CUSPIDAL COHOMOLOGY CLASSES FOR GLn (Z)GEORGE BOXER, FRANK CALEGARI, AND TOBY GEETo Laurent Clozel, in admiration.

Abstract. We prove the existence of a cuspidal automorphic representation π
for GL79 /Q of level one and weight zero. We construct π using symmetricpower functoriality and a change of weight theorem, using Galois deformation
theory. As a corollary, we construct the first known cuspidal cohomology
classes in H ∗ (GLn (Z), C) for any n > 1.

1. Introduction
It is a well-known fact that there do not exist any cuspidal modular forms oflevel N = 1 and weight k = 2. From the Eichler–Shimura isomorphism, this isequivalent to the vanishing of the cuspidal cohomology groupsi
Hcusp(GL2 (Z), C) = 0

for all i (particularly i = 1). It is natural to wonder what happens in higher rank.
i(GLn (Z), C) ̸= 0 for
Problem A. Does there exist an n > 1 such that Hcusp
some i?

Higher rank analogues of the Eichler–Shimura isomorphism (see Remark 1.2)
imply that Problem A is equivalent to the existence of cuspidal automorphic representations π for GLn /Q which have level one and weight zero. Here level one
means that πp is unramified for all primes p and weight zero means that π∞ has thesame infinitesimal character as the trivial representation. The work of Fermigierand subsequently of Miller ([Fer96, Cor. 1] for n ≤ 23, [Mil02, Thm. 1.6] for n < 27)
showed that there are no such π for all 1 < n < 27; their methods are analytic andare related to the Stark–Odlyzko positivity technique [Odl90] for lower bounds on
discriminants of number fields. (In fact [Fer96] and [Mil02] formulate their main
results in terms of the vanishing of the cuspidal cohomology of GLn (Z) and SLn (Z)respectively; for the equivalence of these statements with each other and with the
non-existence of such π, see Remark 1.2.)
Problem A has subsequently been raised explicitly by a number of people, including [Clo16, §2.5], [Kha10], and [CR15, §1.2], where it is referred to as a “wellknown” problem. One motivation for this question, emphasized by Khare, is thatithe vanishing of the Hcusp(GLn (Z), C) for a given n could provide the base casefor an inductive proof of the analogue of Serre’s conjecture in dimension n. It wasG.B. was supported by a Royal Society University Research Fellowship.
F.C. was supported in part by NSF Grant DMS-2001097.
T.G. was supported in part by an ERC Advanced grant. This project has received funding
from the European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement No. 884596).
1

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G. BOXER, F. CALEGARI, AND T. GEE

unclear to many people (including some of the authors of this paper) whether it was
reasonable to hope for this vanishing for all n, although in recent years the workof Chenevier and Taı̈bi on self-dual automorphic representations of level 1 (see e.g.
the introduction to [CT20]) had made this seem unlikely. Another reason to expect
an affirmative answer to Problem A is by comparison to the aforementioned discriminant bounds of Odlyzko, which for a number field K/Q give positive constantlower bounds for the root discriminant δK = |∆K |1/[K:Q] as the degree of K tends
to infinity. One may ask whether there might exist a lower bound which tended
to infinity in [K : Q]. The answer to this question is no by the Golod–Shafarevich
construction; the existence of class field towers gives an infinite sequence of fields
of increasing degree such that δK is constant.
Our main theorem resolves Problem A in the affirmative:
Theorem B (Theorem 2.4, Corollary 3.2). There exist cuspidal automorphic representations for GLn /Q of level one and weight zero for n = 79, n = 105, and n =∗106. In particular, Hcusp(GLn (Z), C) ̸= 0 for these n.
Our argument works for other values of n (presumably infinitely many, although
we do not know how to prove this; see Remark 3.3). In light of Theorem B, thereis the obvious variation of Problem A:i(GLn (Z), C) ̸= 0 for
Problem C. What is the smallest n > 1 such that Hcusp
some i?

