DEFORMATIONS OF SAITO-KUROKAWA TYPE AND THE
PARAMODULAR CONJECTURETOBIAS BERGER1 AND KRZYSZTOF KLOSIN2
WITH AN APPENDIX BY CRIS POOR3 , JERRY SHURMAN4 , AND DAVID S. YUEN5

Abstract. We study short crystalline, minimal, essentially self-dual deformations of a mod p non-semisimple Galois representation σ with σ ss = χk−2 ⊕ρ ⊕ χk−1 , where χ is the mod p cyclotomic character and ρ is an absolutely irreducible reduction of the Galois representation ρf attached to a cusp
form f of weight 2k − 2. We show that if the Bloch-Kato Selmer groups
Hf1 (Q, ρf (1 − k) ⊗ Qp /Zp ) and Hf1 (Q, ρ(2 − k)) have order p, and there exists
a characteristic zero absolutely irreducible deformation of σ then the universal
deformation ring is a dvr. When k = 2 this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable
congruence of Siegel modular forms. As an example we prove the modularityof the abelian surface of conductor 731. When k > 2, we obtain an Rred = T
theorem showing modularity of all such deformations of σ.

1. Introduction
In analogy with the Taniyama-Shimura Conjecture (proved by Wiles et al.), in
the 1980s Yoshida proposed a conjecture postulating that abelian surfaces over Q
should correspond to Siegel modular forms. The most important progress on thisconjecture is due to Tilouine and Pilloni ([Til06, Pil12]). Their results show p-adic
modularity of abelian surfaces A (see also forthcoming work of Boxer, Calegari,Gee, and Pilloni on modularity) under the assumption that the residual Galois
representation A(Q)[p] has image containing PSp4 (Fp ) and, more significantly, isitself modular. The latter assumption essentially calls for an analogue of Serre’s
conjecture for σ A , which at present appears out of reach.
Yoshida himself proved some cases of his conjecture in [Yos80] when EndQ (A) 6=
Z. In 2014, Brumer and Kramer proposed a refinement of Yoshida’s Conjecture by
specifying the level of the Siegel modular form [BK14]. This is sometimes referred
to as the Paramodular Conjecture.
Conjecture 1.1 (Brumer - Kramer). There is a one-to-one correspondence between
isogeny classes of abelian surfaces A/Q of conductor N with EndQ A = Z and weight
2 Siegel modular forms F , which are not in the space spanned by the Saito-Kurokawa
lifts, have level K(N ) and rational eigenvalues, up to scalar multiplication, whereK(N ) is the paramodular group of level N defined by K(N ) = γM4 (Z)γ −1 ∩Sp4 (Q)with γ = diag[1, 1, 1, N ]. Moreover, the L-series of A and F should agree and the
The first author’s research was supported by an LMS Research in Pairs Grant and the EPSRC Grant EP/R006563/1. The second author was supported by the Young Investigator Grant#H98230-16-1-0129 from the National Security Agency, a Collaboration for Mathematicians Grant#354890 from the Simons Foundation and by a PSC-CUNY award jointly funded by the Professional Staff Congress and the City University of New York.
1

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TOBIAS BERGER, KRZYSZTOF KLOSIN

p-adic representation of Tp (A) ⊗ Qp should be isomorphic to those associated to F
for any p prime to N where Tp (A) is the p-adic Tate module.
Brumer-Kramer [BK14] and Poor-Yuen [PY15] verified Conjecture 1.1 in several
cases by proving the non-existence of an abelian surface for certain (small) conductors N for which there are no non-lift Siegel modular forms of level K(N ). When A
acquires extra endomorphisms over a quadratic field the paramodularity of A has
in some cases been proven using theta lifts, see [JLR12] and [BDPS15].
To the best of our knowledge the only genuinely symplectic case (i.e., concerning
surfaces whose associated Galois representations are absolutely irreducible and not
induced from 2-dimensional representations) of Conjecture 1.1 in which the abeliansurface exists that has so far been fully verified is the case of the abelian surface ofconductor 277 (paper in preparation by Brumer, Pacetti, Poor, Tornaria, Voight,
and Yuen).
In this article we study Galois representations arising from abelian surfaces (as inConjecture 1.1) with rational torsion. We propose a method to prove modularity ofsuch representations which we carry out here under certain assumptions. This caseis not covered by the work of Tilouine and Pilloni et al. It has the advantage of not
requiring modularity of σ A and replacing this assumption with a construction ofcongruences between paramodular Saito-Kurokawa lifts and stable Siegel modular
forms. In other words we do not have a need for an analogue of Serre’s Conjecture.
While potentially more accessible, proving the existence of such congruences is stilla considerable problem. In this article however we mostly focus on the Galois
representation side (deformation theory) of the modularity problem, and make useof results where the required congruence has been constructed by others (mostly
by Poor and Yuen). For the purposes of this article Poor, Yuen and Shurman havekindly agreed to prove the existence of such a congruence in the case of forms ofparamodular level 731 (see Appendix). See also [PY15] for other examples. A fullmodularity result would require the existence of such congruences in a more generalcontext, a problem which is a subject of joint work in progress with J. Brown.
As a consequence of the results proved in this article we are able to establish themodularity part of the Paramodular Conjecture in new (genuinely symplectic) cases(in particular the first composite level case) and provide a way to verify more, when
the appropriate ingredients are supplied.
Let us explain the results and the methods of this paper in more detail. Let
p > 3 be a prime such that p - N . Let A be as in Conjecture 1.1 and suppose that
A has a rational p-torsion point, semi-abelian reduction at ` | N , and a polarizationof degree prime to p. This implies that the semisimplification σ ss
A of the residual
Galois representation σ A : GQ → GL4 (Fp ) attached to A (i.e., afforded by the
p-torsion Galois module A(Q)[p]) has the form
ss
∼
σ ss
A =1⊕χ⊕ρ ,where χ is the mod p cyclotomic character and ρ is a two-dimensional representation. Assume that ρ is absolutely irreducible (it is automatically odd). Serre’s
Conjecture (now a theorem of Khare and Wintenberger) implies that ρ arises as a
mod p reduction of the Galois representation ρf attached to a modular form f ofweight 2 and level Γ0 (N ). If the sign of the functional equation for f is −1, thenσ ss
A is the mod p reduction of the Galois representation attached to a weight 2, level
K(N ) Siegel Hecke eigenform SK(f ) which is the paramodular Saito-Kurokawa lift

