomial. In the opposite direction,
the most trivial example is to take the element 2 which has normalized norm 2, but
I wonder if there is an easy improvement on that bound. There is an entire circleof questions here that seems interesting but may well have easy answers.
Comment 136.4 (Noah Snyder). Ooh, this is the calculus argument in Lemma 5of our joint paper [CMS11]! Really cool to see the bigger picture that it’s part of.
My memory is playing tricks on me, I distinctly remember (and our emails seemto confirm it) that I’d come up with this argument myself (while on a train along

PERSIFLAGE: MATH BLOG POSTS275

the Hudson on Christmas Eve). But we cite the Smyth paper elsewhere so surely
it must somehow have come from there indirectly?
Comment 136.5 (Persiflage). I think you indeed did come up with an idea moreor less equivalent to this independently (although not in the ’80s!), but I don’t
think we fully realized it. By the time of Zoey Guo’s thesis paper [CG18b], there isa more direct reference to Smyth’s work.
Comment 136.6 (Noah Snyder). Ah great, yes that turn of events is spelled out
nicely in Guo’s paper. Glad my memory isn’t totally gone, though disappointed that
now all my “original” number theory arguments turned out to be known, because
my other number theory argument here (see [Sny00]) is in some lectures of Osterlefrom the late 80s.
Notes 136.7. Alex’s paper [Smi24] is actually one of only two papers I ever solicited
for the Annals during my time as an associate editor (so far). The other was byLue Pan, but by then he had already submitted it [Pan22].

137. ArXiv × 3
Fri, 04 Mar 2022
Three recent arXiv preprints this week caught my interest and seemed worth
mentioning here.
The first is a paper by Oscar Randal-Williams (see [RW22]) which considers(among other things) the cohomology of congruence subgroups of SLN (Z) in thestable range. (See Note 111.2). This is definitely something I have talked on the blog
about a number of times, including § 64 and § 111. To recall; Matthew Emerton
and I proved that the completed cohomology groupse d (Fp ) = lim H d (SLN (Z, pn ), Fp )H
are independent of N for N sufficiently large with respect to d, and are moreover
finite vector spaces with a trivial action of G = SLN (Zp ). I later explained moreover how these groups are the cohomology groups of the homotopy fibre of themap from SK(Z; Zp ) to SK(Zp ; Zp ). But now the Quillen–Lichtenbaum conjecture
shows (thanks to Blumberg and Mandell, see [BM20]) how the homotopy groupsof these spaces are identified with Galois cohomology groups, which allows one tocompute the maps between homotopy groups and understand (at the very least)
the cohomology groups in degrees less than p. Since one has a Hochschild–Serre
spectral sequencee j (Fp )) ⇒ H i+j (SL(Z, p), Fp ),E i,j = H i (G(p), H
2

this allows one to compute the cohomology of SL(Z, p) over Fp in low degree by
analyzing this spectral sequence. I later came to suspect that for regular primes p
this spectral sequence degenerated immediately at least in degrees less than p orso, which would allow one to compute the cohomology groups in degree d explicitly
for all large regular p. Actually the prediction was slightly stronger: in the range ofcohomology degrees at most d one only had to avoid a finite set of primes (those
dividing B2k for small k together with the set of primes p which divided the finitely
many zeta values ζp (3), ζp (5), . . . ζp (2k + 1) also for small k). Oscar not only proves
this but goes one step further, by showing that it degenerates in small degrees

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F. CALEGARI

for any prime p, even as a SL(Fp )-module. This implies, for example, that, with
H 1 (G(p), Fp ) = M being more or less the adjoint representation, that
H 4 (SLN (Z, p), Fp ) = Fp ⊕ ∧2 M ⊕ ∧4 M
for p > 5 if and only if p does not divide the p-adic zeta function ζp (3), andH 4 (SLN (Z, p), Fp ) = Fp ⊕ Fp ⊕ ∧2 M ⊕ ∧4 M
otherwise. Note this condition implies that p is irregular but is much more restrictive. But it does actually happen! The only known primes with this property are
p = 16843 and p = 2124679.
Part of my original interest in this problem came from Benson Farb and Tom
Church — they noted that these groups should be stable in the weaker sense thatthey should be “independent of N ” more or less exactly in the sense that there
is a uniform description as above (proved later by Andrew Putman, see [Put15]),
but this left open the question of what the groups actually were. Of course my
feeling is that the completed cohomology groups are more “fundamental” and thecohomology at finite level is really just a frothy mix of unwinding what happens in
the limit, but one has to admit that this new result is pretty satisfying.
————————The second is a paper by Will Sawin and Melanie Wood (see [SW24]). I remember 20 years ago or so being one of three BPs at Harvard asked to give asmall presentation to the Harvard “Friends of Math” (Will Hearst and the gang),along with William Stein and Nathan Dunfield. One memory was that my talk
was a chalk talk and theirs were both involved much snazzier technology. But I
also remember that Nathan talked about his very nice paper with Bill Thurston(see [DT06]) on random 3-manifolds. In Melanie and Will’s new paper, they beautifully exploit many of the recent progress on “random groups” (much of it due to theauthors themselves) to show that the profinite completion of a random 3-manifold(in the sense of a random Heegaard splitting for larger and larger genus) itself has
a limiting distribution.
Here is just one immediate corollary of their results which ties into previous
problems considered both by Nathan and me and also Nigel Boston and Jordan
Ellenberg. (Actually I say corollary, but I am just guessing that this should easily
be a corollary without actually doing any of the computation so any error here isdue to me!)
Corollary 137.1 (Expected). For a fixed prime p > 2 and a “random” 3-manifold
M , there is a positive probability that:(1) There is a surjection: π1 (M ) → SL2 (Zp ),(2) The corresponding tower of covers Mn coming from congruence subgroups
all have trivial first Betti number.
The point of course being that (as in Boston–Ellenberg [BE06]) one can deduce
this from the more restrictive condition that the kernel N of the map
π1 (M ) → SL2 (Fp )p3

has N/N = (Fp ) and no larger, and hence it can be phrased as the pro-finite
completion of π1 (M ) surjecting onto one pro-finite group but not some other finitegroup. (Here N/N p = (Fp )3 can I think be weakened to N/N p [N, N ] = (Fp )3 by
an argument of Simon Marshall). I guess another way of saying this is that the

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pro-p completion of the cover N can be described explicitly as the p-congruencesubgroup of SL2 (Zp ). Of course, this work also raises the very natural question:
Question 137.2. What is the distribution of π\
1 (M ) on arithmetic 3-manifolds?
What about congruence arithmetic 3-manifolds?
The main point of course is that the existence of Hecke operators imposes a lotof extra structure, which one certainly expects (and can be numerically observed)
changes the distribution of any given finite group occurring. Here I think the sensible
question is to ask for a conjecture rather than a theorem, of course! (Maybe thefirst sensible question is actually to give a good conjecture for the distribution of
the abelianization of these groups. . . )————————The last paper is this one by Peter Kravchuk, Dalimil Mazáč, and Sridip Pal [KMP24],
which I am even less qualified to talk about, which gives remarkable upper bounds
for the smallest Laplacian eigenvalue of a (closed) hyperbolic orbifold of fixed genus.
For example, when g = 2, they give the bound λ1 < 3.8388976481, which is not too
shabby given that there is an example with λ1 = 3.83888725 . . .! The paper has anumber of other gems, including more or less identifying the complete spectrum ofall λ1 as comprising a set of isolated points combined with the entire interval [0, α]
for some α = 15.8 . . ..

138. A random curve over Q
Sat, 02 Apr 2022
Let X/Q be a smooth projective curve. I would like to be able to say that the
motive M associated to X “generally” determines X. That is, I would like to say it
in a talk without feeling like I’m telling too much of a fib. But is this true? There
are two issues. Recall that, by the Torelli Theorem, the Jacobian together with aprinciple polarization determines X/C. So there are two things to worry about:(1) Knowing M only recovers the Jacobian up to isogeny, and you can certainly have two different curves with isogenous Jacobians, even isomorphic
Jacobians with different polarizations.
(2) Knowing X/C does not determine X/Q.
To overcome the second issue, it is sufficient and necessary to assume that
AutC (X) is trivial. Let me ignore the first point, since I both assume it generically doesn’t happen but since I can’t even address the second point yet I haven’t
thought about it yet.
Perhaps this is obvious to a geometer, but I don’t see why a “random” curve
X/Q doesn’t have automorphisms. My model of a random curve is to take, forexample, an embedding of Mg into projective space and then count points by theambient height function and see what ratio of points has trivial automorphisms.
(Presumably any other counting function like Faltings height or whatever will moreor less be the same.) Certainly a generic C-point of Mg has no automorphisms (atleast for g > 2), but since Mg is of general type for large enough g I don’t whether
one can find enough rational points which are generic!
Probably the most natural way to answer this is to give a positive answer to thefollowing question:

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F. CALEGARI

Question 138.1. Does Mg contain a subvariety X which is unirational over Q andhas dimension strictly greater than the hyperelliptic locus?
Or, to put it more naturally, can you just explicitly write down enough genericcurves which don’t have any automorphisms to see that they dominate any point
count?
Comment 138.2 (Felipe Voloch). Trigonal curves. See the comments (especiallythe one from Jason Starr) to this answer of mine: this answer.
Comment 138.3 (Will Sawin). I don’t think anyone can rule out that there isa unirational subvariety of dimension 3g − 4 for g arbitrarily large, for example.
Even if that was known, that wouldn’t answer your question about heights, since alower-dimensional subvariety can have a larger count of points with bounded height
than a higher-dimensional one. I don’t think there’s much hope to formulate theclaim for anything but a specific family — plane curves, n-gonal curves for n from 2
to 5, complete intersections in higher-dimensional space . . .
There’s a general philosophy related to the Bombieri–Lang and Manin conjectures – I don’t remember exactly which combination of conjectures one could assume
to draw this as a conclusion but probably there is one — that most of the rational
points on a general type variety should lie on its Fano subvarieties, whichever ones
have the most Manin-predicted points, if it has any Fano subvarieties at all.
The point being that all but finitely many rational points should be explainedby varieties that have a lot of rational points (Fano, abelian varieties, Calabi-Yaus)
but Fano varieties have vastly more points than the others (except Calabi-Yaus
which themselves contain Fanos). For example, any rational curve contained in any
variety has vastly more rational points than any abelian subvariety, no matter theample height function you choose.
Thus, Manin-type predictions could come from looking for Fano subvarieties
and trying to find the ones that have the most predicted rational points. I don’t
remember my idea above but I think it had to do with assuming that if there aremany ration points of Mg with automorphisms then there must be a Fano variety
parameterizing rational points of Mg with automorphisms and then showing that
this maps to a Fano variety parameterizing points of Mg′ for g ′ < g and trying tocompare the numbers of rational points.
Comment 138.4 (Andrew Sutherland). How about this example, the genus 3
curvesy 2 + (x4 + x3 + x2 + 1)y = x7 − 8x5 − 4x4 + 18x3 − 3x2 − 16x + 8
andx3 z + x2 yz + x2 z 2 + xy 3 − xy 2 z + y 4 − y 3 z − yz 3
are both generic in the sense that the have no extra endomorphisms, but one is
hyperelliptic and the other is not.
While I have not tried to run a Faltings-Serre computation to prove it, I’m
morally certain these two genus 3 curves have the same L-function (their Frobenius
traces agree out to 228 and the conductor is only 8233), hence they have isogenous
Jacobians. But these two curves really are different in meaningful sense.