We know from [Mil02] and Theorem B that the answer satisfies 27 ≤ n ≤ 79.
The work of Chenevier and Taı̈bi [CT20] suggests that the real answer is muchcloser to the lower bound than the upper bound.
While the formulation of Problem A makes no reference to motives or Galois
representations, according to standard conjectures in the Langlands program it isequivalent to the existence of irreducible rank n pure motives (with coefficients)
over Q with everywhere good reduction and Hodge numbers 0, 1, . . . , n − 1, or to
the existence of irreducible Galois representations ρ : GQ → GLn (Qp ) unramified
away from p and crystalline with Hodge–Tate weights 0, 1, . . . , n − 1 at p. In fact,
we will proceed by producing such Galois representations.
Our approach to proving Theorem B is ultimately based on the conjecture of
Serre [Ser87] predicting the existence of congruences between modular forms ofdifferent weights. If f is a cuspidal eigenform of level 1 and weight k and the mod p
Galois representation ρf,p : GQ → GL2 (Fp ) is irreducible, then Serre predicts that
there exists a modular form g of weight 2 and level 1 with ρg,p ≃ ρf,p if and only
if ρf,p |GQp admits a crystalline lift with Hodge–Tate weights 0 and 1. Of course
this cannot actually occur as no such g exists! The natural generalization of Serre’s
conjecture for larger n predicts that if π is a regular algebraic essentially self dual
cuspidal automorphic representation for GLn /Q of level 1 and arbitrary weight,
and the mod p Galois representation ρπ,p : GQ → GLn (Fp ) has “large” image, thenthere exists an essentially self dual π ′ of level 1 and weight 0 with ρπ′ ,p ≃ ρπ,p if andonly if ρπ,p |GQp admits a crystalline lift which is either symplectic or orthogonal(depending on π) up to twist, and with Hodge–Tate weights 0, 1, . . . , n − 1. In
many instances, these “change of weight” congruences may in fact be produced
using automorphy lifting theorems and the Khare–Wintenberger method, as in[Gee07, GG12, BLGGT14].

CUSPIDAL COHOMOLOGY CLASSES FOR GLn (Z)3

It remains to explain how we find the π to which the above strategy can beapplied. For this, we need a supply of π for which ρπ,p |GQp may be readily understood. Our idea is to take π to be Symn−1 f (up to twist) for f a modular formof level 1; this symmetric power lift is now available thanks to the recent work of
Newton–Thorne (see [NT21, Thm. A] for the version we use). If f is a cuspidal
eigenform of level 1 and weight k < p, then typically f will be ordinary at p and theGalois representation ρf,p |Ip will be a nonsplit extension of ε1−k by 1, where ε denotes the mod p cyclotomic character. In this case no twist of Symn−1 ρf,p |GQp will
have a crystalline lift of Hodge–Tate weights 0, . . . , n − 1, at least for n ≤ p. On theother hand in the less typical situation that ρf |GQp is semisimple (or equivalently
tamely ramified) we are sometimes able to succeed. Here there are two possibilities,
either f is still ordinary at p but the extension splits and ρf,p |GQp is a sum of two
characters, or f is non-ordinary at p and ρf,p |GQp is irreducible.
As an illustration, if f is ordinary at p, ρf,p |GQp splits, and (k − 1, p − 1) = 1,
then as ε has order p − 1, we find that
Symp−2 ρf,p |Ip = Symp−2 (1 ⊕ ε1−k ) =

p−2
M
i=0

εi(1−k) =

p−2
M

εi ,

i=0

and hence Symp−2 ρf,p |GQp has a crystalline lift of Hodge–Tate weights 0, 1, . . . , p−2which on inertia is simply a sum of powers of the cyclotomic character. This leads
to the case n = 106 of theorem, taking f to be the cusp form of level 1 and weight 26
and p = 107, while the case n = 105 comes from a similar consideration of Sym104 f .
Our “change of weight” theorem is proved by extending the techniques introduced
in [Gee07] and developed further by Gee and Geraghty in [GG12], combining the
Khare–Wintenberger method with automorphy lifting theorems for Hida families
on unitary groups due to Geraghty [Ger19] (and refined by Thorne [Tho12]). The
case n = 79 comes from considering Sym78 f for a modular form f which is nonordinary at p = 79. Here the change of weight theorem is more involved and closer
to the arguments of [BLGGT14], using the Harris tensor product trick.
Remark 1.1. While we expect that a cuspidal automorphic representation of GLnof level one and weight zero should exist for all sufficiently large n, we do not know
how to prove this, even conditionally on Langlands functoriality. We can however
give such a conditional argument for the existence for infinitely many n. Indeed,if π is cuspidal automorphic of level one and weight zero for GLn /Q with n odd,
then for each m ≥ 1, there is conjecturally a cuspidal automorphic representationof level one and weight zero for GLnm /Q. Indeed, for each level one cuspidal
eigenform f of weight n + 1 (such an f exists because n > 26), the conjectural
tensor product π ⊠ Symm−1 f should be automorphic and cuspidal of level one andweight zero.
Remark 1.2. Encouraged by one of the referees, we now make some clarifying
remarks about the cuspidal cohomology groups of GLn (Z) and SLn (Z) and theirrelationship with cuspidal automorphic representations. A precise statement is asfollows:∗∗(1) If n is odd, then Hcusp(GLn (Z), C) = Hcusp(SLn (Z), C).
∗∗(2) If n is even, then dim Hcusp (SLn (Z), C) = 2 dim Hcusp(GLn (Z), C).