DEFORMATIONS OF SAITO-KUROKAWA TYPE

3

of f . In fact, regardless of the sign of f , σ ss arises from a congruence level SaitoKurokawa lift of f , see section 9.2 for examples.
Our first result shows that one can construct a Galois-invariant lattice L inthe space Q4p with GQ -action via σA with respect to which the residual Galois
representation is non-semisimple. More precisely, we can ensure that it is blockupper-triangular with a specific order of the Jordan-Hölder factors on the diagonal(1, ρ, χ) - this order plays an important role in controlling the deformations (seebelow). For the modularity argument it is crucial to know that there is only one
isomorphism class of such residual representations as long as we require that they
be short crystalline, minimally ramified and give rise to desired extensions of the
Jordan-Hölder factors of σ A . The existence of such a lattice can be proved by essentially following the standard method of Ribet (cf. Proposition 2.1 in [Rib76])
adapted to a higher-dimensional setting. The uniqueness of the residual representation is trickier and requires us to study iterated Fontaine-Laffaille extensions aswell as to control ramification at the primes dividing N .
We then study short crystalline, minimal and essentially self-dual deformationsof the residual representation σ A . Let R denote the reduced universal deformation
ring. Its structure relevant for our purposes can be controlled by two sets of data:
its ideals of reducibility {IP }P corresponding to all possible partitions P of the setof Jordan-Hölder factors of σ A and the quotients {R/IP }P . Roughly speaking, theformer control trace-irreducible deformations, while the latter the trace-reducible
ones. In the case when IP is the total ideal of reducibility (i.e., corresponds to themost refined partition), the quotient R/IP can be shown to be related to the BlochKato Selmer group H−1 , where Hi := Hf1 (Q, ρf (i) ⊗ Qp /Zp ), by a generalizationof the approach used in [BK13]. On the other hand understanding of the ideals of
reducibility IP requires an entirely new approach as the methods used in [BK13]
cannot be extended to our current situation.
To overcome this problem we draw on ideas of Bellaïche and Chenevier contained
in sections 8 and 9 of their indispensable book [BC09] which has served us as
an inspiration on many occasions. There a problem concerning representations
with multiple Jordan-Hölder factors is studied in characteristic zero (as opposed
to our characteristic p situation). We, too, are able to prove that in our situation(assuming that the divisible Bloch-Kato Selmer group H0 is of corank at most 1)
all the (a priori different) ideals of reducibility in fact coincide and as a consequence
we are able to deduce that they are principal. This approach poses a considerable
amount of technical difficulties because, among other things, we are not able tocontrol ramification in the same way as Bellaïche and Chenevier do, one reason
being that their proof of the splitting of extensions at ramified primes (see proof of
Proposition 8.2.10 in [BC09]) does not extend to characteristic p.
So, while following the general strategy of [BC09] we need a different way of
manufacturing the necessary ingredients, and in particular are led to working witha specific minimality condition. Happily this turns out to also be the correct condition for yielding deformations corresponding to Siegel modular forms of squarefreeparamodular level (see Proposition 8.5). In some sense section 6, where the idealof reducibility is studied, comprises the technical heart of the paper.
We show that if H0 is of corank ≤ 1 and #H−1 = p (that #H−1 ≥ p is automatic
thanks to the existence of the lattice L), then R is a discrete valuation ring (seeTheorem 7.8). It means that the deformation σA of σ A is the unique characteristic

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TOBIAS BERGER, KRZYSZTOF KLOSIN

zero deformation. To conclude modularity of A we now need a non-lift Siegel
modular form F of weight 2 and level K(N ) whose Galois representation σF reduces(after semisimplification) to σ ss
A . This is achieved by exhibiting the existence ofa congruence between a candidate F and the paramodular Saito-Kurokawa liftSK(f ) (see section 9 and the Appendix). Irreducibility of σF (which we prove) anduniqueness of the non-semisimple residual representation (discussed above) now
guarantee that σF is a characteristic zero deformation of σ A . So, we must haveσA ∼= σF , proving the modularity of A. This can be viewed as the main applicationof our method in this paper, stated as Theorem 8.6. If a matching bound on T/J
were available (see next paragraph) our method would in fact be enough to provea full R = T theorem. We plan to address this problem in a future paper.
Let us note that we prove our results for a residual representation σ such that∼ χk−2 ⊕χk−1 ⊕ρ for any integer k ≥ 2. The case k = 2 (discussed above) is theσ ss =
most interesting because it corresponds to abelian surfaces and thus to Conjecture
1.1. In fact for even k > 9 and N = 1 we prove a full modularity result R = T using
a congruence result of Brown [Bro11], which provides us with the desired matchingbound on T/J in this case.
Let us now comment on the restrictiveness of our results. The most serious
assumptions are the ones on the Selmer groups. It is worth noting that while for
H−1 we require a specific bound on its order, for H0 we only require that is of
corank 1. In fact, it is important that for our method the corank assumption of thelatter group is sufficient since H0 is infinite if the central L-value of f is zero. It
is here that the order of the Jordan-Hölder factors on the diagonal of the residual
representation is crucial - a different order could swap the conditions on H0 and
H−1 making our theorem empty in this case.
On the other hand the Selmer group H−1 is non-critical which poses certain
difficulty in computing its order. We overcome this by using Kato’s result [Kat04]
towards the Main Conjecture of Iwasawa Theory and slightly adapting a control
theorem of [SU14] to relate the order to a special value of a p-adic (rather thanclassical) L-function of f (cf. section 2.3).
The assumptions on the Selmer groups are central to our method and in fact
without them R cannot be a dvr.
We also require that p is not a congruence prime for f and that each ` | N satisfies
p - 1+wf,` `, where wf,` is the local (at `) Atkin-Lehner sign of f . The first condition
allows us to relate the order of R/IP to the order of H−1 , while the latter allows
us to prove that the classes in H 1 (Q, ρf ⊗ Qp /Zp ) are automatically unramified at` | N (see Proposition 2.7). It is conceivable that both of these conditions could
be relaxed, but we do not know of a way to do it. What is important is thatthese conditions are often satisfied for small primes which has been our goal since
it is often the case that abelian surfaces possess rational p-torsion for such primes.
Unfortunately, the method to prove principality of the ideal of reducibility requiresthat p does not divide d!, where d is the dimension of the Galois representations (cf.
section 1.2 of [BC09]), which forces us to exclude p = 2, 3. As for elliptic curves thesize of the torsion subgroup of rational points on abelian surfaces is conjectured to
be bounded, but so far this has only been proven in special cases. Importantly for
us it is known though by work e.g. of Flynn [Fly90] that there are infinitely many
abelian surfaces with rational torsion points for certain primes p.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

5

While we hope that this article is only the first half of a larger undertaking ofproving a full R = T theorem for abelian surfaces in Conjecture 1.1 with rational
torsion (the second half being the mentioned work in progress on constructing
congruences), it is worth noting that even without the matching work on the Heckeside, it allows us to prove modularity of several new examples. In section 9 we workout the details for the abelian surface of conductor 731, and discuss other examples.
The paper is organized as follows. We begin by assembling necessary results
concerning Selmer groups in section 2 and define the relevant deformation problem
in section 3. The lattice L is constructed in section 4 and the uniqueness of theresidual representation is proved in section 5. Section 6 is devoted to the studyof the ideals of reducibility IP , while in Section 7 we study the quotients R/IP
and conclude the proof that R is a dvr. In section 8 we prepare the ground for
applications to examples of abelian surfaces satisfying our conditions and state theapplication to the Paramodular Conjecture (Theorem 8.6). Finally, in section 9 we
prove the modularity of an abelian surface of conductor N = 731 and discuss otherexamples. We conclude the paper by proving an R = T theorem in the case ofk > 2 and N = 1 in section 10.
We would like to thank David Savitt for helping us with the proof of Proposition
5.1 for the prime p and Jim Brown, Armand Brumer, Kenneth Kramer, Chris
Skinner and Eric Urban for helpful conversations related to the topics of this article.