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139. What would Deuring do?
Wed, 13 Apr 2022
This is an incredibly lazy post, but why not!
Matt is running a seminar this quarter on the Weil conjectures. It came upthat one possible way to prove the Weil conjectures for elliptic curves over finite
fields is to lift them to CM elliptic curves using Deuring’s theorem. But after somediscussions we couldn’t quite work out whether this was circular or not.
Certainly if you can lift to a CM elliptic curve and lift Frobenius to an endomorphism ϕ of the lift you get Weil immediately; the degree of ϕ is p which implies
the norm of ϕ is p, but for imaginary quadratic fields the norm coincides with theabsolute value. But how did Deuring prove his theorem?
The most obvious way to lift an (ordinary, say) elliptic curve E/Fp to characteristic zero is to note that, by the Weil conjectures, the order O = Z[ϕ] generated
by Frobenius lies inside an imaginary quadratic field K (this is equivalent to theWeil conjectures), and so one can consider C/Z[ϕ]. To make things simple, if theorder is maximal, then this is defined over the Hilbert class field H of K, and since
p splits principally in K (since ϕ has norm p) it follows that p splits principally in
H as well by class field theory, and so the CM elliptic curve is also defined over Zp
and gives a lift. Of course, this argument uses the Weil conjectures! Without that,the ring O lives inside a real quadratic field and it’s not clear what one can do.
One approach is to prove the existence of the canonical lift, which automaticallywill have extra endomorphisms and thus be CM since it lives in characteristic zero.
This doesn’t depend on the Weil conjectures. But the canonical lift is a construction
I associate more with Serre–Tate than with Deuring. But it’s certainly possible that
Deuring’s argument was via the canonical lift.
Some might say that the easy way to solve this is simply to look in one of
Deuring’s papers. But instead I will try to call upon my readers (possibly either
number theorists who speak German or Brian Conrad) to save me the work andtell reveal all in the comments!
Comment 139.1. Dick Gross] Frank, You may still need to brush up on yourGerman, but I believe that the first proof of the Riemann hypothesis (Artin’s conjecture) for elliptic curves over finite fields via lifting to characteristic zero is due
to Hasse [Has36]. Surely you are not too lazy to have a look!
What Deuring did was to refine these arguments to describe all possible endomorphism rings of elliptic curves in char p ≥ 0. They are: Z (which cannot occur
over finite fields), an order of conductor prime to p in an imaginary quadratic field
where p splits, a maximal order in the quaternion algebra over Q ramified at p and∞.

140. Murphy’s law for Galois deformation rings and Ozaki’s theorem
Sat, 23 Apr 2022
A theorem of Ozaki from 2011 [Oza11], perhaps not as widely known as expected,
says the following:
Theorem 140.1. [Oza11] Let p be prime, and let G be a finite p-group. Then there
exists a number field F and an extension H/F such that:

280

F. CALEGARI(1) H/F is the maximal pro-p extension of F which is everywhere unramified.
(2) Gal(H/F ) = G.
Since any non-trivial p-group G has a non-trivial center, it can be written as acentral extension of a smaller p-group G′ by Z/pZ, and thus the proof is (as one
might imagine) by induction. But the structure of the argument is quite tricky, andit’s a little hard to absorb all the ideas at once.
This post is to report on two extensions of Ozaki’s result. The first is a new
preprint by Hajir, Maire, and Ramakrishna [HMR24] which gives both a simplification and also an extension of Ozaki’s result (the extension being that one has more
explicit control over the degree of F ).
But this post will actually be about a somewhat different generalization due tomy student Andreea Iorga. Let me give her result now:
Theorem 140.2. Iorga Let Φ be a finite group of order prime to p, and let G bea finite p-group with an action of Φ. Assume there exists an extension L/K with
Galois group G such that:(1) L/K is Galois with Galois group Φ,(2) L contains ζp ,(3) L has trivial p-class group.
Then there exists number fields H/F/E such that:(1) H/F is the maximal pro-p extension of F which is everywhere unramified.
(2) Gal(H/F ) = G.
(3) Gal(H/E) = G ⋉ Φ, where the semi-direct action is the given action of Φ
on G.
When Φ is trivial one recovers Ozaki’s theorem in the case when p is a regularprime. In fact, Ozaki’s first proof also has a similar hypothesis. Most likely Iorga’s
argument extends to the more general case where one does not need to assume thatζp ∈ E. (Of course, in order not to accidentally solve the inverse Galois problem,the other two conditions on L and K will be necessary! One nice consequence(and a motivating example) of Iorga’s theorem is as follows. Consider an absolutely
irreducible residual representation:
ρ : GK → GL2 (Fp )to a finite field. What possible rings R can occur as deformation rings of ρ? In thissetting, let R denote the deformation ring of everywhere unramified representations.
Let’s also assume that the image (to be absolutely concrete) has image which hasorder prime to p, say projectively Φ = A4 or S4 . The Fontaine–Mazur conjecture
predicts that the only Qp -points will have finite image, and thus correspond to
the natural lift (assuming the characteristic of k is p > 5. An argument with classgroups then implies that one should expect R[1/p] = Qp , or equivalently that R isa ring admitting a map
R → Zpwith finite (as a set) kernel I. A consequence of Iorga’s theorem is the following:
Theorem 140.3. Iorga Let R be any local ring admitting a surjection to Z5 withfinite kernel. Then R is a universal everywhere unramified deformation ring.
The key idea to reduce this theorem to the previous one is a follows. Supposee Since this has order prime to p, it lifts to a representationthat the image of ρ is Φ.

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e ⊂ GL2 (Zp ). Then denote Γ denote the inverse image of this group inside GL2 (R),Φ
so it lives inside an exact (split) sequence:e →1
1 → 1 + M2 (I) → Γ → Φ
The group Γ admits a natural residual representation via ρ, and clearly Γ admits adeformation to GL2 (R) by construction. The point is that one can show that this R
is the universal deformation ring, and hence providing one has extensions H/F/E
with Gal(H/E) = Γ and H/F the maximal everywhere unramified pro-p extensionof F (using the previous theorem) one is in good shape. (There is a trick to reduce
the problem in this case to Φ in order to make the “base case” easier, since one has
e and if Φ = A4 andfields F/Q and Fe/Q with Gal(F/Q) = Φ and Gal(Fe/Q) = Φ
p = 5 then proving that F (ζ5 ) of degree 96 has class number prime to 5 is easierthan the same clam for Fe(ζ5 ) of degree some multiple of 192.
One way to view this result is as an example of “Murphy’s Law” for moduli
spaces. Here Ravi Vakil proves [Vak06] in a different setting that all possible singularities occur inside deformation spaces. The analog is to say that all possible
local rings (subject to some obvious constraints) occur as Galois deformation rings.
Another natural class of rings one might expect to arise in this way (still considering everywhere unramified deformation rings) is all finite rings. Of course for suchrings one would have to consider residual representations whose images have orderdivisible by p, requiring a further modification of the theorems of Ozaki and Iorga.
In a different direction, one can ask what happens for deformation conditions withother local conditions at p. Here are two natural such questions:
Problem 140.4. Let (R, m) be any complete local Noetherian ring with finite
residue field which is finite over W (k). Then does R occur as the finite flat deformation ring of some absolutely irreducible residual representation?
Problem 140.5. Let (R, m) be any complete local Noetherian ring with finite
residue field over W (k), and assume that:(1) R is a complete intersection, namely that there is a presentation:
R ≃ W (k)[[x1 , . . . , xd ]]/(f1 , . . . , fr )where d ≥ r.
(2) p ∈ R is a regular element.
Then does R occur as the universal deformation ring (with fixed determinant) of
some absolutely irreducible residual representation?
I would guess the first problem has a positive answer but I’m honestly not evensure about the second one!
Comment 140.6 (Persiflage). There are some countability issues with Problem 140.5.

141. Joël Bellaïche
Tue, 14 Jun 2022
Very sad to hear that Joël Bellaïche has just died. He got his PhD at the sametime as me, and I first got to know him during the Durham conference in 2004 andlater at the eigenvarieties semester at Harvard (was that in 2005 or 2006?).

282

F. CALEGARIJoël was an original mathematician, and his papers (many written with Gaëtan
Chenevier) contain many really good ideas. As a postdoc, I was totally immersedin thinking about Galois deformations of reducible representations when the paper lisseté de la courbe de Hecke de GL2 aux points Eisenstein critiques [BC06]
appeared on the arXiv. In that paper, they study the ideal of reducibility for certain Galois deformation rings (or pseudo-deformation rings). By studying the ringtheoretic properties of this ideal, they proved the Eigencurve was smooth at theevil Eisenstein points. It clarified immediately a number of the phenomena I had
been thinking about, but it was also simply the “right” way to think about thesethings. I also learnt from Joël at Durham the problem of proving the non-vanishingof p-adic zeta values like ζp (3) ̸= 0, which remains 18 years later one of my favourite
problems.
Another really beautiful idea was the approach by Joël and Gaëtan to Bloch-Kato
type conjectures (including the Selmer group part of the Birch–Swinnerton-Dyer
conjecture) via the geometry of eigenvarieties (including those associated to U (3)).
This is of course related to the ideal of reducibility. Their joint asterisque paper
Families of Galois representations and Selmer groups [BC09] is a very nice read on
this topic, as are Joël’s notes for the Clay summer school as well as his recent bookon Eigenvarieties (see [Bel21]).
In more recent times, Joël had been exploring ideas in some interesting directions,
including his intriguing work on self-correspondences on curves [Bel23]. What wasconsistent about his research was that his primary motivation always seemed to
be rooted in coming to an original understanding of interesting math rather thansimply making incremental improvements on work of others.
Last but not least, one should not forget his sense of humor with a decidedly
irreverent streak. This is probably best appreciated with a beer or a glass of wine
in a summer evening in Luminy, but to take a quote from on of Joël’s own papers:
Let p be a prime number that, we shall assume, splits in E. We
shall also assume that p ̸= 13. I don’t think this is really useful, but
who knows?
My thoughts are with his family.
Comment 141.1 (Clémentine Fauré-Bellaïche). I am Joël’s wife and I came uponthis blog post by chance — merci beaucoup, it deeply touched me.
Comment 141.2 (Danièle Kuzmanovic). Extrêmement peinée par le décès deJoël ! Je l’ai connu enfant, il venait jouer chez nous à la maison. Notre fils, Djordje
Kuzmanovic, en avait fait son meilleur ami lors de la rentrée scolaire au collège
Montaigne de Paris (1984). Nos pensées affectueuses vont à ses parents et à sonépouse. Danièle et Dejan Kuzmanovic.
Comment 141.3 (Régis Lebrun). J’apprends le décès de Joël avec une infinie
tristesse. Il était mon meilleur ami lors de l’année passée en TC1 au lycée Louisle Grand en 1990/1991 et dans les années qui ont suivi. Nous avons partagé desmoments inoubliables ensemble dans les gorges du Verdon, puis les concours etdes parcours professionnels différents nous ont fait nous perdre de vue. Je voudrais
exprimer mes sincères condoléances à ses parents et son épouse. Cette disparition
prématurée enlève une personne exceptionnelle à ses proches et à la communauté.
Régis Lebrun

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142. Locally induced representations
Thu, 16 Jun 2022
Today is a post about work of my student Chengyang Bao [Bao22]. Recall that
Lehmer’s conjecture asks whether τ (p) ̸= 0 for all primes p, where
∆=q

∞
Y

(1 − q n )24 =Xτ (n)q nn=1

is Ramanujan’s modular form. You might recall that Naser Talebizadeh Sardari andI (in [CTS21], see also § 120) studied a “vertical” version of Lehmer’s conjecture
where instead of fixing a modular form, we fixed a prime p and a tame level N andshowed that there were only finitely many normalized eigenforms f of level N andeven weight k with ap (f ) = 0 which were not CM. We exploited the fact that suchforms give rise to Galois representations
ρ : GQ → GL2 (Qp )which are crystalline at p but also locally induced at p from the unique unramified
quadratic extension K/Qp . As explained in § 122, it’s hard to see this method being
able to say much more to this (for example, to say anything about Lehmer’s actual
conjecture), since there do actually exist non-CM forms with ap (f ) = 0.
In practice, we don’t even know in level N = 1 whether there exist infinitely
many normalized eigenforms f with ap (f ) ≡ 0 mod p. As mentioned in § 69, one
source of such representations comes from modular forms with exceptional image.
For example, if f = ∆E4 is the normalized eigenform of weight 16, then as firstobserved by Serre and Swinnerton-Dyer, the mod-59 representation
ρf : GQ → GL2 (Fp )has projective image S4 coming from the splitting field of x4 − x3 − 7x2 + 11x + 3.
But the local residual representation in this case is induced, which implies thata59 ≡ 0 mod 59. As explained in that post, standard conjectures about primes
predict that there should be infinitely many S4 -representations unramified outside
a single prime p giving rise to modular Galois representations which will then comefrom level one modular forms f with ap (f ) ≡ 0 mod p.
Chengyang’s work concerns examples precisely of this sort. From my work with
Naser, we can deduce that there are at most finitely many f of level one witha59 (f ) = 0. Chengyang proves that there are no such forms. More precisely:
Theorem 142.1 (Bao [Bao22]). Suppose that f is a modular form of level one, andsuppose that ap (f ) = 0. Then all of the residual mod-p representations ρ associatedto f have big image, that is, image containing SL2 (Fp ).
In other words, none of the (presumably many) infinite examples of S4 representations giving rise to f of level one with ap (f ) ≡ 0 mod p can ever give an f withap (f ) = 0.
Chengyang also proves some further results about the deformations of representations with exceptional image. For example, for the mod-59 representation above,
the only deformations to characteristic zero unramified outside p = 59 which arelocally induced are the representations which up to twist are the ones which up to
twist coincide with the unique lift with finite image and order prime to p = 59.