4

G. BOXER, F. CALEGARI, AND T. GEE(3) In either case, the spaces are nonzero if and only if there exists a cuspidal
automorphic representation for GLn /Q of level one and weight zero.
We now recall some definitions. There are two locally symmetric spaces of level
one associated to GLn /Q given by the quotientsbGLn (Q)\ GLn (A)/K∞ GLn (Z)where K∞ is either O(n) or SO(n). These can be identified with the quotient of
the connected contractible symmetric space GLn (R)/O(n) by GLn (Z) and SLn (Z)respectively. When n is odd, we have GLn (Z) = SLn (Z) × {±1} and these two
spaces are equal, but when n is even, one is a double cover of the other. The∗∗
cuspidal cohomology groups Hcusp(GLn (Z), C) and Hcusp(SLn (Z), C) are defined
to be the subspaces of the cohomology groups of the symmetric spaces of classes
represented by harmonic cusp forms, and (see for example [HR20, §3.3] and [Clo90,
Lem. 3.15]) we have a commutative diagram as follows:
L
≃
∗
∗
Hcusp
(GLn (Z), C)
π H (sln , O(n); π∞ )
L

πH

∗

(sln , SO(n); π∞ )

≃

∗
Hcusp
(SLn (Z), C),

where the sums on the left hand side range over the cuspidal automorphic representations π for GLn /Q of level one and weight zero, and the cohomology is therelative Lie algebra cohomology [BW00, I.5]; it is immediate from the definition
that we have H ∗ (sln , O(n); π∞ ) = H ∗ (sln , SO(n); π∞ )O(n)/ SO(n) .
If n is even, there is a unique tempered cohomological π∞ , and the (gln , SO(n))cohomology is free as a C[O(n)/ SO(n)] ≃ C[Z/2Z]-module. (See [Clo90, Lem.
3.14] and its proof.) This results from the fact that the restriction of π∞ to theidentity component GL(R)◦ decomposes into the sum of two irreducible representations. In particular, there always exist O(n)/ SO(n)-invariants; more precisely,∗∗(GLn (Z), C), as claimed.
(SLn (Z), C) = 2 dim Hcusp
dim Hcusp
If n is odd, the cohomologies of GLn (Z) and SLn (Z) agree as explained above.
The interpretation for this in terms of the above diagram is as follows. There are
now two tempered cohomological π∞ which differ by a twist by the sign characterof GLn (R), and the action of O(n)/ SO(n) on (gln , SO(n))-cohomology is either
trivial or by −1. (Again see [Clo90, Lem. 3.14].) However, only the π∞ with trivial
central character can arise from a cuspidal automorphic representation of GLn
with level one and weight zero, as the central character of such an automorphic
representation is necessarily trivial.
1.3. Acknowledgements. We have been aware of Problem A for some time, but
it was most recently brought to our attention at a lecture [Che23] by Gaëten Chenevier at the conference Arithmétique des formes automorphes at Orsay in September, 2023, in honour of Laurent Clozel’s 70th birthday. In light of this, together
with the obvious connections between the methods of this paper and Clozel’s work
(Galois representations associated to self-dual automorphic representations, modularity lifting theorems for self-dual Galois representations, and symmetric powerfunctoriality for modular forms, to name but three), it is a pleasure to dedicate
this paper to him. We would also like to thank James Newton, A. Raghuram, Will
Sawin, Joachim Schwermer, Olivier Taı̈bi and Jack Thorne for helpful comments