2. Selmer groups
For each prime ` of Q we fix embeddings Q ,→ Q` ,→ C. Let p > 3 be a prime.
Throughout this paper E will denote a sufficiently large finite extension of Qp ,O its valuation ring with uniformizer $ and residue field F. Let  be the p-adic
cyclotomic character. We will write χ for its mod $ reduction.
2.1. Definitions and first properties. In this section we define (local and global)
Selmer groups which will be in use throughout the paper and recall some of theirbasic properties - for a more detailed treatment we refer the reader to section 5 of
[BK13]. For Σ a finite set of finite places of Q containing p we write GΣ for the
Galois group of the maximal extension of Q unramified outside of Σ and infinity.
We write GQ` for the absolute Galois group of Q` . Let M be an O-module withan O-linear action of G = GK or GΣ . We call M a p-adic G-module over O if oneof the following holds:(1) M is finitely generated, i.e. a finitely generated Zp -module and the G-action
is continuous for the p-adic topology on M ;(2) M is discrete, i.e. a torsion Zp -module of finite corank (i.e. M is isomorphic
as a Zp -module to (Qp /Zp )r ⊕M 0 for some r ≥ 0 and some Zp -module M 0 of
finite order) and the G-action on M is continuous for the discrete topology
on M ;(3) M is a finite-dimensional Qp -vector space and the G-action is continuous
for the p-adic topology on M .
M is both finitely generated and discrete if and only if it is of finite cardinality.
Given a p-adic GΣ -module M we assume that we have a finite/singular structure
S on M in the sense of [Wes00], i.e. for each prime ` ∈ Σ a choice of O-submodule

6

TOBIAS BERGER, KRZYSZTOF KLOSIN

1
Hf,S(Q` , M ) ⊆ H 1 (Q` , M ). We then define two global Selmer groups for M :
Y
1
1
(2.1)
Hf,S
(Q, M ) = ker(H 1 (GΣ , M ) →H 1 (Q` , M )/Hf,S(Q` , M )`∈Σand a “relaxed” Selmer group (no conditions at primes ` ∈ Σ \ {p})

1
1
HΣ,S
(Q, M ) := ker H 1 (GΣ , M ) → H 1 (Qp , M )/Hf,S(Qp , M ) .
1
We consider the following local finite/singular structures Hf,S(Q` , M ) (dropping the subscript S for the place at infinity and at p, where we fix the choice offinite/singular structure):
We always take Hf1 (Q` , M ) = 0 for ` | ∞.
For ` = p we define the crystalline local finite-singular structure as follows. Let
T ⊆ V be a GK -stable Zp -lattice and put W = V /T . For n ≥ 1, put
Wn = {x ∈ W : $n x = 0} ∼= T /$n T.

Following Bloch and Kato [BK90] we define Hf1 (Q` , V ) = ker(H 1 (Q` , V ) →H 1 (Q` , Bcrys ⊗ V )), denote by Hf1 (Q` , T ) its pullback via the natural map T ,→ V
and let Hf1 (Q` , W ) = im(Hf1 (Q` , V ) → H 1 (Q` , W )). Finally, we set Hf1 (Q` , Wn )to be the inverse image of Hf1 (Q` , W ) under the map H 1 (Q` , Wn ) → Hf1 (Q` , W ).
For finitely generated p-adic GQp -modules we recall the theory of FontaineLaffaille [FL82], following the exposition in [CHT08] Section 2.4.1. Let MF O(“Dieudonné modules”) denote the category of finitely generated O-modules M
together with a decreasing filtration Fili M by O-submodules which are O-direct
summands with Fil0 M = M and Filp−1 M = (0) and Frobenius linear maps
P i i
Φi : Fili M → M with Φi |Fili+1 M = pΦi+1 andΦ Fil M = M . They define anexact, fully faithful covariant functor G of O-linear categories from MF O (in theirnotation Gṽ and MF O,ṽ ) to the category of finitely generated O-modules with continuous action by GQp . Its essential image is closed under taking subquotients, finite
direct sums and contains quotients of lattices in short crystalline representations
defined as follows: We call V a continuous finite-dimensional GQp -representation
over Qp short crystalline if, for all places v | p, Fil0 D = D and Filp−1 D = (0) for
the filtered vector space D = (Bcrys ⊗Qp V )Gv defined by Fontaine.
For any p-adic GQp -module M of finite cardinality in the essential image of G
(we will call such modules short crystalline) we define Hf1 (Qp , M ) as the image of
Ext1MF O (1FD , D) in H 1 (Qp , M ) ∼= Ext1O[GQp ] (1, M ), where G(D) = M and 1FD isthe unit filtered Dieudonné module defined in Lemma 4.4 of [BK90].
For any complete Noetherian Zp -algebra we define a representation ρ : GQp →GLn (A) to be crystalline if for each Artinian quotient A0 of A, ρ ⊗ A0 lies in theessential image of G.
Remark 2.1. Note that for short crystalline representations V we gave two different definitions for Hf1 (Qp , Wn ) (via pullback of Hf1 (Qp , W ) and via the G-functor).
That these agree follows from the proof of Proposition 2.2 in [DFG04] or [BK13]
Lemma 5.3.
For primes ` 6= p we define the unramified local finite-singular structure on any
p-adic GQ` -module M over O as
1
1
Hf,ur
(Q` , M ) = Hur(Q` , M ) = ker(H 1 (Q` , M ) → H 1 (Q`,ur , M )),DEFORMATIONS OF SAITO-KUROKAWA TYPE

7

where Q`,ur is the maximal unramified extension of Q` .
Let V be a continuous finite-dimensional GQ` -representation over Qp and T ⊆ V
be a GQ` -stable Zp -lattice and put W = V /T . Bloch-Kato then define the following
minimal finite-singular structures on V , T and W :1
1
Hf,min(Q` , V ) = Hur(Q` , V ),i