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F. CALEGARI

In contrast, one might ask what happens for p = 79, the next case where there
exists a form f of level one with ap (f ) ≡ 0 mod p. I suspect that in this casea (possibly quite complicated computation) should show that there should be at
most one form with ap (f ) = 0, but that it might be quite difficult to prove using
p-adic methods that there are no such forms. The problem will be that there will
exist a deformation which will have infinite image and be locally induced, but now itwill have generalized Hodge–Tate weights [0, κ] for some p-adic number κ for which
it will be very hard to show is not an integer. This is analogous to the family ofEisenstein series of level one with p = 37. One knows that the p-adic zeta function
will have a unique zero, but it is very hard to probe the arithmetic nature of thatzero and to rule out it occurring at some arithmetic weight. To put this slightly
differently, is there an integer k ≥ 1 such that
ζ37 (−31 + 36k) = 0?
Presumably not, but this seems extremely difficult; the difficulty of course is that
there will be a solution with k ∈ Z37 .

143. The future is now; recap from Cetraro
Sun, 31 Jul 2022
I’ve just returned from the second Journal of Number Theory biennial conference in Italy. It’s always nice to get a chance to see slices of number theory one
wouldn’t otherwise see at the conferences I usually go to (although this was thefirst conference of any kind I attended in person since 2019). Here is a brief andincomplete recap.
There were more talks that mentioned the Manin-Mumford conjecture and its
various generalizations (particularly to uniform bounds in families) than I haveever previously attended in my life. There were probably equally many talks which
mentioned Ax–Schanuel as well. It was nice to see these subjects and I learnt quite
a lot from these talks.
143.1. Modularity of elliptic curves over imaginary quadratic fields. In
my ICM talk, I claimed that the modularity of elliptic curves over the Gaussian
integers is “within our grasp”; well, the future is now! James Newton talked about
his work in progress with Ana Caraiani [CN23] where they prove modularity of
all curves over√ imaginary quadratic fields F such that #X0 (15)(F ) < ∞, which
includes Q( −1). One of the key tools in their proof is a suitable local-global
compatibility statement for Galois representations coming from torsion classes in
the crystalline setting where one is not in the Fontaine–Laffaille range (because of
ramification in the base, for example). This was a situation where I had even been
hesitant to make a precise conjecture. The problem is that the natural conjecture
one might want to make is that the map of Hecke algebras factors locally through the
Kisin deformation rings. But the construction of Kisin deformation rings as closures
which are flat over W (k) by default might make one worried whether it is thecorrect integral object for torsion representations. But Caraiani and Newton showthat such concerns are unfounded, and the W (k)-flat deformation rings are indeed
the correct objects. One key point of their argument is showing that the (possibly
torsion) representations ρ ⊕ ρ∨ (for suitable twists of ρ occur inside the cohomology

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of the Shimura variety in such a way where (using some notion of ordinary for aparabolic other than the Borel) the local representations in characteristic zero are
reducible and realize the required crystalline lifts of each factor. One remaining
annoyance is that one would like to find points over twists X(ρE , ∧) of the Klein
quartic X(7) over F corresponding to E[7] which lie on solvable CM fields (inorder to do a switch at the prime 7. You could (for example) start with the point
E and the 15-isogenous curve E ′ and connect them via a line. This line will go
through two further points defined over a quadratic extension H/F , but there is
no reason to suppose a quadratic extension of an imaginary quadratic field will be
CM. I had some idea related to a half-forgotten fact I learnt from John Cremona
walking in the woods near Oberwolfach, but upon further consideration this halfforgotten fact was sufficiently ephemeral that it could not be reconstructed anddidn’t appear to correspond to any actual facts (See Note 143.6 below). I did learn
from Tom Fisher the nice fact that the four curves 3-isogenous to E are collinear
on the curve X(ρE ), 3, ∧) corresponding to the same mod-7 representation with theother choice of symplectic form (that is, ∧ scaled by a quadratic non-residue).
143.2. BSD for abelian surfaces. This was my first chance seeing a talk on thework of Loeffler–Zerbes [LZ21] on BSD for abelian surfaces. The most difficult condition to verify in their theorem is that a certain (characteristic zero) deformation
problem is unobstructed. It seems very plausible to me that one could numericallyverify some interesting examples and get truly unconditional results on BSD forsome autochthonous abelian surfaces, or at least autochthonous elliptic curves overimaginary quadratic fields. The idea is that to prove a certain ordinary (of someflavour) deformation problem is unobstructed it suffices to prove that it is unobstructed modulo p, which reduces to a computation with ray class groups in the
splitting field of ρ : GQ → GSp4 (Fp ). This seems within the realm of practicality.
Moreover, once one verifies the condition for A, one immediately deduces it for all
the twists of A as well. It is important here to take p small, that is at most 3. Certainly if the mod-3 image is surjective the extension will be too large, but the caseof a representation induced from an imaginary quadratic field seems completely
manageable. The other possibility is to work with p = 2. Here I think one should
work with H 1 (Q, ad0 (ρ)) where ad0 is the quotient of the 11-dimensional adjoint
representation by the diagonal (so slightly different from trace zero matrices). HereI’m imagining starting with a modular abelian surface which has good ordinary at
2 and whose mod-2 image is S5 . It might also be convenient if the local factor at
2 is congruent to (1 + T + T 2 )T 2 mod 2 so that the local deformation problem has
good integral properties. Anyone interested in computing such an example?
143.3. The David Goss Prize. During the conference the second “David Goss
Prize” was awarded. This prize is for work done in the past two years in number
theory by someone at most 35(!) and also by someone who has not (yet!) won any
other major prizes. (I joked that it might be nice to have a prize for people 50 orolder who have not won any prizes but there is something nice about no longer being
eligible for any prize except those one has no hope of winning.) This year’s winners
were Ziyang Gao and Vesselin Dimitrov. The laudatio is here. Congratulations toboth!
Talking of prizes, I can’t quite work out whether there are far more prizes thanthere used to be, or whether I was simply oblivious about them when I was younger.

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Comment 143.4 (David Hansen). Do you know the smallest d for which X0 (15)(Q( −d))is infinite? Glancing at [LMF24], it looks this set is finite for d = 1, 2, 3 and infinite
for d = 7, but the info there doesn’t seem to cover the cases d = 5, 6 . . . Anyway,
it’s a great theorem!
Comment 143.5 (Persiflage). James mentioned it in his talk. I think d = 6?
For full √disclosure, I also believe that the theorem also includes the condition thatF ̸= Q( −55). I assume this is because one also has to rule out elliptic curves withother level structures at 3 and 5 which are too small, and this leads to a numberof curves of genus g ≥ 2, and one of those curves has an exceptional point overF . In principle, one can check “by hand” that this E is modular. However, if it
has enormously large level, that could plausibly be an annoying computation. I did
think that James missed a trick by not calling the talk “imagining Elliptic Curves
over imaginary quadratic fields, but particularly not those involving the squarerootof −55.”
Notes 143.6. Now that I can search through posts, the “half-forgotten fact” wasdiscussed here § 102. It is not helpful here.

144. Potential modularity of K3 surfaces
Tue, 15 Nov 2022
This post is to report on results of my student Chao Gu who is graduating this(academic) year.
If A/F is an abelian surface, then one can associate to A a K3 surface X (theKummer surface) by blowing up A/[−1] at the 16 singular points (corresponding
to 2-torsion points of A. If F is a totally real field, then one knows that A is
(potentially) automorphic, and it follows that X is also (potentially) automorphic,
which in particular implies the Hasse-Weil conjecture for X. It also proves that
ρ(X/F ) = −ords=1 L(H 2 (X/F , Qp (1)), s),where H 2 (X/F , Qp (1)) is the etale cohomology group considered as a Galois representation of GF ; this was conjectured by Tate in the same paper where he makes
the “usual” Tate conjecture on algebraic cycles. Not all K3 surfaces arise in thisway. For a start, if A has (geometric) Picard rank ρ(A) ≥ 1, then X has geometric
Picard rank 16 + ρ(A) ≥ 17. If the Picard rank of X is at least 19, then X also has
to arise (at least in the category of Motives) as a Kummer surface, but more subtly
this is not true in rank 17 and 18, where there are further obstructions relating
to the structure of the transcendental lattice (as first observed by Morrison in thispaper, see [Mor84]). What Chao does is prove the following:
Theorem 144.1 (Gu). Let X/Q be a K3 surface of Picard rank at least 17. Then
X is potentially automorphic, and the Hasse-Weil conjecture holds for X.
In the most interesting case of rank 17, the approach is to lift the compatible
family of 5-dimensional orthogonal representations associated to the transcendental
lattice to a compatible family of 4-dimensional symplectic representations which one
hopes to prove is potentially automorphic. Finding (motivic) lifts of K3 surfaces
is a well-studied problem, and a nice analysis of what happens arithmetically canbe found in Patrikis’ thesis. From the Kuga–Satake construction, one can certainly

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reduce to considering certain abelian varieties. The question is then narrowing down
the precise endomorphism structures of these varieties as well as their fields ofdefinition. It turns out that for the problem of interest, there are more or less three
types of abelian varieties one might want to consider beyond abelian surfaces overa totally real field F :(1) Abelian varieties A/F of dimension 2d, where A admits endomorphisms byan order in the ring of integers of a totally real field E of degree [E : Q] = d.
(2) Abelian surfaces A/H over some Galois extension H/F where the conjugate
of A by Gal(H/F ) are all isogenous over H.
(3) Abelian 4-folds A/F with endomorphisms by an order in a quaternion algebra D/Q.
More generally, one needs to consider the “cross-product” where several (or all)of these phenomena may occur at once. For those more familiar with the story of
two-dimensional Galois representations over Q, these three extensions correspondto replacing elliptic curves over Q by abelian varieties of GL2 -type, to Q-curves, andto fake elliptic curves respectively. It turns out that the last case doesn’t happen
over totally real fields but the analog for abelian surfaces does, see § 98.
The optimal generalization of [BCGP21] to this setting would be to prove that
all of these varieties are potentially modular. However, it turns out that there
is an obstruction to proving this: namely, is not always possible to prove thatthese varieties have enough ordinary primes (one needs something slightly stronger,
namely ordinary primes whose unit eigenvalues are distinct modulo p). This puts
some restrictions on what can be proved unconditionally, but everything works
as long as there are enough ordinary primes. Chao’s proof requires a number ofmodifications from [BCGP], in particular to the Moret-Bailly part of the argument.
In our original paper, we exploited the fact that we were working only with abeliansurfaces which allowed us to use some tricks to simplify this step. In particular, theproblem of finding an appropriate point on the desired moduli space over Qp was
made much simpler by virtue of the fact that the original abelian surface produced
such a point. In Chao’s generalization, however, this trick doesn’t work, and one
must use more subtle arguments using Serre–Tate theory. Fortunately, enough tricks
are available concerning ordinary primes to settle the general case of K3 surfaces of(geometric) Picard rank at least 17 when they are defined over Q. But note there doexist many such K3 surfaces (not related to abelian surfaces) over Q that one canwrite down explicitly; see the examples due to Nori discussed in [BCGP21, §9.4].
Note that this result is new even for Picard rank 18. For Picard rank 19 and20, the (potential) modularity of any X/F for a totally real field F reduces to thecorresponding problem for elliptic curves. The case of Picard rank 16 appears as
hopeless as the case of generic genus three curves.