CUSPIDAL COHOMOLOGY CLASSES FOR GLn (Z)5on earlier versions of this paper, and the anonymous referees for several helpfulcomments and corrections.
2. The ordinary caseWe fix once and for all for each prime p an isomorphism ı = ıp : C ∼= Qp ,
and we will accordingly sometimes implicitly regard automorphic representations
as being defined over Qp , rather than C. In particularly we will freely refer to“the” p-adic Galois representation associated to a (regular algebraic) automorphic
representation. We write ρf : GQ → GL2 (Qp ) and ρf : GQ → GL2 (Fp ) for thecohomologically normalized representations associated to an eigenform f . Let εdenote the p-adic cyclotomic character and ε its mod-p reduction.
Theorem 2.1. Let f be an eigenform of level SL2 (Z) and weight k ≥ 2, andlet p > 5 be a prime such that:(1) ρf (GQ ) ⊇ SL2 (Fp ).
(2) (p − 1, k − 1) = 1.
(3) f is ordinary at p.
(4) ρf |GQp is semisimple.
Then, for both n = p−1 and n = p−2, there exists a self-dual cuspidal automorphic
representation π for GLn /Q of level one and weight zero whose mod p Galois
representation ρπ : GQ → GLn (Fp ) is isomorphic tok−2

Symn−1 (ρf ⊗ ε 2 ) = ε(n−1)(k−2)
2⊗ Symn−1 ρf .

Proof. Let n = p − 1 or p − 2, and write Gn = GSpn if n = p − 1 (equivalently, if
n is even), and Gn = GOn if n = p − 2 (equivalently, if n is odd). Let F/Fp be a
finite extension such that ρf (GQ ) ⊆ GL2 (F), and writek−2ρ := Symn−1 (ρf ⊗ ε 2 ) = ε(n−1)(k−2)
2⊗ Symn−1 ρf : GQ → GLn (F).
k−2

Since ρf is symplectic with multiplier ε1−k , the twist ρf ⊗ ε 2 is symplectic withmultiplier ε−1 , and so we can and do regard ρ as a representation GQ → Gn (F)with multiplier ε1−n . In particular, we have an isomorphism ρ ≃ ρ∨ ε1−n .
By the hypotheses that f is ordinary at p and ρf |GQp is semisimple, we can write−1
ρf |GQp ∼= ψ ⊕ ψ ε1−k

for some unramified character ψ, so that
ρ|GQp ∼
=

n−1
M

ψ

n−1−2i (n−1)(k−2)/2−(k−1)i

ε

.

i=0

Since (p − 1, k − 1) = 1, either n = p − 1 or n = p − 2, and ε has order (p − 1), it
follows easily that there are unramified characters ψ i for i = 0, . . . , n − 1 such that
ρ|GQp ∼=n−1M

ψ i ε−i ;−1

ψ n−1−i = ψ i .

(2.1.1)i=0Since SL2 (Fp ) ⊆ ρf (GQ ), the representation ρ is absolutely irreducible (see also
Lemma 2.2.) Let E/Qp be a finite extension with ring of integers O and residue
field F. Recall that Gn = GSpn if n is even, and Gn = GOn if n is odd. Write R
for the complete local Noetherian O-algebra which is the universal deformation