1
Hf,min(Q` , T ) = i−1 Hf1 (Q` , V ) for T ,→ V

and1
Hf,min(Q` , W ) = im(Hf1 (Q` , V ) → H 1 (Q` , W )).
1
Remark 2.2. Following [Rub00] Definition 1.3.4 we define Hf,min(Q` , Wn ) as the11
inverse image of Hf,min (Q` , W ) under the map H (Q` , Wn ) → H 1 (Q` , W ).
1
1
By [Rub00] Lemma 1.3.5 we have Hf,min(Q` , W ) = Hur(Q` , W )div and if W I` is1
1
1
1
divisible we further have Hf,min (Q` , W ) = Hur (Q` , W ), Hf,min (Q` , T ) = Hur(Q` , T )1
1
and Hf,min (Q` , Wn ) = Hur (Q` , Wn ) (for the latter using also [Rub00] Lemma 1.3.8
and its proof).
ijLemma 2.3 ([Wes00] Lemma II.3.1). Let 0 → M 0 → M → M 00 → 0 be an exact
sequence of p-adic GΣ -modules over O. Assume that the modules have been given the11
induced finite/singular structures, i.e. such that Hf,S(Q` , M 00 ) = j∗ Hf,S(Q` , M )1
1
and Hf,S(Q` , M 0 ) = i−1
H
(Q
,
M
)
for
all
`.
`
∗
f,S
Then there is an exact sequence
1
1
1
0 → H 0 (Q, M 0 ) → H 0 (Q, M ) → H 0 (Q, M 00 ) → Hf,S(Q, M 0 ) → Hf,S(Q, M ) → Hf,S(Q, M 00 ).

The following proposition summarizes the facts we will need about the Selmer
groups we have defined:
Proposition 2.4.
(1) Suppose W GΣ = 0. Then HΣ1 (Q, Wn ) ∼= HΣ1 (Q, W )[$n ]11nand Hf,min(Q, Wn ) ∼
H
(Q,
W
)[$
].
= f,min
1(2) If the module W I` is divisible for every ` ∈ Σ \ {p}, then Hf,ur(Q, Wn ) =1
1
1
Hf,min (Q, Wn ) and Hf,ur (Q, W ) = Hf,min (Q, W ).
(3) If W is short crystalline at p then one can identify Hf1 (Qp , Wn ) as theimage of Ext1MF O (1FD , D), where D maps to Wn by the Fontaine-Laffaillefunctor.
Proof. The first statement follows from Lemma 2.3 applied to the sequence 0 →×$ n
W [$n ] → W → W → 0 since the minimal and crystalline finite/singular structures on Wn are induced from the one on W by definition. (This statement is also
stated as [Rub00], Lemma 1.5.4 (where this Selmer group is denoted S ∅ (Q, W ))upon noting that our assumption forces injectivity of the map H 1 (GΣ , Wn ) →H 1 (GΣ , W )[$n ] by [Rub00], Lemma 1.2.2(i)).
The second statement follows immediately from Remark 2.2, and the final one
from Remark 2.1.

1
In the following we will write Hf1 (Q, M ) for Hf,ur(Q, M ) and HΣ1 (Q, M ) for1
HΣ,ur (Q, M ) for any p-adic GΣ -module M .

8

TOBIAS BERGER, KRZYSZTOF KLOSIN

2.2. More properties of Selmer groups. Assume that k ≥ 2 is an even integer
such that p > 2k − 2. Note that under this assumption the p-adic Galois representation ρf associated to an eigenform of weight 2k − 2 of level not divisble by p isshort crystalline. This assumption will be in force throughout the paper.
Proposition 2.5. Let ρf : GΣ → GL2 (E) be the Galois representation attached to a
newform f ∈ S2k−2 (N ) for N a square-free integer with p - N and Σ = {` | N }∪{p}.
Suppose also that the residual representation ρf is absolutely irreducible and ramified
at every prime ` | N . Then for i = 1, 2(i) (ρf (i − k) ⊗ E/O)I` is divisible for all ` 6= p(ii) (ρf (i − k) ⊗ E/O)GΣ = 0.
Proof. For the convenience of the reader we include the proof, however one can also
consult the proof of Lemma 3.4 in [PW11].
(i) Set % := ρf (i − k) (i = 1, 2). We choose a basis {e1 , e2 } for the space V = V%
so that % with respect to that basis is valued in GL2 (O). We set T = Oe1 ⊕ Oe2(a GΣ -stable O-lattice in V ) and set W = V /T .
One has by our assumption and by Theorem 3.26(3)(b) of [Hid00] that


ψ ∗
(2.2)
ρf |GQ` ∼, ρf |I` 6= 1,=Eψwhere ψ : GQ` → E × denotes the unramified character sending Frob` to a` (f ). So,
there exists A ∈ GL2 (E) such that
 i−k+1ψ
∗
A%|G` A−1 =.
ψi−kFirst let’s note that % is only tamely ramified at ` and so the image of I` is procyclic. Suppose that g ∈ I` is such that %(g) topologically generates the image ofI` . Conjugating %(g) by a matrix of the form diag(u$n , 1) for a suitablen and
1 1
a b−1unit u we may assume that A%(g)A =∈ GL2 (O). Write A =
. By
1
c d
√
enlarging E (so that det A ∈ E) we may also assume that det A = 1. So,




1 1
1 + cdd2%(g) = A−1
A=
.
1
−c2
1 − cdSince % is valued in GL2 (O) we must have c2 ∈ O and d2 ∈ O. Since O is integrally
closed in E this implies c, d ∈ O.
Since ρf is ramified at ` so is %, i.e., we must have that either c ∈ O× or d ∈ O× .
I`
Let v ∈ V . Then v + T ∈ W if and only if g · v − v ∈ T. Writing  v in the basisxxO{e1 , e2 } as v =the above relation assumes the form:+∈ W I` if andyyO
only if     
cdx + d2 yxx
O= %(g)−∈−c2 x − cdyyyOwhich happens if and only ifd(cx + dy) ∈ O×and− c(cx + dy) ∈ O.