145. Check the arXiv regularly!
Sat, 18 Feb 2023
In § 136 I discussed a new result of Smith which addressed the following question:
given a measure µ on R supported on some finite union of intervals Σ, under what
conditions do there exist polynomials of arbitrarily large degree whose roots all lie
in Σ whose distribution (in the limit) converge to µ?

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A natural generalization is to replace Σ by a subset of C subject to certain natural
constrains, including that µ is invariant under complex conjugation. I decided that
this had a chance of being a good thesis problem and scheduled a meeting with oneof my graduate students to discuss it. Our meeting was scheduled at 11AM. Then,around 9:30AM, I read my daily arXiv summary email and noticed the preprint(https://arxiv.org/abs/2302.02872) (see [OS23]) by Orloski and Sardari solving thisexact problem! There are a number of other very natural questions along these lines
of course, so this was certainly excellent timing. When I chatted with Naser overemail about this, he mentioned he had become interested in this problem (in part)by reading my blog post!
There is of course a general danger of giving my students problems related to myblog posts, and indeed I have refrained from posting a number of times on possible
thesis problems, but in this case everything turned out quite well.
Notes 145.1. My student Kapil Chandran is currently thinking about an adelic
generalization of these theorems.

146. What the slopes are
Sat, 25 Feb 2023
Let f be a classical modular eigenform of weight k, for example, f = ∆. The
Ramanujan conjecture states that the Hecke eigenvalues ap satisfy the bound|ap | ≤ 2p(k−1)/2 .
A slightly fancier but cleaner way of saying this is as follows. Associated to f ofweight k, level N prime to p and finite order Nebentypus character χ is a polynomial
X 2 − ap X + pk−1 χ(p).
(For ∆ one has χ(p) = 1 for all p, but in general it can be some other root of unity.)This is the characteristic polynomial of Frobenius on the ℓ-adic representation associated to f for ℓ ̸= p, or the characteristic polynomial of crystalline Frobenius forℓ = p. The Ramanujan conjecture is now that the Frobenius eigenvalues αp and βpsatisfy|αp | = |βp | = p(k−1)/2 .
This conjecture was famously proved by Deligne. Having determined the complex
valuation of these eigenvalues, one might ask about their ℓ-adic valuations as well.
If ℓ ̸= p, then since the polynomial above has constant term prime to ℓ, then αpand βp are ℓ-adic units. So the remaining case is p = ℓ.
When p = ℓ, the p-adic valuation of the two roots αp and βp certainly satisfyv(αp ), v(βp ) ≥ 0 and vp (α) + vp (β) = k − 1. Given f , we call these valuations theslopes of f . The first observation is that the slopes depend on more than just theweight k. For example, suppose that f is the weight 2 eigenform associated to a(modular) elliptic curve E with good reduction at p, then either E has ordinary
reduction, in which case the slopes are 0 and 1, or E has supersingular reduction,and the slopes are both 1/2.
Instead of fixing f and varying p, one can fix both p and the tame level N andask how the slopes vary as the weight changes. For example if N = 1 and p = 2,PERSIFLAGE: MATH BLOG POSTS

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then the first interesting case is when k = 12 and f = ∆. In this case τ (2) = 24,
and the two slopes are 3 and 11 − 3 = 8.
I’ve already tried to suggest by analogy to the Ramanujan conjecture why determining the p-adic valuations (the slopes) might naturally be an interesting question.
But let me mention two other natural reasons. The first is that, from p-adic Hodge
Theory, the crystalline eigenvalues αp and βp determine the restriction of the p-adic
Galois representation associated to f to the local Galois group at p. The valuationof these slopes while containing less information than the eigenvalues themselves
still tell you a lot about the p-adic representation, although determining exactly
what is still a question of active and open interest. Secondly, as Coleman observed(following Hida in the case when one of the slopes is 0), one can deform the Galois
representations associated to these finite slope forms into continuous p-adic families
of Galois representations associated to cuspidal eigenforms, which leads to the storyof the eigencurve of Coleman–Mazur and beyond.
Gouvêa was one of the first people to undertake a numerical study of the roots.
In this paper where the slopes are (see [Gou01]), Gouvêa observed a number ofinteresting behavior of the slopes which were all somewhat mysterious. First, theslopes were almost all integers. This is not surprising when ap (f ) ∈ Z, but ingeneral ap (f ) will be a random algebraic integer. The slopes also seemed to be
distributed in a number of surprisings way. For example, although the slopes inweight k associated to f add up to k − 1, they tended to be concentrated in theintervals [0, (k − 1)/(p + 1)] and [p(k − 1)/(p + 1), k − 1].
Kevin Buzzard went one step further and looked more closely at the slopes when
N = 1 and p = 2. Ostensibly, according to Kevin, this was to test William Stein’s
latest magma code for bugs! Kevin found that the slopes in this case satisfied amuch more regular pattern. As mentioned above, when k = 12, there is a single
eigenform whose smallest slope is 3. When k = 12 + 64, there are six eigenforms
whose smallest slopes are 3, 7, 13, 15, 17, 25, and when k = 12 + 210 , there are 86
forms whose slopes are
3, 7, 13, 15, 17, 25, 29, 31, 33, 37, 47, 49, 51, 57 . . .
where all of these sequences are given explicitly by the 2-adic valuation of 2((3n)!/n!)2(explained in our paper [BC05]). Note that the Gouvêa–Mazur conjecture saysthat these sequences should have some initial segment in common, but in fact theGouvêa–Mazur conjecture only implies that the slopes in weights 16+26 and 16+210
should be the same up to slope 6, and in practice they agree ridiculously further than
this. This was all very mysterious. Kevin found a general algorithm (see [Buz05])
which conjecturally computed by an inductive procedure all the slopes in all weights
for a fixed tame level N . (In what I always regarded as a missed opportunity, he
did not call the paper “What the slopes are”).
In fact, Kevin’s conjectural answer required an assumption on N and p which
for p > 2 was equivalent to asking that the local residual representations associatedto all low-weight forms are locally reducible. For N = 1, the first case for which
this does not happen is p = 59, and this story is related to our counterexample tothe original form of the Gouvêa–Mazur conjecture (see [BC04]).
Kevin Buzzard was a speaker at the Arizona Winter School in 2001, and for his
project he outlined a special case of his conjecture corresponding to overconvergent
modular forms of weight k = 0. Of course there are no classical modular cuspidal
forms of weight k = 0, but by Coleman’s theory there is a direct link between the

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questions about classical forms in all weights and overconvergent forms of finite
slope in all weights. Kevin’s problem in its original formulation is described here.
I was a graduate student at the time, and although I wasn’t actually assigned to
Kevin’s group, I did see his problem and had an idea which more or less amounted
to the idea of using a slightly different explicit basis for this space than Kevin had
considered and for which the U2 operator has a much nicer explicit form. In fact thisbasis was related to computations I had done in the summer of 93-94 and shortly
afterwards during my first year at Melbourne uni, using what Matthew Emerton
and I had come to definitely know as the “f ” functionY
f =q(1 + q n )24 .
n=1(When It came to writing my paper with Kevin, I still felt strongly enough to insist
we call it by this letter.) Kevin and I put quite a bit of effort into proving his
conjecture for more general k, still with N = 1 and p = 2, but only succeeded ina few cases, including k = −12, but also k = −84 which was somewhat randomly
chosen as a case where our approach failed but only by a little bit which could beovercome. There are a few other cases where X0 (p) has genus zero and one can dosomething similar, but otherwise very little progress was made on these conjecturesin this situation.
There are a plethora of related conjectures which also came out (at least indirectly) from similar calculations. For example, Kevin’s algorithm certainly always
produces integers, so, in light of the p-adic theory, there is the natural question ofwhether a crystalline representation with Hodge–Tate weights [0, k − 1] for k even
whose residual representation is reducible always has finite slope, Then there arequestions of the exact relationship between the slope and the Galois representation,and so on.
Concerning the exact slopes themselves, perhaps the biggest advance over time
were refinements of Kevin’s conjecture. The Ghost Conjecture, formulated by Bergdall
and Pollack [BP19], is some “master conjecture” of a combinatorial nature generalizing a number of previous conjectures concerning the slopes of classical overconvergent modular forms over the centre of weight space. (Although, perhaps amusingly,
it’s not clear that these conjectures do actually imply Buzzard’s original conjecture.)
Now cut to the present day. In a recent preprint, Ruochuan Liu, Nha Xuan
Truong, Liang Xiao, Bin Zhao [LTXZ23] have now proved all of these conjectures,at least up to some genericity hypotheses (excluding the case N = 1 and p = 2!). The
authors certainly employ technologies that didn’t exist 20 years ago (p-adic Langlands, for example), but that was not the only obstruction to previous progress: thepaper contains a number of very original and clever ideas. Very amazingly and satisfyingly, it resolves a large number of the open problems discussed above, includingGouvêa’s conjectures about the distribution of slopes, the integrality properties ofslopes for locally reducible representations, and even a version of the Gouvêa–Mazur
conjecture. It is also satisfying that the arguments use p-adic local Langlands, giventhat some of the initial computations of slopes served at least in part as inspiration
for some aspects of this program. I myself have not really worked on this circleof problems for almost 20 years but I am still very happy to see these questions
answered!

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147. Deciphering QuantaThu, 23 Mar 2023
Sometimes it is claimed that Quanta articles are so watered down of mathematical content that they become meaningless. That presents a challenge: do I
understand the quanta article on my own work? Let’s consider New Proof Distinguishes Mysterious and Powerful ‘Modular Forms’. I can confirm that I did not see
this article in any form before it appeared. Overall I would say that it is faithful tothe facts and I can interpret what everything means. I’m not quite sure why there
is an Alex Kontorovich explainer about the Langlands Program in there but why
not? I did, however, have to stop and wonder when I saw the following picture:

Figure 14. Congruence modular forms (left) have additional
structure that noncongruence modular forms (right) lack
This appeared with the caption “Congruence modular forms (left) have additional structure that noncongruence modular forms (right) lack.” OK, here is thechallenge: can I work out what this picture actually represents?
In both pictures, there is clearly some color gradient between yellow and blue,
and this has discontinuities along some regions we call L and R for left and right.
These are also clearly pictures inside H2 in the Poincaré disc model. Here L looks atleast superficially like a fundamental domain for some Fuchsian group. The rays of
L going towards the point i on the boundary do look like geodesics in the hyperbolic
metric (which are circular arcs meeting the boundary at right angles). There is acorresponding invariance for the figure by the parabolic element (with some cusp
width) through i. Consider a conformal map from the upper half plane to this model
which sends ∞ to i: Using the standard conformal map with the upper half plane:
ϕ : H → D(0, 1),z 7→

1 + iz1+zFrom the picture, we see the images of the geodesics from the infinite cusp to nα
for n odd. All parabolic elements are conjugate, but if we are to guess what Γ isby looking at the picture then working out α for this specific model is important. I
tried to eyeball it for a while before printing it out and trying to compute the angle
using continued fractions, which wasn’t so accurate but gave tan θ ∼ 1/(1 + 1/3)
with θ in the mid 30s. Then using some angle tools in keynote it came out tosomewhere between 36 and 37 degrees, closer to the latter. So maybe it was exactly
a 10th root of unity, which would make α live in a degree 4 field, or (much better!)maybe α = 1/2 in which case the angle is 36.8698 . . ., and then L (or rather ϕ(L)292