6

G. BOXER, F. CALEGARI, AND T. GEEring for Gn -valued deformations of ρ which have multiplier ε1−n , are unramified
outside p, and whose restrictions to GQp are crystalline and ordinary with Hodge–
Tate weights 0, 1, . . . , n − 1.
By [BG19, Prop. 4.2.6], every irreducible component of R has Krull dimension at least 1. (We are applying [BG19, Prop. 4.2.6] with l equal to our p, andthe local deformation ring Rp being the union of those irreducible components ofthe corresponding crystalline deformation ring which are ordinary, as in [FKP22,
Lem. B.4]; this is indeed a nonempty set of components because (2.1.1) shows that
ρ|GQp admits an ordinary crystalline lift, by lifting the characters ψ i to their Teichmüller lifts and the ε−i to ε−i . In order to apply this proposition, we need toverify that H 0 (Q, (g0n )∗ (1)) = 0, where g0n is respectively spn or son according towhether n is even or odd. To see this, it suffices to check that there are no invariants after taking the semi-simplification. But (g0n )∗,ss ⊂ gl∗,ss≃ glssnn (such an
inclusion need not exist before taking semi-simplifications) and the latter module
Lp−1
Ln−1
is isomorphic to i=0 (Sym2i ρf )ss ⊗ det(ρf )−i ⊂ i=0 (Sym2i ρf )ss ⊗ det(ρf )−i .
From the representation theory of SL2 (F), we see that the only characters occurring in each of these factors occur with multiplicity at most one and only for i = 0,2i = p + 1, and 2i = 2p − 2 (the second case only occurring when F = Fp ). The
characters that arise are in particular self-dual, and so distinct from ε−1 since p > 3.
ss∗0
It follows that H 0 (Q, (g∗,ssn ) (1)) ⊆ H (Q, gln (1)) = 0, as required. The remaining hypotheses of [BG19, Prop. 4.2.6] hold because the multiplier character ε1−n isodd/even precisely when Gn is symplectic/orthogonal, and the Hodge–Tate weights
0, 1, . . . , n − 1 are pairwise distinct.)
Let F/Q be an imaginary quadratic field in which p splits and which is disjoint
from (Q)ker ρ (ζp ). As in [CHT08] we let Gn denote the semi-direct product of
Gn0 = GLn × GL1 by the group {1, ȷ} whereȷ(g, a)ȷ−1 = (ag −t , a),with multiplier character ν : Gn → GL1 sending (g, a) to a and ȷ to −1. Following [BLGGT14, §1.1], given a homomorphism ψ : GQ → Gn (R), we have anassociated homomorphism rψ : GQ → Gn (R), whose multiplier character is thatnof r multiplied by δF/Q
, where δF/Q is the quadratic character corresponding to
the extension F/Q. Explicitly, if An is the matrix defining the pairing for thegroup Gn (so An = 1n if n is odd and An = Jn if n is even, where Jn is thestandard symplectic form), then rψ can be defined as the composite
ψ×pr

GQ −→ Gn (R) × GQ /GF → Gn (R),where pr is the projection GQ → GQ /GF ∼= {±1}, and the second map is theinjection
Gn × {±1} ,→ Gn(2.1.2)
given byr((g, 1)) = (g, ν(g)),
n+1r((g, −1)) = (g, ν(g)) · (A−1)ȷ.
n , (−1)