Now suppose c ∈ O . Then from the second relation we get cx + dy ∈ O. Since
this condition also implies that the first expression is in O, we get that in this case

DEFORMATIONS OF SAITO-KUROKAWA TYPE

9

 
x
+ T ∈ W I` if and only if cx + dy ∈ O. Similarly if d ∈ O× , then we again getythe exact same condition. Hence we conclude that   
xO
W I` =+| x, y ∈ E, cx + dy ∈ O .
yO
Suppose c ∈ O× . ThenW

I` −1   

−c dy
O
=
+
|y∈E .
yO

Consider the map from the module on the right to E/O given by −1   
−c dy
O
+
7→ y + O.
yO −1 
−c dy
It is clearly a map of O-modules and clearly surjective. Suppose that
+
y
 
O
∈ ker. Then y ∈ O, which implies that c−1 dy ∈ O, so the map is also injective,Ohence an isomorphism. Analogous map works in the case when d ∈ O× . Since E/O
is divisible this proves that W I` is divisible.
(ii) Let T, W be as in the proof of (i). Clearly if v ∈ W GΣ , then some O-multipleof v is a non-zero element in W [$]. Hence it is enough to show that W [$]GΣ = 0.
However, we have (see e.g. [Rub00], Lemma 1.2.2(iii)) W [$] ∼= T /$T ∼= %, so theclaim follows from the irreducibility of ρf .
Corollary 2.6. Let ρf be as in Proposition 2.5. For i = 1, 2 one has
11(Q, ρf (i − k)).
Hf,min(Q, ρf (i − k)) = Hf,ur

Proof. This follows from Propositions 2.5 and 2.4(1)+(2).



Proposition 2.7. Let ρf : GΣ → GL2 (E) be the Galois representation attached toa newform f ∈ S2k−2 (N ) with p - N . Suppose also that the residual representation
ρf is absolutely irreducible and ramified at every prime ` | N . One has HΣ1 (Q, ρf (2−k) ⊗ E/O) = Hf1 (Q, ρf (2 − k) ⊗ E/O) if N is square-free, Σ = {` | N } ∪ {p}, andp - 1 + wf,` ` for all primes ` | N . Here wf,` ∈ {1, −1} is the eigenvalue of the
Atkin-Lehner involution at `.
Remark 2.8. Note that when N is prime wf,N = (−1)k−1 Wf , where Wf ∈ {1, −1}
is the sign of the functional equation of f (see e.g. [Shi71] Theorem 3.66).
Proof. We use Lemma 5.6 in [BK13] with V = ρf (2 − k). Then V ∗ = V ∨ (1) = V .
By (2.2) we have that (V ∗ )I` = ψ(3 − k) and hence we get (using the notationof [BK13])
P` (V ∗ , 1) = det(1 − Frob` |(V ∗ )I` ) = 1 − a` (f )`3−k .
By Theorem 3 in [AL70] (but with Hecke action normalized as in [Miy89], see[Miy89] Theorem 4.6.17(2) for the correct power of ` in our normalization) we knowthat a` (f ) = −`k−2 wf,` . This shows that under the conditions of the Proposition
we have valp (P` (V ∗ , 1)) = 0 for all ` - N . To be able to use Lemma 5.6 in [BK13],
we also need to know that W I` is divisible, where W = V /T with V as above (i.e,the two dimensional vector space over E on which GΣ acts via ρf (2−k)) and T ⊂ V
a GΣ -stable O-lattice. However, this follows from Proposition 2.5(i). We note here

10

TOBIAS BERGER, KRZYSZTOF KLOSIN

that the order of the Selmer group does not depend on the choice of T since ρ isabsolutely irreducible (cf. e.g., Lemma 9.7 in [Klo09]).
Proposition 2.9. One has
Hf1 (Q, F(1)) = Hf1 (Q, F(−1)) = 0.
Proof. This is essentially Proposition 6.16 along with Theorem 6.17 in [Was97],
which tell us that the p-part of the class group of the splitting field of χ±1 is trivial(for χ−1 this follows from the fact that the numerator of B2 (the second Bernoulli
1
number) is 1). This implies that Hf,min(Q, W1 ) = 0 for W = Qp /Zp (±1 ). By
1
1
Proposition 2.4 we also have that Hf,min (Q, W1 ) = Hf,ur(Q, W1 ) = Hf1 (Q, W1 )since W I` = 0 for all ` 6= p.

2.3. Computing the non-critical Selmer group. We will be interested in bounding from above the order of Hf1 (Q, ρf (1 − k) ⊗ E/O). In the case that k = 2 themotif ρf (1 − k) is not critical in the sense of Deligne. However, we will show how
it can still be bounded by a certain p-adic L-value using Iwasawa theory for p 6= 3.
Let ω be the Teichmüller lift of χ. Suppose f ∈ S2 (Γ0 (N )) (with p - N ) isa newform ordinary at p and that ρf is absolutely irreducible and ramified at all−1
primes ` | N . We will denote by Lan) ∈ O[[T ]] the Manin-Vishik p-adic
p (f, ω−1
L-function for f ⊗ ω (see e.g. [Kat04] Theorem 16.2 where the p-adic L-function
is defined for the p-stabilisation of f as an element of Λ̃ = O[∆][[T ]]; we take its
projection onto the component of Λ̃ where ∆ acts by ω −1 ).
Proposition 2.10. For p 6= 3 we have−1
valp (#Hf1 (Q, ρf (−1) ⊗ E/O)) ≤ valp #(O/Lan, T = p)).
p (f, ωProof. Write Tf for a choice of a GΣ -invariant lattice inside the Galois representation Vf = ρf : GΣ → GL2 (E). Since f is ordinary there is a unique Dp -stable line
Vf+ of Vf on which Ip acts via . Set Tf+ := Tf ∩ Vf+ . Following [SU14] (p. 34) for
any continuous character ξ : GF,Σ → O× we define the Greenberg Selmer group
n
o
+
1
1
SelΣ
F,E (f, ξ) := ker H (GF,Σ , Tf ⊗ ξ ⊗ E/O) → H (Ip , (Tf /Tf ) ⊗ ξ ⊗ E/O) ,where F = Q or F = Q∞ , the cyclotomic Zp -extension of Q. We are interested in
the case where ξ = ω m−1 −m for m = 1, while Skinner and Urban are mainly interΣ
ested in the case where ξ = ω m −m . We set XF,E
(f, ξ) := HomO (SelΣF,E (f, ξ), E/O)
to be the Pontryagin dual. By a theorem due to Kato ([SU14], Theorem 3.15) weΣ
(f, ξ) is a torsion Λ-module, where Λ = O[[Gal(Q∞ /Q)]] =know that XQ∞ ,EΣΣ(f, ω −1 ). The
O[[T ]]. Set ChQ∞ ,E (f, ω −1 ) to be the characteristic ideal of XQ
∞ ,E
Σ
−1
Main Conjecture of Iwasawa Theory states that ChQ∞ ,E (f, ω ) is a principal ideal−1, T ) (cf. Conjecture 3.24 in [SU14]). By Theorem 17.14 ingenerated by Lanp (f, ω[Kat04] (but see also Theorem 3.25 in [SU14], whose notation is consistent with−1−1ours) the ideal ChΣ) divides Lan, T ) in Λ. Note that the asp (f, ω
Q∞ ,E (f, ωsumption in Theorem 17.14 in [Kat04] that the image of ρf contains SL2 (Zp ) canbe replaced by ρf being ramified at some `kN , as explained in [Ski16] pages 187/8.
To relate this to our Hf1 (Q, ρf (−1) ⊗ E/O) we need a control theorem which wasproved by Skinner and Urban in the case ξ = ω m −m (Corollary 3.21(i) in [SU14]
which should say m 6= 0 instead of m 6= 1). Let us only indicate here what changes
are needed to adapt it to the case ξ = −1 = ω −1 · (ω/)m with m = 1.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