F. CALEGARI

is invariant under exactly



1 1
A=
.
0 1
Maybe I should have guessed this in retrospect! Let’s look at the geodesic from i∞
to 1/2, which in the picture goes from i to 4/5 − 3i/5. There appears to be a pointwith 4-fold symmetry. That suggests invariance under some order four element
corresponding to an elliptic point. But this is a little worrying, since PSL2 (Z) does
not have any elements of order 4! Now there are some ways around this. For example,
this could be the level set of a function which satisfies an extra invariance property
under the normalizer of some congruence subgroup, and so does not itself comefrom an element of SL2 (Z) but GL2 (Q)+ . For example, the Fricke involution on
X0 (N ) is τ 7→ −1/N τ which corresponds to. a matrix in GL2 (Q) with determinant
N , although that has order 2. Any element of order 4 has to have eigenvalues ofthe form α and αi, with α + iα ∈ Z, and thus eigenvalues 1 + i and 1 − i up torational multiples. That means the determinant should be 2 times a square. The
unique element such element up to conjugacy is


1 1
S=
1 −1
which has order 4 and fixes i. If we want to fix a point on the geodesic 1/2 + it, we
can just make a hyperbolic scaling and then conjugate to get the matrix
1 − 1/2t t + 1/4t
St =
.
−1/t
1 + 1/2t
Here I started to run into a problem because it’s hard to approximate t to any
precision from the picture, although t ∼ 0.1 numerically. There are a few natural
integral matrices one can write down but it wasn’t clear what was going on.
Next I turned to the picture around the cusp −i. This looks kind of weird because
the biggest curves decidedly do not look like geodesics. But if we ignore this, wemight decide that −i is another cusp. It’s also disturbing that the fundamental
domain appears to contain enough of the boundary to suggest that Γ is a thin group,
but let’s ignore this as well. So what is the cusp width (without normalizations) at−i? Here numerically I simply get nonsense because if there really were geodesics
around i translated by a fixed parabolic element and the first one was of the size
indicated in the picture then there would be many fewer visible ones. As a typicalexample, one should expect a picture like this:which doesn’t look anything like the original picture. So I’m ready to give up
now. What about the picture on the right? Well here I don’t even know how to
begin. There is not any obvious evidence of parabolic elements, which are not only
present in congruence subgroups of SL2 (Z) but of any non-congruence subgroup as
well. Perhaps there is a cusp at i with some large cusp width. There also seem tobe singularities inside the disc. But that just suggests this might be a level set ofa meromorphic form. But why choose a meromorphic form rather than one thatis holomorphic away from the cusps? I wonder if that gives hints about the first
picture; perhaps the transition from yellow to blue occurs when Im(f ) goes frompositive to negative, and where it might be 0 along some geodesic arc (For example:
the Fourier coefficients are rational so the form is real along the purely imaginaryaxis) but the form can also have imaginary part 0 along some non-geodesic regions
and that is what one is seeing around −i. That might allow one to reinterpret the

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Figure 15. Geodesics through i with fixed cusp widthgraphs as something related to Im(f ). This suggests that the point on the left thatappears to have degree four instead is related to a “local” symmetry coming froma vanishing point of the derivative of the modular function, but I’m just guessingnow.
I should probably spend no more time on this, so instead I open up speculations
to the comments!
Full Disclosure: The picture comes with an attribution to David Lowry-Duda
but I did not try to follow that lead.
147.1. Update. So here is an new example. Like the graphs above, it plots theargument of a two modular forms on (different) finite index subgroups of SL2 (Z).
I have normalized both pictures so that the cusp width at the cusp i is the same inboth cases (under the normalizations above invariant under τ → τ + 3; making thecusp width too small makes the behavior at τ = i∞ dominate the picture as thecovolume goes up. There are dome differences with this example however:(1) I have used (holomorphic) modular functions rather than modular forms.
(2) For these particular choices of functions, they are non-zero everywhere.
(3) I made the picture myself so it is not as pretty as the ones above.
(4) The functions I chose have the property that they are real precisely on
geodesics, and thus the singularities here do form a tessellation.

Figure 16. Which one is congruence?

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F. CALEGARI

Now one of the forms is defined on a congruence subgroup and has a Fourier
series in Z[[q]], the other has a Fourier series in Q[[q]] but not in Q ⊗ Z[[q]] and is
only defined on a non-congruence subgroup. But which is which? The pictures here
are clearly much more similar than the pair of pictures above!
Comment 147.2 (Will Sawin). In the spirit of your update let me mention what
I thought upon seeing the image, before seeing Peter’s link:
The color ranges from blue to yellow. If we think of the color spectrum from blue
to yellow as an interval, the function taking a point to its color looks continuous,
except for the lines of discontinuity where it switches from completely blue tocompletely yellow. In other words, this function is continuous after we glue bothends of the interval, forming a circle.
How do we get a continuous function to the circle from a modular form, or othercomplex-valued function? The only obvious thing is to take the argument. So thecolor in the picture must depict the argument of the modular form.
Now something I figured out after seeing the links, but feel like I could havefigured out purely mathematically/visually: The argument does have another typeof discontinuity, at the roots, where all colors collide around a single point. Thesecan be distinguished from the potentially similar-looking points where the function
is negative real and the derivative is zero because those points only have brightblue and yellow near that point. If the root has order 1, we will see a line of blueyellow discontinuity ending at that point. The left picture only appears to have thebrightly-colored kind of singularity, i.e. no roots, which could have been the clue totell us it was ∆, while the right picture has clearly visible roots.
The roots in the right picture are significant because, while the argument is
not invariant under the group of symmetries of the modular form (and unlike theabsolute value can’t even be normalized to become invariant), the set of rootscertainly is, so examining the roots in the disc might give us a clue to the noncongruence group (but this is far beyond my visual-calculational abilities).
Comment 147.3 (David Lowry-Duda). I can’t say that this is “speculation” as I
guess I hold all the cards, but I thought I would comment.
Will is exactly right about how the left picture is (essentially) the argument ofthe ∆ function.
I first used those colors because the colormap works well for various types of
insensitivity to certain colors. When Quanta asked me for images. But it’s truethat the sharp color discontinuity when across values when arg(z) = π suggests animportant feature even where there isn’t one.
The specific Cayley transform that I use to relate the disk to the upper halfplaneis ϕ(z) = (1 − iz)/(z − i), which preserves the “apparent” vertical orientation of theimaginary axis. That is, the points (−i, 0, i) in the disk correspond to the points(0, i, ∞) in the upper halfplane.
The form on the right was computed using the work of Berghaus, Monien, andRadchenko here (see [BMR24]). It’s one of the forms of weight 4 on the group withsignature (8, 0, 1, 2, 2) (using their notation and their data).
I also didn’t see the article (or even really know much about what it was going
to be about) when they asked me for an image. It just happens to be that when you
google how to visualize modular forms, my little paper here comes up (see [LD21]).

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148. Quadratic reciprocity
Tue, 02 May 2023
I accidentally proved quadratic reciprocity in class today, or at least three quarters of a proof. Can you finish it off? Here’s the proof: start with a real quadratic
field K, and the sequence×
1 → OK→ K × → K × /O× → 1

then take cohomology. If PK is the group of principal ideals of K, then from Hilbert
Theorem 90 you deduce that×GPK
/PQ ≃ H 1 (G, OK).
G
If IK is the group of all ideals of K, the left hand side is a subgroup of IK/PQ which√is a product of groups of order two for each ramified prime. Now if K = Q( pq)with p ≡ q ≡ 3 mod 4, there is no unit of norm −1 because (−1/p) = (−1/q) = −1,
so you deduce that the cohomology of the unit group has order 4 and so from orderconsiderations that the prime ideals of norm p and q are principal. Write p = (α)
and q = (β). One has x and y withx2 − pqy 2 = N (α)
Here N (α) = ±p. But the right hand side must be a square modulo q, and so
N (α) = p if (p/q) = +1 and −p if (p/q) = −1. Equivalently, N (α) = (p/q)p, and√√similarly N (β) = (q/p)q. But pq = ( pq), so αβ = ε pq for some unit ε, and since
all units in K have norm one, it follows that√pq(p/q)(q/p) = N (αβ) = N (ε)N ( pq) = −pq,
which is quadratic reciprocity! There is a similar argument for p ≡ 1 mod 4 andq ≡ −1 mod 4 (though now using facts about (2/p) rather than (−1/p)). However,
it is not so clear if one can prove the case p ≡ q ≡ 1 mod 4 using this argument.
Is there one? Of course the challenge here is to “only” use Hilbert 90 and uniquefactorization of ideals, but never to make any arguments about how primes are
splitting.
Notes 148.1. Actually, the “similar” argument doesn’t quite work. So only one
quarter of a proof!

149. Clozel 70, Part I
Sun, 24 Sep 2023
I recently returned home from a trip to Paris for Clozel’s 70th birthday conference. Naturally I stayed in an airbnb downtown, and the RER B gods smiled onme with a hassle free commute for the entire week. Tekés was an interesting find,a fun (and surprisingly cheap) Israeli vegetarian restaurant right near where I wasstaying. But surely the food highlight of the week was the lunch spreads during
the conference at Orsay — certainly the best conference food I’ve ever had! Great
vegetarian food with amazing eggplant dishes, feta, figs, all the good stuff. (Rumor
was it was chosen by Valentin Hernandez and paid for by Vincent Pilloni; I’m notsure if that’s true but a great job all round.)There were quite a number of interesting talks, although as mentioned before I
don’t like singling out because that sometimes seems like an implicit criticism of296

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the other talks. But a few thoughts spurred by some of the talks (which you canmore or less guess if you wanted to), some of which were already raised by othersin conversation during the conference:
149.1. (Global) modules for Galois deformation rings. As a result of Taylor–Wiles patching, one usually constructs a CM-module M∞ defined over some local
deformation ring R. Often quite a bit of mileage can be gained by exploiting thering theoretic structure of R to conclude something about the module M∞ and viceversa. Perhaps the ur-version of this argument is Diamond’s argument showing (insome circumstances) that the formal smoothness of R implies that M∞ is free. A
more recent example is in the work of Jeff Manning where he exploits the geometryof some particular R (by relating to a more geometric situation where one canperhaps understand the Picard group) to restrict the possible M∞ to a very small
number of possibilities from which one can then get some mileage. But one question
raised is the extent to which this one can always do this. As one considers more
and more complicated R, is there some constraint on possible R which means that
there are only going to ever exist a small number of faithful maximal rank one
CM-modules M , or are there going to be situations where R is very complicatedand one can’t hope classify all such M , but only (for some mysterious situation) avery small number of them turn up in global situations. Note that globally there isoften a few possibilities of the type of cohomology one patches, and even for GL(2)you can be in situations where you can force M to be free or self-dual by working in
coherent cohomology or etale cohomology respectively and these modules are notalways the same.
149.2. The work of Arthur. (Some) experts are at the point where they no longer
expect Arthur to publish proofs of results he has claimed, leaving a huge gap in
the literature. The summary seems to be that many very smart people are putting
lots of effort into filling in some of these details, and that this seems to require
new arguments. For example, it seems to be the case that one of Arthur’s proposed
inductive arguments will not (at least naively) work. The mathematical community
should be immensely grateful to people working on this!
149.3. Shimura Varieties. I once joked that today’s generation is more likely tolearn about Galois representations associated to elliptic curves and modular curves
before learning any class field theory. Well that generation has passed! We may beapproaching a moment where people learn about Shimura varieties without everthinking about modular curves, let alone getting close and personal with X0 (11).
(Note: I don’t think that Richard knows the genus of X0 (11) and that never stopped
him, of course.) Some people can look at the abstract definition of a Shimura variety
and then start proving things; I am certainly not one of those people. Fortunately
there are still many interesting open questions even about classical modular forms.
149.4. A result of Garland. (at least) two talks reminded me of a vanishing
result of Garland. Suppose that Γ is (for example) a lattice in SLn (Qp ). Then thecohomology (with coefficients in Q) vanish in positive degrees below (n − 1). But
I think that much more should be true, namely, that the cohomology should all
have a “trivial” Hecke action in the expected ways, i.e. the completed cohomology
groups should all be finite in this range, as they are for SLn (Z) (more or less, let’s
not be precise about ranges and what exactly is known). It feels like conjectures ofPERSIFLAGE: MATH BLOG POSTS

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this sort are not completely out of reach. Is this too optimistic? This is already aninteresting problem in the case of H 2 (Γ, Fp ).