In particular we can apply this construction to ρ, and we write r := rρ : GQ →Gn (F).
We let RF be the complete local Noetherian O-algebra which is the universal den
formation ring for Gn -valued deformations of r which have multiplier ε1−n δF/F
+ , areCUSPIDAL COHOMOLOGY CLASSES FOR GLn (Z)7unramified outside p, and whose restrictions to the places above p are crystalline
and ordinary with Hodge–Tate weights 0, 1, . . . , n − 1. The association ψ 7→ rψ
induces a homomorphism RF → R, which is easily checked to be a surjection. (Indeed, it suffices to show that the map RF → R induces a surjection on reduced
cotangent spaces. It in turn suffices to see that the induced map of Lie algebras from (2.1.2) is a split injection of GQ -representations, or equivalently (sincep > 2) a split injection of GF -representations, which is clear.) The polarized representation (r|GF , µ) is ordinarily automorphic by [NT21, Thm. A] applied to f
(together with quadratic base change), and the group r(GF (ζp ) ) is adequate by
Lemma 2.2. Applying [Tho12, Thm. 10.1], we see that RF is a finite O-algebra(see [BLGGT14, Thm. 2.4.2] for a restatement in the precise form we use here; inthe notation of that statement, we are taking l = p, n = p − 1, S = {p}, µ = ε1−n ,Hτ = {0, 1, . . . , n − 1}). Thus R is a finite O-algebra, and since it has dimension
at least 1, it has a Qp -valued point. The corresponding lift ρ : GQ → Gn (Qp ) of ρis unramified outside p, has multiplier ε1−n , and is crystalline and ordinary with
Hodge–Tate weights 0, 1, . . . , n − 1.
The representation ρ is automorphic by [BLGGT14, Thm. 2.4.1] (taking F = Q,nl = p, n = p − 1, r = ρ, and µ = ε1−n δF/F+ ). More precisely, there is a selfdual regular algebraic cuspidal automorphic representation π of GLn (AQ ) whose
corresponding p-adic Galois representation ρπ : GQ → GLn (Qp ) is isomorphic to ρ.
By local-global compatibility (e.g. [BLGGT14, Thm. 2.1.1]) we see that π has level
one and weight zero, as claimed.
□
Lemma 2.2. Let p > 5 and let r : GQ → GL2 (Fp ) be a representation withSL2 (Fp ) ⊆ r(GQ ). Then for p − 2 ≤ n ≤ p, the group (Symn−1 r)(GQ(ζp ) ) isadequate in the sense of [Tho17, Defn. 2.20].
Proof. Since SL2 (Fp ) is perfect, we have SL2 (Fp ) ⊆ r(GQ(ζp ) ), so it follows from
Dickson’s classification that for some power q of p, we have SL2 (Fq ) ⊆ r(GQ(ζp ) ),and p ∤ [r(GQ(ζp ) ) : SL2 (Fq )]. By [GHT17, Rem. 6.1], it suffices to check that for U
the standard 2-dimensional Fp -representation of G = SL2 (Fq ), V := Symn−1 U isadequate. It is absolutely irreducible (because n ≤ p), and is therefore adequate
by [GHT17, Cor. 9.4], noting that since p > 5 we have n ≥ p − 2 > (p + 1)/2. □2.3. The case p = 107. We now prove the cases n = 105 and n = 106 of Theorem B
as an application of Theorem 2.1.
Theorem 2.4. There exist self-dual cuspidal automorphic representations π for
GLn /Q of level one and weight zero for n = 105 and n = 106. In particular,∗(GLn (Z), C) ̸= 0 for these n.
Hcusp
Proof. Let f = ∆E42 E6 = q − 48q 2 − 195804q 3 + . . . be the unique normalized
cuspidal Hecke eigenform for SL2 (Z) of weight k = 26. Let p = 107, and ρ :GQ → GL2 (F107 ) denote the mod 107 Galois representation associated to f (inits cohomological normalization). By [SD73, Cor., p.SwD-31], the image of ρ is25
exactly GL2 (F107 ) (note that (F×
= F×
107 )
107 ). Sincea107 (f ) = 35830422465487817813321292 ≡ −1 mod 107,
f is ordinary at 107.
Certainly (106, 25) = 1, so in view of Theorem 2.1 we only need to checkthat ρf |GQp is semisimple. That this is indeed the case is a consequence of a

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G. BOXER, F. CALEGARI, AND T. GEE

computation of Elkies, recorded in [Gro90, §17]: the form f admits a companion
form of weight p + 1 − k = 82, i.e. an eigenform g of level one and weight 82 withρf ∼= ε−25 ρg . The semisimplicity of ρf |GQp is an immediate consequence of the existence of g (see e.g. [Gro90, Prop. 13.8(3)]). By Theorem 2.1 we deduce the existenceof the desired automorphic forms π for GLn /Q for n = 105, 106 respectively. The
existence of such π implies the non-vanishing of the cuspidal cohomology groups(see Remark 1.2).
□
Remark 2.5. Combining Theorem 2.4 with the descent result [CKPSS04, Thm.
7.2], we see that there is a globally generic, non-endoscopic, cuspidal automorphic
representation for Sp104 /Q of level one and weight zero. If Ag is the moduli space of∗principally polarized abelian varieties of dimension g, we deduce that Hcusp(A52 , C) ̸=
0. However, as Olivier Taı̈bi explained to us, one can construct cuspidal cohomology
classes of Ag for much smaller g coming from endoscopic representations, and one
can even arrange that these endoscopic representations are tempered; see [CR15, §
1.24] for a closely related discussion.
3. The non-ordinary caseWe now explain how to improve n = 105 to n = 79, at the cost of a slightly
more involved construction. The idea behind the proof is again quite simple: wereplace the ordinary eigenform f in Theorem 2.1 by a non-ordinary form, where
one can hope to use the change of weight results of [BLGGT14]. It turns out that
there is no local obstruction to the existence of a weight zero lift of (a twist of)
Symn−1 ρf if n−1 = p−1 or p. However, in the latter case the global representation
Symn−1 ρf is reducible, and we do not know whether to expect a congruence toexist in level one, while in the former case it has dimension p, which is excluded by
the hypotheses of [BLGGT14]. Nonetheless, in the case n − 1 = p − 1, we are able
to use a simplified version of the arguments of [BLGGT14], since we do not need tochange the level and only need to make a relatively simple change of weight, andindeed our arguments are very close to those of [BLGG11].
Theorem 3.1. Let p > 5 be a prime, and let f be an eigenform of level SL2 (Z)and weight 2 ≤ k < p, such that:(1) (k − 1, p + 1) = 1.
(2) f is non-ordinary at p.
Then there exists a self-dual cuspidal automorphic representation π for GLp /Q oflevel one and weight zero whose mod p Galois representation ρπ : GQ → GLp (Fp )is isomorphic to Symp−1 ρf .
Proof. Where possible, we follow the proof of Theorem 2.1. We begin by showingthat ρf has image containing SL2 (Fp ). Since (k − 1, p + 1) = 1, the projective
image of ρf (GIQp ) contains a cyclic subgroup of order p + 1 > 5, so ρf does nothave exceptional image (that is, projective image A4 , S4 , or A5 ). Since ρf |GQp isabsolutely irreducible, so is ρf . Hence it remains to rule out the possibility that ρf
has dihedral image. If this were the√ case, then since it is unramified outside p, it
would have to be induced from Q( p∗ )√where p∗ = (−1)(p−1)/2 p. But this would
imply that ρf |GQp is induced from Qp ( p∗ ), which would in turn imply that it is(p2 −1)/2