11

Theorem 2.11 (Control Theorem). One has
Σ−1#XQ(f, −1 ) | #Λ/(T − p, ChΣ
)).
Q∞ ,E (f, ω∞ ,EProof. Skinner and Urban prove the corresponding statement (for ξ = ω m −m )by a series of results culminating in Proposition 3.20 which is then used to prove
Corollary 3.21 (here we only care about one divisibility as stated above instead ofequality). The assumption that f be p-stabilized in [SU06] Section 3.3.13 is notnecessary. The first result is Proposition 3.10. The proof of the only part relevant−1−1 GΣ
for us, the injectivity of the map SelΣ) → SelΣ) , carries overQ,E (f, 
Q∞ ,E (f, unchanged to our situation. As the next step we need an analogue of PropositionΣ
3.20 (stating that XQ(f, ω −1 ) has no non-zero pseudo-null Λ-submodules) for∞ ,E
which it is sufficient to prove corresponding analogues of Lemmas 3.18 and 3.19.
Let us explain what changes are needed to the proof of Lemma 3.18 which for us
only holds for p 6= 3. While in [SU14] the module denoted there by M − [x]|Ip isisomorphic to E/O(ω m −m ) we have M − [x]|Ip ∼= E/O(ω m−1 −m ) (and m = 1).
The assumption that p 6= 3 is needed in our case to ensure that M − [m]∗ (1) = F(ω 2 )is ramified at p (note  = ω mod $ and m = (x, $)), where ∗ denotes the Pontryagin
dual. Also in our case M − [x]Ip = 0, hence in particular finite. Lemma 3.19 follows
the same way as in [SU14]. Having all this, Corollary 3.21 is also proved the same
way as in [SU14].

We continue with the proof of Proposition 2.10. As a corollary of the control
theorem we obtain the following inequality:Σ−1#XQ(f, −1 ) ≤ #O/Lan, T = p)p (f, ω∞ ,E(cf. Theorem 3.36 in [SU14]). Note that Skinner and Urban work within the criticalrange of the relevant L-function, while we are outside of this range, so in particular
we do not relate the p-adic L-value to a classical one.
Σ(f, −1 ) ≥ #Hf1 (Q, ρf (−1) ⊗ E/O)). First note
Let us now show that #XQ
∞ ,E
−1
Σ
(f, −1 ) = # SelΣ) since the group on the left is finite. We
that #XQ
Q,E (f, 
∞ ,E
Σ
−1
1
will show that SelQ,E (f,  ) ⊃ Hf (Q, ρf (−1) ⊗ E/O). The conditions at ` 6= p
defining both groups coincide by combining Propositions 2.5(i) and 2.4(2), so it
is enough to compare conditions at p. There we have Hf1 (Qp , ρf (−1) ⊗ E/O) ⊂Hg1 (Qp , ρf (−1) ⊗ E/O) which follows from (3.7) in [BK90] by taking images in
the cohomology with divisible coefficients. On the other hand the diagram at thebottom of p. 87 in [Och00] tells us that Hg1 (Qp , ρf (−1) ⊗ E/O) injects into1
HGr
(Qp , ρf (−1)⊗E/O) := ker(H 1 (Qp , Tf (−1)⊗E/O) → H 1 (Ip , Tf (−1)⊗E/O)).

Indeed, Hg1 (Qp , ρf (−1) ⊗ E/O) is defined as the image of Hg1 (Qp , Vf (−1)) in
H 1 (Qp , Tf (−1) ⊗ E/O) and according to the diagram the defining map factors
1(Qp , ρf (−1) ⊗ E/O). The corresponding inclusions of global Selmer
through HGrgroups follow.

3. Deformations
Let ρ : GΣ → GL2 (F) be a continuous absolutely irreducible odd Galois representation of determinant χ2k−3 ,short crystalline at p and such that for every prime1 ∗` ∈ Σ \ {p} one has ρ|I` ∼6= I2 . Consider a residual (short crystalline)
=
0 1

12

TOBIAS BERGER, KRZYSZTOF KLOSIN

representation
 k−2
χ
σ := aρbc  : GΣ → GL4 (F),χk−1where a, c are assumed to be non-trivial classes in H 1 (Q, ρ(1 − k)) (no assumption
on b).
Remark 3.1. We have
Hom(ρ, χk−2 ) = ρ∨ ⊗ χk−2 = ρ ⊗ χ3−2k ⊗ χk−2 = ρ(1 − k) = Hom(χk−1 , ρ),so a and c are indeed both classes in H 1 (Q, ρ(1 − k)).


0

0 1 
 and for any prime ` 6= p let tp : I` → Zp (1) be theSet N1 = 0 0tame character. We will always assume that σ as above is semi-abelian by which
we mean that for each ` ∈ Σ \ {p} one has(3.1)∼ exp(tp N1 ).
σ|I` =

Lemma 3.2. Let x ∈ GΣ be such that σ(x) generates the image of I` . Then σ isisomorphic to a representation of the form

 k−2
χ
∗1
∗2
ρ
∗3 
σ0 = 
χk−1
and such that0
00
σ (x) − I4 = 
0
0

0
0
0
0


0 0
∗ 0
=
6 0,
0 0
0 0

i.e., the matrix conjugating σ|I` to exp(t` N1 ) does not change the matrix form of
σ.


1 α
Proof. Without loss of generality assume that ρ(x) =with α 6= 0. Write
0 1
αi,j for the (i, j)-entry of σ. Then
0 α1,2 (x) α1,3 (x) α1,4 (x)00αα2,4 (x)
.
σ(x) − I4 = 000α3,4 (x)
0
0
0
0


α1,3 (x) α1,4 (x)Since rank(σ(x)−I4 ) = rank(N1 ) = 1, we must also have that M :=αα2,4 (x)has rank 1. Thus there exists β ∈ F such that α1,3 (x) = βα and α1,4 (x) = βα2,4 (x).
Hence we have






0 0
1 −β
1 −α2,4 (x)/α−1−1
AM D =, where A =,D =∈ GL2 (F).
α 0
0 1
0
1

DEFORMATIONS OF SAITO-KUROKAWA TYPE

13

Thus

A


D


(σ(x) − I4 )

−1

A
D


0
0 α1,2
(x)
0
0
=
0
0
0
0

0
α
0
0


0
0 
.
0
α3,4
(x)
0

0
However, since the last matrix has rank one and α 6= 0, we must have α1,2(x) =0
α3,4 (x) = 0. Furthermore note that

0


A

 
A
σ

σ :=
D
 k−2

 χ
∗
A

ρ
=
D

(3.2)

−1
D



∗
A
∗ 

χk−1

−1
D

=


A
D


 χk−2


∗0
ρ



∗0
A
∗0 

χk−1

−1
.