150. Clozel 70, Part II
Sat, 30 Sep 2023
Many years ago, Khare asked me (as I think he asked many others at the time)
whether I believed their existed an irreducible motive M over Z (so good reduction
everywhere) with Hodge–Tate weights [0, 1, 2, . . . , n − 1] for any n > 1. (Here theMotive is allowed to have coefficients.) When n = 2, the answer is no. Assuming all
conjectures, such an M must be modular associated to a cusp form of weight 2 andlevel one, but no such cuspform exists. But the answer is also no unconditionally(for any notion of motive), and this fact is intertwined with the (inductive) proof
by Khare and Wintenberger of Serre’s Conjecture. The hope might be that if nosuch motive existed for all n, it could serve as the inductive basis for a more generalform of Serre’s Conjecture.
My response at the time was that I guessed that no such motive existed for any n.
I generally feel that my intuition is quite good in these matters, so it was surprising
to learn some time later a convincing meta-argument that such motives should reallyexist. This idea, which I can’t now remember whether I learned from Chenevier or
Clozel, is related to trying to construct such forms which are in addition self-dualand so come from a classical group. In favourable situations, there exists a compact
inner form on this group, so that computing these forms “reduces” to computing
on certain finite sets. One such finite set turns out to be the set of positive definite
lattices of discriminant one and dimension n. As is well-known, they only occur indimensions a multiple of 8. For n = 8 there is just E8 , and for n = 16 there are two,and for n = 24 there turn out to be exactly 24, as classified by Niemeier, and which
include the famous Leech lattice whose automorphism group is a central extensionof the first sporadic group discovered Conway. Easier to compute is the weighted
sum of such lattices by automorphisms; for n = 24, for example, this weighted sum
is1027637932586061520960267
129477933340026851560636148613120000000
which is very small, and of course is related to the fact that these lattices are quite
symmetric. For n = 32, however, the weighed sum is bigger than 107 , and so there
are lots of lattices. You might then think that the existence of these lattices (evenjust E8 when n = 8) implies the existence of automorphic forms which then should
give rise to the desired automorphic forms on GL(n). But there are issues. One
concerns the technical issue of transferring forms between groups which is of course
a subtle problem. But there is another. A form which is cuspidal on some group needno longer be cuspidal after transferring to GL(n). So to see which forms are cuspidal
you really need to do a computation. But these objects are of large complexity —already computing Hecke actions on supersingular points for X0 (11) is a non-trivial
exercise; here the objects involve lattices of enormously high dimension. Chenevierand his co-authors, (including Lannes, Renard, and Taïbi, [CR15, Che20, CT20])
have done a remarkable job understanding what is going on here. The most basic
example of the type of theorem they prove is as follows. When n = 16 so there

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are two lattices; one can try to compute the action of a Hecke operator Tp , and itturns out (see for example Theorem A here) that the answer involves Ramanujan’sfunction τ (p). But this also tells you that the transfer to GL16 will have some
explicit isobaric decomposition corresponding to twists of the modular form ∆, andin particular the associated π will clearly not be cuspidal.
At the same time, there are some automorphic arguments which show that cusp
forms of level one (and cohomologically trivial weight) cannot exist. Here the idea
goes back to the (automorphic) proof of lower bounds for discriminants of number
fields by Stark and Odlyzko. The idea in that case to use the explicit formula forζK (s) to construct an expression (and in particular the normalized version ΛK (s)which satisfies the functional equation and involves N s where N is the level which is
directly related to the discriminant of K) and ultimately arrive at some expression
which is provably non-negative unless the root discriminant of K is larger thansome explicit constant (minus some explicit o(1) depending on the degree). Mestré
generalized this argument [Mes86] to automorphic forms corresponding to other
Motives, in particular proving that, assuming conjectures of Langlands type, that
there did not exist any abelian varieties over Z of dimension at least one (which wasproved unconditionally by Fontaine [Fon85]) but also that the conductor of such an
abelian variety had to be at least 10g . This was then later generalized by Fermigier(a student of Mestré) and then by Stephen Miller (Rutgers!) [Fer96, Mil02] toprove that there are no automorphic forms π for GLn /Q of level one which arecohomological for the trivial representation when n < 27. These are exactly theforms associated (conjecturally) to the motives of weight [0, 1, . . . , n − 1].
Returning to the conference at Orsay: Chenevier gave a talk on understanding
automorphic representations π of level one and low motivic weight, and once again
raised the automorphic version of Khare’s question. Now I have known about thisquestion for a long time, but somehow being reminded of a problem can sometimes
be the spark to help one think about the question again.
Correcting what was a past failing of my own intuition, I was very happy that
George Boxer, Toby Gee, and I were able to come up with a very simple argument [BCG23a] to answer both questions; there does exist a compatible family of
crystalline Galois representations with Hodge–Tate weights [0, 1, 2, . . . , n − 1] forsome n; for example one can take n = 105. Moreover, this compatible system iseven automorphic and associated to a cuspidal π of level one and cohomologicalweight zero for GLn . (With work, it is even “motivic” in the sense that the compatible system can be found inside an explicit algebraic variety over Q, so it is inparticular also pure.) Now while the argument is very simple, it must also be said
that is uses some extremely hard theorems; for a start, it uses both the full modularity lifting results of [BLGGT14] (Barnet-Lamb, Gee, Geraghty, Taylor), following
Clozel–Harris–Taylor and many others, and it uses the even more recent full symmetric power functoriality result [NT21a, NT21b] for classical modular forms by
Newton and Thorne. (Since the paper is only nine pages and the proof only half of
that, I won’t explain it here.)
One would still like to prove, of course, that there are a huge number of self-dual
forms for all sufficiently large n. And one can naturally ask what is the smallest
such n, which we now know satisfies 27 ≤ n ≤ 105. The expectation is certainlythat n is probably close to around 32. It would be nice to know!

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Of course, there is an endless list of other tricky problems one can pose of thisform. For example, does there exist a regular motive (with coefficients) over Z with
Hodge–Tate weights [0, 1, . . . , n − 1] for some n which is note essentially self-dual?
Comment 150.1 (Will Sawin). Discussing, I think, exactly this question, I believe I heard a long time ago the argument that if one counts cuspidal self-dual
automorphic representations of GLn with Hodge–Tate weights consecutive integers
via the trace formula for the corresponding classical group, the main term comingfrom the adelic volume should grow so fast that all the other contributions, including the counts of Eisenstein series on GLn arising from cusp forms on the classicalgroup, should be dominated, and hence such representations should exist for all
large enough n. Of course actually doing this kind of analysis with the trace formula in the large n limit seems essentially impossible, though the work of Deligne
and Flicker does give a function field analogue.
I’m confused about one point in the Chenevier research report you link — why
do the Galois representations considered there have a repeated Hodge–Tate weight,
while the ones you are interested in do not?
Let me see if I see what is going on here. All forms on SO2n arising from functions
on even unimodular lattice have the same infinity type. Since the trivial representation is obviously among them, they all have the infinity-type of the trivial representation. Transferring them along any sort of Langlands functoriality will preserve
these infinity-types, so you will get forms which have the infinity-type of the transferof the trivial representation. However, the Arthur parameter of the trivial representation of SO2n is the representation Sym2n−2 + trivial of SL2 into the dual SO2n ,and transferring to GL2n gets you a representation with Arthur parameter Sym2n−2+ trivial, i.e. not the trivial representation which would be Sym2n−1 .
————————————————————————. . . I was tempted to try to, rather than find an explicit eigenvalue of the set ofrank 32 even unimodular lattices and check it gives a cusp form on GL32 , boundthe number of possible Eisenstein series on GL32 that contribute to eigenforms on
the set of rank 32 even unimodular lattices, but now I see why this is too hard:
The worst case is perhaps the direct sum of Delta with a cusp form on GL30 thathas exactly two gaps in its sequence of Hodge–Tate weights matching the weights
of Delta, so we can sum them with Delta to get a problematic form on GL32 . Suchforms on GL30 surely can’t be ruled out by L-function methods as their weights
are similarly spaced to forms that can’t be ruled out, and counting them with thetrace formula is clearly out of reach.
On the lower bounds side, I wonder if the results of Miller can be improved at
all if one (a) assumes GRH and/or (b) restricts attention to self-dual representations. Given a lower bound of 27 and a potential upper bound of 32, even a tiny
improvement would do a lot to close the gap.
With regards to the existence of modular forms of weight k and level 1 with
reducible mod p reductions, I guess by Serre’s conjecture this is equivalent to theexistence of certain Galois representations into GL2 (Fp ). By a natural extension of
Bhargava’s heuristics for counting number fields to counting GL2 (Fp ) extensions
with fixed GL1 (Fp )-part I get that the probability one should exist for each k is1/(p−1) so the probability none exists for any k is (1−1/(p−1))ϕ(p−1) which agrees
closely with your heuristic. (The local factor at each non-archimedean prime is one
so one only needs the local factor at infinity, which is the proportion of elements of

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GL2 (Fp ) with determinant −1 that have order 2 and could be complex conjugation,
which is 1/(p − 1) for centralizer reasons.)
Comment 150.2 (Persiflage). Concerning GRH: In Chenevier, the use of GRH
does make a difference but only a very slight one in this range (a more general
result for m ≤ 23 can be improved to m ≤ 24 but no more). So my guess that forthis specific problem GRH probably wouldn’t move the bar much, quite possibly
even not at all.

151. Magma instability
Fri, 13 Oct 2023
I had occasion to return to some magma scripts I wrote in 2012. I the script used
a number of pre-computed auxiliary files with computations, and was a little complicated, but didn’t use anything particularly complicated. So I was really surprised
to run them in 2023 and find that they no longer worked. That is, they compiled,
but the results they gave were different (and also incompatible with the truth). It
was quite confusing to understand what has gone wrong, but eventually I traced it
to the following. Early on in the file one has (having defined t as a variable usingcode that’s easy to write but which is somehow causing issues with wordpress):
F := NumberField(t^2 - 5);
ZF:=Integers(F);So far, so good. But later on, the script called upon elements of F of the form
ZF![a,b]. But it turns out that if x:=ZF[0,1] thatx2 + x = 1
in 2012, but
x2 − x = 1
in 2023; that is, x was replaced by −x, which is not an automorphism. I have noidea how or why that changed, but it certainly broke everything and took several
days to fix. Annoying!
Comment 151.1 (Nathan Dunfield). I get x2 − x = 1 in an old copy of Magma
2.21-8. As the first version of Magma 2.21 came out in 2014, your scripts might
have broken not long after you wrote them.
My guess is the cause is a change in the algorithm that computes an integral
basis for a number field, which has the side-effect of giving a different answer for thisone. I suspect the change was introduced in 2.21, where the release notes include:
There is a new implementation of size reduction of an element by
multiplication by units of the maximal order. This size reduction
is now done in all situations where one would expect it to be done,for example in choosing a generator for a principal ideal and so on.
given that the two answers do differ by −1.