invariant under twisting by ε(p−1)/2 = ω2

p(k−1). Since ρf |Ip ≃ ω2k−1 ⊕ ω2,

CUSPIDAL COHOMOLOGY CLASSES FOR GLn (Z)9this can only happen if k ≡ (p + 3)/2 (mod p + 1), contradicting the assumptionthat (k − 1, p + 1) = 1.
Let F/Fp be a finite extension such that ρf (GQ ) ⊆ GL2 (F), and write ρ :=Symp−1 ρf , so that ρ : GQ → GOp (F) has multiplier ε1−p = 1, and ρ(GQ(ζp ) ) isadequate by Lemma 2.2.
×Let ε2 , ε′2 : GQp2 → Zp be the two Lubin–Tate characters trivial on ArtQp2 (p),and write ω2 for the reduction modulo p of ε2 . For any n, m ≥ 1 we let ρn,m denote
the representation
GQ: GQp → GLn (Zp ),Symn−1 IndGQp ε−m2p2which is crystalline with Hodge–Tate weights 0, m, . . . , (n − 1)m.
We have(p−1)/2

ρp,m ∼= εm(p−1)/2 ⊕

M
i=1

GQ

m(1−p)i

IndGQp ω2.

p2m(1−p)

Suppose that (m, p + 1) = 1 (so that in particular m is odd). Then ω2hasm(1−p)i−m(1−p)iorder exactly p + 1, and the Gal(Qp2 /Qp )-conjugate of ω2
is ω2
. It
follows, under this assumption on m, that ρp,m does not depend on m, so there
is an isomorphism of orthogonal representations ρp,m ∼= ρp,1 . Our assumptions
that f is non-ordinary, that k < p, and that (k − 1, p + 1) = 1 therefore imply that
ρ|GQp ∼
= ρp,1 , which admits the weight 0 crystalline lift ρp,1 .
Write R for the complete local Noetherian O-algebra which is the universal
deformation ring for GOp -valued deformations of ρ which have multiplier ε1−p , areunramified outside p, and whose restrictions to GQp are crystalline of weight 0,
and lie on the same component of the corresponding local crystalline deformation
ring as ρp,1 . By [BG19, Prop. 4.2.6], every irreducible component of R has Krull
dimension at least 1 (the verification that H 0 (Q, so∗p (1)) = 0 is exactly as in the
proof of Theorem 2.1).
Let F + /Q and F/F + be quadratic extensions, with F + real quadratic and F
imaginary CM, such that p is inert in F + , the place of F + above p splits in F ,and F/Q is disjoint from (Q)ker ρ (ζp ). As in the proof of [BLGGT14, Prop. 4.1.1],
using [BLGGT14, Cor. A.2.3, Lem. A.2.5] we can find a cyclic CM extension M/F of′×
degree (k − 1), and characters θ, θ′ : GM → Qp with θ = θ , such that the represen′Ftation s := IndG
GM (θ ⊗ ρ|GF ) is absolutely irreducible. Furthermore we choose θ, θGF ′F
so that θθc = ε2−k , θ′ (θ′ )c = εp(2−k) , and IndG
GM θ, IndGM θ are both crystalline,with all sets of labelled Hodge–Tate weights respectively equal to {0, 1, . . . , k − 2},{0, p, . . . , p(k − 2)}.
By construction, after possibly replacing F + by a solvable extension, we can anddo assume that for each place v|p of F , we have