D


∼ 1 ⊕ ρ|I ⊕ 1. In particular, theCorollary 3.3. For ` ∈ Σ \ {p} we have σ|I` =`extensions given by the entries a and c of σ are both split when restricted to I` .
Let LCN(E) be the category of local complete Noetherian O-algebras withresidue field F.
Definition 3.4. For an object A of LCN(E) we will say that σ : GΣ → GL4 (A) is
minimal if σ|I` ∼=Tot(A) exp(tp N1 ) for every prime ` ∈ Σ \ {p}. Here Tot(A) denotes
the total ring of fractions of A.
Note that any minimal σ : GΣ → GL4 (A) is tamely ramified at all ` ∈ Σ \ {p},so σ(I` ) is a pro-cyclic group for all such `.
We consider the deformation problem for σ where to every object A of LCN(E)we assign the set of strict equivalence classes of representations σ : GΣ → GL4 (A)which are short crystalline, have determinant 4k−6 , are minimal and reduce to σ
modulo the maximal ideal mA of A.
Theorem 3.5. The above deformation problem is representable.
Proof. It is easy to check that σ has scalar centralizer, so by e.g. [CHT08] Proposition 2.2.9 it is enough to verify that short crystallinity and minimality are deformation conditions. The former is due to Ramakrishna (cf. Theorem 1.1 of [Ram93]).
The latter can be proven similar to [Maz97] Section 29.

We will write R0 for the universal deformation ring and (σ 0 )univ : GΣ → GL4 (R0 )for the universal deformation.
Consider the following antiinvolution τ : R0 [GΣ ] → R0 [GΣ ] given by τ (g) =2k−3(g)g −1 . Clearly τ induces the following permutation on the set I of Jordan
Hölder factors of σ:χk−1 7→ χk−2 ,ρ 7→ ρ,χk−2 7→ χk−1 .

For A ∈ LCN(E) we will say that a short crystalline deformation σ : GΣ → GL4 (A)is τ -self-dual if tr σ = tr σ ◦ τ . Note that for semisimple σ and fields A this isequivalent to σ ∨ ∼= σ(3 − 2k).

14

TOBIAS BERGER, KRZYSZTOF KLOSIN

Proposition 3.6. The functor assigning to an object A ∈ LCN(E) the set ofstrict equivalence classes of τ -self-dual short crystalline (at p), minimal deformations to GL4 (A) is representable by the quotient of R0 by the ideal generated by{tr (σ 0 )univ (g) − tr (σ 0 )univ (τ (g)) | g ∈ GΣ }. We will denote this quotient by R andwill write σ univ for the corresponding universal deformation.
We write Rred for the quotient of R by its nilradical and σ red for the corresponding (universal) deformation, i.e., the composite of σ univ with R  Rred .
4. Lattice
Theorem 4.1. Let E be a finite extension of Qp with valuation ring O, uniformizer$ and F = O/$O. Let σ : GΣ → GLn (E) be an absolutely irreducible Galois
representation. Suppose there exists a GΣ -stable O-lattice Λ in the space of σ such
that


ρ1 ∗
∼
σΛ =
0 ρ2
with ρi : GΣ → GLni (F) (for i = 1, 2) having scalar centralizer, ρ1 semisimple
and such that none of the irreducible constituents of ρss
1 is isomorphic to any of the0irreducible constituents of ρss
.
Then
there
existsaGΣ -stable lattice Λ in the space2of σ such that
ρ1 ∗ ∼σ Λ0 ∼
6 ρ1 ⊕ ρ2 .
==0 ρ2Let us note that if ρ2 is semisimple, then Theorem 4.1 follows from a generalization of Theorem in [Urb99] - cf. Remark (a) in [Urb99].
We begin the proof of Theorem 4.1 with the following lemma.
Lemma 4.2. Let ρ1 , ρ2 be as in Theorem 4.1. Assume that σ : GΣ → GLn (F)given by


ρ1 f
σ=
ρ2
is isomorphic to ρ1 ⊕ ρ2 . Then there exists a matrix
I n1 BM=∈ GLn (F)
In2


ρ1
−1
such that M σM =
.
ρ2


A B
Proof. By assumption there exists h =∈ GLn (F) (with correct dimenC D
sions) such that




ρ
f
ρ
0
(4.1)
h 1
h−1 = 1
ρ2
ρ2which implies that ρ2 C = Cρ1 , i.e., that C ∈ HomGΣ (ρ1 , ρ2 ). The image of C
is a quotient of ρ1 which is a submodule of ρ2 . However, by our assumption onss
the irreducible constituents of ρss
1 and ρ2 , this image must be zero, so C = 0.
Then (4.1) implies that A centralizes ρ1 and D centralizes ρ2 , hence A = αIn1 andD = δIn2 with α, δ ∈ F× . By simply scaling h by α−1 we may assume without

DEFORMATIONS OF SAITO-KUROKAWA TYPE

loss of generality that α = 1. Finally (with C = 0), the B-entry of h

15


ρ1


f
h−1
ρ2

equals(4.2)−αρ1 α−1 Bδ −1 + αf δ −1 + Bρ2 δ −1 = ρ1 Bδ −1 + f δ −1 + Bρ2 δ −1 = 0.

From this we see that the expression in (4.2) equals zero regardless of the value ofδ, so we may assume that δ = 1.

Proof of Theorem 4.1. In this we follow Ribet [Rib76], the proof of Proposition 2.1.
Let the notation be as in the statementof the Theorem. Suppose no Λ0 exists. Then
I
Bby Lemma 4.2 there exists h = n1
(where B here is any lift to Mn1 ×n2 (O)In2of B in lemma) such that for all g ∈ GΣ one has
α $βhσΛ (g)h−1 =with α ≡ ρ1 (g) mod $, δ ≡ ρ2 (g) mod $$γ
δand α, β, γ, δ with entries in O. Then