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152. The horizontal Breuil–Mezard conjectureThu, 19 Oct 2023
Today I wanted to talk about Chengyang Bao’s thesis. Fix a local mod-p representation, sayρ : GQp → GL2 (Fp )given on inertia by ω2 ⊕ ω2p . Associated to this residual representation is a Kisin
deformation ring R corresponding to fixed determinant crystalline lifts of weights[0, k − 1], for some fixed positive integer k ≡ 2 mod (p − 1). The special fibres R/pof these rings have dimension one, and so, if one denotes their maximal ideal bym), then the Hilbert seriesX
Hk (x) =dim(mk /mk+1 )X k
has the form
Pk (x)
1−x
where Pk (x) is a polynomial. The Hilbert-Samuel multiplicities of these rings are
given by the Breuil–Mezard conjecture (also proved by Kisin). These numbers are
explicitly given by the values Pk (1). It seems quite surprising that understanding theseemingly simple number Pk (1) is so intimately linked to the proof of the Fontaine–
Mazur conjecture. At the same time, we know very little about these rings R (ortheir special fibres) when k is large.
In the example above, the unrestricted (fixed determinant) local deformation
ring Rloc is is formally smooth of dimension three over Zp . Although the rings R/ponly have dimension one, one expects that for larger and larger k they start to “fillout” the unrestricted deformation ring. It is natural to wonder: how fast does thishappen?
More explicitly, the Hilbert-Samuel series of the special fibre of the unrestricted
deformation ring with fixed determinant is1= 1 + 3x + 6x2 + 10x3 + . . .
(1 − x)3
Hk (x) =So one can ask: what weight does one have to go to see all three dimensions of thetangent space? How far does one have to go to see all of Rloc /(p, mn )?
This was the thesis problem of Chengyang Bao, which grew out of (in part)questions arising during her work [Bao22]. In this particular case, it seems that one
has to go to weight k = p2 + 1 to see the entire tangent space. Actually, an evenmore basic question is whether there exists surjective map
Rk+p−1 /p → Rk /p,this seems very tricky and is still open (but Chengyang’s work strongly suggeststhat it is true).
One of the difficulties in this project is that close to nothing was known about
the rings R for k anything larger than k = 2p or so (although there has certainly
been quite a bit of work understanding the link between ap and the residual representation, including Buzzard–Gee [BG13], Sandra Rozensztajn [Roz18, Roz20],
and many others using p-adic Langlands for larger k).
Chengyang’s approach was, perhaps surprisingly, to use global methods. The
basic summary of the Taylor–Wiles method as formulated by Kisin is that via

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patching one finds that a patched Hecke ring may be identified with a power series
ring over R. By reverse engineering this, if one finds a residual representation with
sufficiently nice global properties, one can use explicit Taylor–Wiles primes to get
arbitrarily close approximations to the Kisin deformation ring R. One of the tricks
here is to be able to do this in a way where one can work efficiently after fixing a
residual representation and then increasing the weight.
By doing these computations, Chengyang generated lots of explicit data about
these rings R from which one can start making conjectures. I said before how
the Breuil–Mezard conjecture amounts to predicting the value of Pk (x) at x = 1.
Chengyang has, at least in this particular case, been able to formulate an exact
conjectural answer for the entire polynomial Pk (x). As a consequence, one can readoff from this the answer of how large a weight one has to go to see all the directionsin Rloc /mn . I mentioned before that for n = 2 the answer is p2 + 1. and my guess
was that the answer in general might be of order pn . But somehow the conjecturalanswer (up to constants which depend on p) turns out to be of order O(n2 ), which
is honestly completely different from anything that I would have guessed. I think ofthis conjecture as a new “horizontal Breuil–Mezard conjecture.” But really, it’s only
half a conjecture; the hope is that one can understand and interpret Chengyang’s
conjecture on the GL2 (Qp )-side, and working this out is an exciting problem.
At the same time, there are lots of other things one can start to guess fromlooking at these explicit rings. Chengyang has a precise conjecture which says when
the rings R in this setting are complete intersections or Gorenstein, and it also seems
that they are always Cohen–Macaulay.
Even though we “know” p-adic Langlands for GL2 (Qp ) much better than in any
other situation, there seems to be a real opportunity here to tease out many more
precise and explicit conjectures from Chengyang’s work, and really to discover new
phenomena which have hitherto never been noticed because computations of theserings has been so limited. (Another basic question: how many components does thegeneric fibre of R have in terms of k?).

153. Unramified Fontaine–Mazur for representations coming fromabelian varieties
Thu, 09 May 2024
Mark Kisin gave a talk at the number theory seminar last week where the following problem arose:
Let W be the Galois representation associated to the Tate module of an abelianvariety A over a number field, and suppose that W = U ⊗ V . Now suppose that
the Galois action on U is unramified at all primes above p. Can you prove that the
Galois action on U has finite image?
Of course this is a special case of the unramified Fontaine–Mazur conjecture.
But here the representation U literally “comes from an abelian variety” although asa tensor factor rather than a direct factor. At first sight it seems like it should bemuch easier than the actual Fontaine–Mazur conjecture if you just find the righttrick, but I don’t see how to do it! Here at least is a very special case.
Lemma 153.1. Suppose that A/K has ordinary reduction at a set of primes ofdensity one, and that U is a representation which is unramified at all primes dividing

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p of odd dimension which occurs as a tensor factor of W = H 1 (A) = U ⊗ V . Then,after some finite extension of K, U contains a copy of the trivial representation.
Proof. One may as well assume by induction that the action of the Galois groupon U is absolutely irreducible of odd dimension d and remains so for every finite
extension (otherwise decompose it into such pieces and take one of odd dimension).
Now choose a prime v of K. Let αi be the eigenvalues of Frobenius at v on U ,and let βj be the corresponding eigenvalues on V . We know that αi βj are algebraic
numbers which are Weil numbers of norm N (v). The ratios of any two roots thus
are also algebraic numbers with absolute value 1 at all real places, and so αi /α1
has this property.
Let’s suppose that the ratios αi /α1 are actually roots of unity for a set v of
density one. Since W is be defined over a fixed finite extension E = Qp , the degreesof these ratios has uniformly bounded order over E, and the the orders of theseroots of unity also have uniformly bounded order. But then (projectively) only
finitely many characteristic polynomials will arise from Frobeneii for a set of density
one, , which would imply that U has finite projective image, from which it easily
follows that U becomes trivial over a finite extension (remember the determinant
is unramified so of finite image). Hence it suffices to show that the αi /α1 are all
algebraic integers and then use Kronecker’s theorem.
For finite places not dividing N (v) this is clear because the valuations of the αi βjare all trivial and so are their ratios. For finite places dividing N (v) now suppose inaddition that A is ordinary. Fix a place above v. If the αi /α1 have valuation given
by ai , and βj /β1 have valuation bi , it follows that the quantities ai + bj take on
precisely two values, zero and either 1 or −1, and they take on each of these values
exactly half the time. But then either ai is constant and thus (considering i = 1)
equal to 0, or the bj are all zero, and then half the ai are zero and half are 1 or −1.
But that’s clearly only possible if U has odd dimension. So done!
□
I suspect the case that dim(U ) = 2, even with an ordinary hypothesis, is probablyquite hard. But I would be happy to be mistaken.
(I did avoid mentioning Pink’s paper [Pin98] in part because in Kisin’s talk heused the Mumford–Tate conjecture as an ingredient to avoid having to address this
Fontaine–Mazur question. Pink also proves some very nice results “at almost allprimes”.)

154. A talk on my new work with Vesselin Dimitrov and Yunqing
Tang on irrationality
Sun, 16 Jun 2024
Here is a video of my talk from the recent 70th birthday conference of Peter
Sarnak. During a talk one always forgets to say certain things, so I realized thatmy blog could be a good place to give some extra context on points I missed.
There are three things off the top that I can add before rewatching the talk. The
first is that I made a typo in one of my collaborator’s name (oops!). The second
is that I didn’t mention the work of Bost–Charles [BC22], whose influence on ourwork is clear. Indeed the m = 0 version of the holonomy theorem (version III) in
this talk is a theorem in their monograph. The third is that my presentation of
known irrationality results for explicit zeta values makes sense in the context of

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framing of my talk, but it’s good to note that the irrationality results of Rivoal,
Ball-Rivoal, and Zudilin [Riv00, BR01, Zud01] (for example, at least (edit: one)of ζ(5), ζ(7), ζ(9), ζ(11) is irrational) in a closely related direction are amazing theorems. There’s probably more to say, and I might add some extra comments if I
watch the video again).
The talk
154.1. Some incidental remarks concerning history I thought about when
preparing my talk. I know from popular accounts [vdP79] that Apéry’s resultcame as a complete surprise. Similarly, the result of Gelfand–Schneider was a complete shock as well. (Hilbert was reputed to say that he didn’t think this problem
would be solved within his lifetime.) Now these two theorems are “recent enough”
so that the memory of their resolution is still within the collective consciousnessof mathematicians. In the first case, I still know a bunch of people (Henri Cohenand Frits Beukers) who were actually at Apéry’s infamous lecture. But what about
Lindemann’s proof that π is transcendental? I have no sense as to what was thereaction at the time, in part due to my lack of historical knowledge but also to thelack (as far as I can see) of easily available informal discussions about contemporary
mathematics from the 19th century (I assume that personal letters would be thebest source). The best (?) I could find was the following (quoted from here):In fact his [Lindemann’s] proof is based on the proof that e is transcendental together with the fact that eiπ = −1. Many historians ofscience regret that Hermite, despite doing most of the hard work,
failed to make the final step to prove the result concerning which
would have brought him fame outside the world of mathematics.
This fame was instead heaped on Lindemann but many feel that
he was a mathematician clearly inferior to Hermite who, by goodluck, stumbled on a famous result.
First, this seems pretty brutal towards Lindemann (to be fair, the continuationof the text does give some more grudging praise of Lindemann). Second, which historians are being referred to here? This seems far too judgmental for the historians
I have ever spoken to in real life. If this text is at all accurate, it seems to suggestthat Lindemann’s result was lauded but perhaps not considered surprising to his
contemporaries? I feel that this is recent enough that one should be able to get afuller idea of what was going on at the time.
Going back in time further, I also wonder what Lambert’s contemporaries thoughtof his proof (in the 1760s) that π was irrational. When I was giving a public talk on
π in Sydney I looked up Lambert’s paper. The introduction is quite amusing, withthe following remark that suggests a modern way of thinking not much different to
how I think about things today:
Démontrer que le diametre du cercle n’est point à sa circonférence
comme un nombre enteir à nombre entrier, c’est là une chose, dont
les géometres ne seront gueres sorpris. On connoit les nombres deLudolph, les rapports trouvés par Archimede, par Metius, etc. demême qu’un grand nombre de suites infinies, qui toures se rapportent à la quadrature du cercle. Et si la somme de ces suites est unq
quantité rationelle, on doit assez naturellement conclure, qu’elle
sera ou un nombre entier, ou one fraction très simple. Car, s’il y

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falloit une fraction fort composée, quoi raison y auroit-il, pourquoi
plutôt relle que telle autre quelconque?
(Or in translation, errors some combination of mine and google translate):
We prove that the ratio of the diameter of the circle to its circumference is not rational; something that geometers will hardly be
surprised by. We know the number π of Ludolph, and expressions
for this number found by Archimedes, by Metius, etc. in terms ofa large number of infinite series of rational numbers, which all relate to the squaring of the circle. If the sum of these sequences wasa rational quantity, we must quite naturally conclude that it will
be either a whole number, or a very simple fraction. For, if a very
complicated fraction were necessary, what reason would there be to
be equal to such a number rather than any other real (irrational)
number?
I guess Occam dates back to the 14th century!
Comment 154.2 (Anonymous). Michel Waldschimdt writes here that all the mainideas for showing the transcendence of π were present in Hermite’s 1873 memoir,
but doesn’t explicitly quote any contemporary reaction de Hermite’s work.
This text of Emile Picard [Pic01] on the work of Hermite doesn’t either.
Incidentally, in a footnote of a text on the correspondence of Lebesgue, there ismention of the tragic loss all of Hermite’s correspondence in a fire of a storage unit
. . . So all handwritten reactions are probably lost.
Comment 154.3 (Vesselin Dimitrov). It seems that Hermite’s own opinion is
captured by a letter to Borchardt from 1873 (from right-after Hermite’s memoiron the exponential function, where he proved the transcendence of e and started
the modern theory of functional rational approximation). The English translation
could be something like this:I will not venture in search of a demonstration of the transcendenceof the number π. Let others try to pull it off. No one would behappier than me in their success. But, believe me, my dear friend,
it will not fail to cost them some effort.
In a nice recent popular (and historically well-researched) book “Tales of Impossibility” (David Richeson, Princeton University Press, 2019), there is the following
discussion where the “historian” cited is none other than the famed mathematician
Hans Freudenthal, but which completely contrasts with how Hermite viewed thesethings:In 1873, the same year that Hermite proved the transcendence of e,a German mathematics student named Ferdinand von Lindemannearned his PhD from Felix Klein in Erlangen. Shortly afterwards,
he headed abroad to visit mathematicians in England and France.
It was during his stop in Paris that Lindemann met Hermite andwas able to discuss with him his methods.
Nine years later, in 1882, Lindemann proved that π is transcendental. Some mathematicians have expressed disappointment that
Hermite, who had devised the key ideas for what would eventually be Lindemann’s proof, did not prove it. Although Lindemann
had a good career, producing 60 PhD students, he was not seen as

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F. CALEGARI

Hermite’s equal. But Hermite was generous, inspiring, and shared
his knowledge far and wide through his correspondences. Freudenthal pointed out this was often to Hermite’s detriment, because it
allowed others to achieve the important results — such as Lindemann’s proof that it was impossible to square the circle — thatwere “rightfully” Hermite’s.
and then follows Freudenthal’s strange and scathing quote in the “Dictionary of
Scientific Biography”:
Freudenthal: In a sense, [the proof of the transcendence of e] isparadigmatic of all of Hermite’s discoveries. By a slight adaptationof Hermite’s proof, Felix Lindemann, in 1882, obtained the much
more exciting transcendence of π. Thus, Lindemann, a mediocre
mathematician, became even more famous than Hermite for a discovery for which Hermite had laid all the groundwork and that hehad come within a gnat’s eye of making.
With all the hindsight, it is all too easy to make strong statements!