GF ′
F
IndG
GM θ |GFv ∼ ρk−1,1 |GFv , IndGM θ |GFv ∼ ρk−1,p |GFv ,where ∼ is the notion “connects to” of [BLGGT14, §1.4]. We let RF be the complete local Noetherian O-algebra which is the universal deformation ring for G(k−1)p valued deformations of (the usual extension of) s, which have multiplier ε1−(k−1)p δF/F + ,are unramified outside p, and whose restrictions to the places above p are crystalline

10

G. BOXER, F. CALEGARI, AND T. GEEwith Hodge–Tate weights 0, 1, . . . , (k − 1)p − 1, and lie on the same irreducible components of the local crystalline deformation rings as∼ ρ(k−1)p,1 |G =∼ (ρp,k−1 ⊗ ρk−1,1 )|G .
(ρk−1,p ⊗ ρp,1 )|G =
Fv

Fv

Fv

F
We have a finite map RF → R, taking a lifting ρ of ρ to IndG
GM (θ ⊗ ρ|GF ).

We claim that the conclusions of [Tho17, Prop. 7.2] apply in our setting, sothat RF is a finite O-algebra by [Tho12, Thm. 10.1]. Admitting this claim for a
moment, we deduce that R is a finite O-algebra, and since it has dimension atleast 1, it has a Qp -valued point. The corresponding lift ρ : GQ → GOp (Qp ) of ρis unramified outside p, has multiplier ε1−p , and is crystalline with Hodge–TateFweights 0, 1, . . . , p − 1. By [Tho17, Thm. 7.1], IndG
GM (θ ⊗ ρ|GF ) is automorphic, so ρ
itself is automorphic by [BLGGT14, Lem. 2.2.1, 2.2.2, 2.2.4].
It remains to show that we can apply [Tho17, Thm. 7.1, Prop. 7.2]. To thisend, we note that the notion of adequacy in [Tho17, Defn. 2.20] can be relaxed toassume only that H 1 (H, ad) = 0, rather than assuming that H 1 (H, ad0 ) = 0; more
precisely, the proof of [Tho17, Prop. 2.21] only uses this weaker assumption. Now,
since ρ(GQ(ζp ) ) is adequate, and since p ∤ (k − 1), we see that s(GF (ζp ) ) is adequate
by [BLGG13, Lem. A.3.1] (whose proof goes over unchanged in this setting), as
required.
□Corollary 3.2. There exists a self-dual cuspidal automorphic representation π for
GL79 /Q of level one and weight zero.
Proof. There exists ([Gou01, CG13]) a modular eigenform f of level 1 and weight k =38 which is non-ordinary at p = 79, and (37, 79 + 1) = 1.
□
Remark 3.3. The prime p = 79 is the second smallest prime for which there exists
a non-ordinary form f of weight k < p. The smallest is p = 59 for which there
exists a non-ordinary eigenform of weight k = 16. However, (k − 1, p + 1) ̸= 1 inthis case, so the construction fails in a number of places. Following [CG13], we seethat there exist modular forms f satisfying the hypotheses of Theorem 3.1 for p =79, 151, 173, 193, . . . and modular forms satisfying the hypotheses of Theorem 2.1
for p = 107, 139, 151, 173, 179, . . .. We expect (but have no idea how to prove) that(in either case) there exist such f for a positive density of primes p.
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Email address: g.boxer@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Email address: fcale@math.uchicago.edu
The University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA
Email address: toby.gee@imperial.ac.uk
Department of Mathematics, Imperial College London, London SW7 2AZ, UK