155. SLn versus GLnThu, 18 Jul 2024
I recently wrote a paper (with Toby Gee and George Boxer [BCG23a]) on constructing regular algebraic automorphic representations π of (cohomological) weightzero and level one, and therefore also cuspidal cohomology classes in the cohomology
of GLn (Z) for some values of n.
There was one slightly subtle point which we had to address concerning the relation between the cohomology of SLn (Z) and GLn (Z), or at least the relationshipbetween the parts of cohomology which come from cuspidal modular forms. I haveobserved this issue turn up in some different contexts, and that is what I wanted totalk about today. The main message is that from the perspective of the Langlands
program, the cohomology of GLn (OF ) is more fundamental than tbe cohomology
of SLn (OF ). When F = Q, these groups are "more or less" the same (more onthat below), but the differences are more pronounced and significant when F ̸= Q.
But let’s start by talking about the case of classical modular forms, where there isalready something a little bit interesting to say. A regular algebraic automorphic
representation π for GL(2)/Q of level one corresponds to a cuspidal modular eigenform of weight k ≥ 2 and level one. We know that cuspidal modular forms of weightk ≥ 2 and level one contribute via Eichler–Shimura to the Betti cohomology groupsof the modular curve. As an orbifold, the modular curve can be realized as H/Γ
where now Γ = SL2 (Z) rather than GL2 (Z). In this situation at least, we understand quite well what is happening. These eigenforms give rise to a two-dimensional
space inside H 1 of the modular curve, and thus inside H 1 (Γ), and we understandwhat the “extra” action of the element


1 0
0 −1
is; namely under the Eichler–Shimura isomorphism, it corresponds to the actionof complex conjugation (so from the perspective of the Hodge filtration, it takes
the holomorphic forms to the antiholomorphic forms and vice-versa). It acts on the

PERSIFLAGE: MATH BLOG POSTS

307

relevant piece of cohomology with trace zero. Note that this no longer holds on noncuspidal cohomology, for example H 0 is one dimensional in both cases. Of course
in cohomological weight zero (which corresponds to weight k = 2), there turn out
to be no such forms, but the point is that the vanishing of the cuspidal cohomology
for GL2 (Z) is equivalent to the same statement for SL2 (Z). (Something similar isalso true in higher weight as well when there really do exist such forms.)
For larger n there is a similar equivalence; but now the behavior depends on theparity of n. For n odd, the cohomology of GLn (Z) and SLn (Z) is (rationally) thesame because GLn (Z) ≃ SLn (Z) × Z/2Z (then use the Künneth formula). But forn even, a level one weight zero π gives rise to two copies of the exterior algebra
∗
^

Cℓ0

in degrees [q0 , . . . , q0 +ℓ0 ], with ℓ0 = (n−2)/2, and the action of the "extra" element
acts freely on these two copies. All this comes down to the differences in the real
representation theory of GLn (R) and GLn (R)+ which is discussed briefly in thepaper but which I won’t talk about here.
But what happens for general number fields F ? There’s a confusion which I haveseen in various places even for n = 2 about whether one should be considering
the cohomology of SLn (OF ) or GLn (OF ). Of course it depends on what exactly
one wants to do. But at least if one is interested in computing automorphic representations conjecturally associated to motives which have level one, one should
really be considering the cohomology of GL2 (OF ). This confusion comes with good
pedigree — It turns up in the Serre–Tate correspondence! Tate mentions (October
15, 1969, [Col15a, p.382]) a√colloquium by Swan who “disappointed everybody” bycomputing that H1 (SL2 (Z[ −14]), Z) has rank three, compared to the lower bound(coming from the√ boundary√tori) of two. (Note: Tate notes in a later letter [Dec15] it should be −10, not −14.) Serre responds [Col15a, p.384] on November 15that he doesn’t find this at all surprising, and in fact:(via la théorie de Weil cela signifait qu’il existe de courbes elliptiques sur le corps en question qui n’ont pas de multiplication complexe — on n’en doute pas). En fait, vu Weil,√ il s’impose d’essayer
de construire une courbe elliptique sur Q( −56) ayant bonne réduction partout;
Now I confess that when I first read this quote I interpreted it as a misapprehension on Serre’s part, because (since this is SL2 not GL2 ) there need not exist
any such elliptic curve. But looking it up again now, I started to have my doubts,
and Serre was perhaps more circumspect than I assumed. Indeed chatgpt tells me:
The phrase “il s’impose d’essayer” in French does not have the same
strict sense of necessity as “it is necessary” in English. A more nuanced translation could be “it is imperative to try” or “it is important to try” It suggests a strong recommendation or importance,
rather than an absolute necessity.
(Possibly Colmez can confirm this; AI has rendered his go playing superfluous
but not yet his skills interpreting for anglophones the nuances of Serre’s correspondence.) That’s also consistent with how Serre continues:
je connais trop mal la théorie de Weil pour être sûr que ça doit
exister; mais il vaut la peine d’essayer

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F. CALEGARI

Later (note the remark on d = −56 versus d = −40 above), Serre says:√
C’est bien Q( −40) le corps où Mennicke a trouve que le rang de
SL2 rendu abélien
√ est nombre de classes. Mais il a un corps encore
plus beau: Q( −109) où le GL2 rendu abélien est infini (c’est une
propriété plus forte S1 que la précédente). Ici aussi, on a envie dechercher des courbes elliptiques à bonne réduction.
Perhaps worth adding the modern footnote as well:«via la théorie de Weil cela signifiait que. . . » je m’avançais beaucoup en disant ça (I was talking through my hat).
Of course, 45 years later things have been clarified, at least conjecturally. (Westill have no general way to produce motives from cohomology, even for Hilbert
modular forms of parallel weight 2.) One perspective which I think is helpful (atleast to those who care more about Galois representations) is thinking about thedifferences between the Galois representations associated to automorphic forms on
SLn versus GLn . Given a π for the former (say cuspidal algebraic of weight zero andlevel one), you should think about this as giving a compatible family of projective
representations:ρ(π) : GF → PGLn (Qp )which are absolutely irreducible and crystalline of the expected weights and unramified outside v|p. Now in this situation,one knows (following for example Patrikis([Pat19]) that there exists for any such ρ a lift to a genuine representation of GF
which is crystalline at v|p of the right weight for all v|p – this generally requires
some parity condition on the weight but we are assuming that here. What is not
automatic, however, is that this lift has level N = 1 any more; that is, the image ofinertia at other primes v may be non-trivial (though of course the image lies in thecenter). Here there is something special which happens only for F = Q; as observed
by Tate, you can globalize these local characters and then twist to eliminate all theauxiliary ramification. (This argument is explained by Serre in his 1975 Durham
paper which is always impossible to find online; it is used to show that a complex
projective representation can be lifted to an Artin representation ramified at thesame set of primes.) For other fields, even if the class number is trivial, you get
global obstructions coming (via class field theory) from the unit group. (Even for
imaginary quadratic fields, where the unit group is not very big, this is still an issue,
and the general problem can only be avoided for fields for which the unit group hasorder 2 and which have a real place, which is quite a restrictive condition when you
think about it.) The direct automorphic argument is ultimately quite similar, butthere are some traps waiting for the unwary (related to Grunwald–Wang); see thediscussion in [LS19].
So for example, it is true that as F ranges over all imaginary quadratic fields,
one has1
Hcusp(SL2 (OF ), C) ̸= 0

for all but finitely many F . But the analogue for GL2 (OF ) is not only unknown,
but we certainly have:
Conjecture 155.1. There are infinitely many imaginary quadratic fields F with1
Hcusp
(GL2 (OF ), C) = 0.

PERSIFLAGE: MATH BLOG POSTS

309

By the way, from the perspective of Galois representations, one can see√ why thegroup above should be non-zero in the case of SL2 (OF ). Let F = Q( −D). All
we need to find are modular forms π of weight two with the property that, locallyat primes p|D, the corresponding Weil-(Deligne)√ representation on restriction toinertia becomes trivial after restriction to Qp ( −D) up to twist. One easy way toachieve this is to take ramified principal series PS(1, χ) for some (local) ramified
quadratic character χ. The problem is this leads (globally) to a sign difficulty; if F
has prime discriminant, then globally you would want the weight of π to be two andthe Nebentypus character to be the quadratic character of conductor ∆F which is√odd, which is a problem. (Sometimes it is not; if F = Q( −p) and p ≡ 1 mod 4
then you can take the real character of conductor p, but if p ≡ −1 mod 4 thisdoesn’t work.) But instead of principal series, one can take certain supercuspidal
representations: Assume that Fp /Qp is a ramified quadratic extension. Then if χ isa totally ramified character of Fp× of order 2m where 2m ∥p−1, then the base changeof this supercuspidal representation will be unramified up to twist, but the original
representation will not be unramified up to twist. It’s now easy to construct suchforms (and even compute how many of them there are), and see there are plentyof them when the discriminant of ∆K gets large (one has to avoid CM forms overK which can become non-cuspidal but these are easy to bound.) It’s also easy to
see that while these base changes are unramified at every place up to a local twist
they are not in general unramified everywhere up to a global twist.
The forms one finds in this way by base change are invariant under complex
conjugation (now acting on the group), and there is another “geometric” way toshow they exist which was originally done by Rohlfs [Roh85], who I believe was
the first person to prove the non-vanishing claim above. (In fact, this is one way tostart proving base change in this situation.)
When it comes to general number fields, one certainly expects (by functoriality!)∗that Hcusp(GLn (OF ), C) should be non-zero for n = 79 say and every number field
F , but this is hopeless for almost all fields. Using our arguments (and NewtonThorne for totally real fields!) One certainly can prove it for many totally realand CM fields (some ramification conditions are required for the arguments towork) using the exact same argument. Of course, when for such fields there exists a
cuspidal Hilbert modular form of weight two and level one then you can just used
Newton–Thorne directly (see [NT22])! For general fields, as usual, the problem ofunderstanding automorphic forms eludes us.
Curiously enough, while writing this post, there appeared a very recent preprint
by Darshan and Raghuram here (see [DR24]) which constructs, for example, cuspidal cohomology classes for GLn /F of (for example) cohomological weight zerofor any number field F which is Galois over a totally real field F + of some deep
enough level). Clozel [Clo87] did something similar when n is even by automorphic
induction, but already for n = 3 this no longer works. Assuming all conjectures, thesimplest way to construct such forms for F = Q or any totally real field is to takesymmetric squares of Hilbert modular forms (these more or less constitute all the
self-dual forms). It seems to me that the forms found by Darshan and Raghuram
must be some shadow of these forms over the largest totally real subfield F + of F
and so one is seeing a hint of non-cyclic base change here which is intriguing! I hope
to return to this later when I understand it better.

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