In1
−1 In1τ1 (g) :=hσΛ (g)h
$In2

−1
$In2



α
=
$2 γ


β
.
δ

The reduction of τ1 still satisfies the assumptions of Lemma 4.2, 
hence by that
In1 B1α1$β 0−1lemma there exists h1 =∈ GLn (O) such that h1 τ1 h1 =,In2$2 γ 0δ00
00
where α , β , γ = γ, δ have entries in O. We continue this way as Ribet does toconclude in the end that σ itself is reducible, which leads to a contradiction. Thisfinishes the proof of Theorem 4.1.
Corollary 4.3. Let σ : GΣ → GL4 (E) be an absolutely irreducible short crystalline Galois representation. Suppose that σ ss = χk−2 ⊕ ρ ⊕ χk−1 with ρ absolutely
irreducible and that σ|I` ∼=E exp(tp N1 ) for all ` ∈ Σ \ {p}. Then there exists aGΣ -stable lattice Λ in the space of σ such that

 k−2
χ
∗1
∗2
ρ
∗3 
σΛ = 
χk−1
is semi-abelian with ∗3 a non-split extension of χk−1 by ρ and ∗1 a non-trivial
extension of ρ by χk−2 .
Proof. Applying Theorem 6.1 in [Bro11] with R = O and I = $O (note thatT ∼= O because any lattice in E is isomorphic to O) we get that there exists alattice Λ0 in the space of σ such that
 k−2

χ
0
∗2
ρ
∗3  6∼σ Λ0 = = χk−2 ⊕ ρ ⊕ χk−1 .
k−1χWe now claim that σ Λ0 cannot be equivalent to a representationof the same shape
 k−2
χ
∗2
where ∗3 = 0. Indeed, suppose it were, then σ Λ0 ∼
. Let
= ρ⊕
χk−1
` ∈ Σ \ {p}. If x ∈ GΣ is such that σ(x) generates the image of I` , then the rank ofσ(x) − I4 must be one. Since σΛ0 ∼=E σ, we also must have rank(σΛ0 (x) − I4 ) = 1,
and so also rank(σ Λ0 (x)−I4 ) = 1. Let us prove this last implication. Since ρ|I` 6= 1,
we must have rank(σ Λ0 (x)−I4 ) ≥ 1. So, we need to prove a rank one matrix cannot

16

TOBIAS BERGER, KRZYSZTOF KLOSIN

reduce mod $ to a matrix of a higher rank. Let A be a rank one matrix with entries
in O. Then every row is a scalar multiple of the first non-zero row. These scalars
are of the form u$n , where n ∈ Z. Pick a row for which n is minimal and by making
this row first (permutation matrix has entries in O), we may now assume that alln ≥ 0, i.e, that all the scalars are in O. Thus every row of the reduction of A mod$ is a scalar multiple ofthe first row. This establishes that rank(σ Λ0 (x) − I4 ) ≤ 1.
ρ
χk−2
∗2 , then this rank condition forces ∗2 to
However, if σ Λ0 ∼
= 
χk−1
be
unramified at `. Thus in this case, σΛ0 has a direct summand isomorphic to
 k−2
χ
∗
with ∗ unramified away from p. This direct summand is short crysχk−1talline at p since σΛ0 is, and so gives rise to an element in Hf1 (Q, F(−1)). Since
this group is trivial (by Proposition 2.9), this implies that σ Λ0 is semisimple, contradicting our assumption.
k−1
So we must have that ∗3 gives a non-trivial
by ρ. Now,
 of χ
 extension
ρ∗3
k−2(note that ρ2 has scalar
apply Theorem 4.1 with ρ1 = χand ρ2 =χk−1
centralizer). This gives us

 k−2


χ
∗1
∗2
ρ
∗3
k−2
∼


ρ
∗3
(4.3)
6 χ
σΛ =
⊕
=
χk−1
χk−1
with ∗3 a non-split extension of χk−1 by ρ. It remains to showthat ∗1 gives rise to k−2
χ
∗1 ∼ k−2
a non-trivial extension of ρ by χk−2 . Suppose that one has
⊕ ρ.
=χ
ρ
Then ∗1 corresponds to a coboundary in H 1 (Q, Hom(ρ, χk−2 )). More precisely,
there exists a matrix f ∈ Mn1 ×n2 (F) (i.e., a map in HomF (ρ, χk−2 )) such that
 

 k−2
χ
∗1 ∼ χk−2 χk−2 f − f ρ
.
=
0
ρ
ρ
However, note that  k−2
In1 f
χ


In2
In3


b
In1
c 

a
ρ

  k−2
χ

In3



 k−2
χ
=

In3

χ

So, it remains to show that
 k−2
χ
0

ρ
Note that we have
0n2 ×n1
In2
 I n10n1 ×n2

k−1

−1

f
I n2



b
ρ
k−2
∼

c
⊕
=χ

χk−1


b
0n2 ×n1c   In1

0
ρ
χ

k−1

∗


0 b + fc
ρ
c .
k−1
χ



χk−1

.

In2

−1

0n1 ×n2




ρ
=

χk−2

In3

By the same argument as before b ∈ H 1 (GΣ , HomF (χk−1 , χk−2 )) must again bea coboundary, i.e., b(g) = χk−2 (g)f − f ρ(g) for some matrix f . But then again


c
b .

χk−1

DEFORMATIONS OF SAITO-KUROKAWA TYPEIn1
the matrix 



In2


ρ
f  conjugates 
χk−2
In3

 
ρ
c
b  to 

17

χk−2


c
0 ∼
=

χk−1
χ

ρ
∗3
χk−2 ⊕, which leads to a contradiction to (4.3). The fact that σ Λ is semiχk−1abelian follows again from the minimality of σ.

k−1



Corollary 4.4. Let σ, τ : GΣ → GL4 (E) be two absolutely irreducible short crystalline at p Galois representations such that their restriction to I` is isomorphic toexp(tp N1 ) for all ` ∈ Σ \ {p}. Suppose that σ ss = τ ss = χk−2 ⊕ ρ ⊕ χk−1 with ρabsolutely irreducible. Suppose that dimF Hf1 (Q, ρ(1 − k)) ≤ 1. Then there existsa GΣ -stable lattice Λ in the space of σ and a GΣ -stable lattice Λ0 in the space of τsuch that
 k−2

 k−2

χ
a
b
χ
a
b0
ρ
c  and τ Λ0 = ρc ,σΛ = 
k−1
k−1
χχboth semi-abelian, with a, c both non-trivial elements of Hf1 (Q, ρ(1 − k)).
Remark 4.5. Note that in Corollary 4.4 we only assume an upper bound on thedimension of Hf1 (Q, ρ(1 − k)), but in fact as argued at the beginning of the proof ofthe corollary our assumptions imply that we always have dimF Hf1 (Q, ρ(1−k)) ≥ 1.
Proof. By Corollary 4.3 we can find lattices, so that both reductions have the above
shape, are short crystalline at p and semi-abelian. By Corollary 3.3 we know thatthe extensions induced by the entries a and c lie in Hf1 (Q, ρ(1 − k)). Since thelatter group is one-dimensional we can conjugate one of them to ensure that both
representations have the same a-entries and c-entries.

5. Uniqueness of iterated residual extensions
In this section we assume that ρ is the mod $ reduction of ρf : GΣ → GL2 (E),the Galois representation attached to a newform f ∈ S2k−2 (N ), where N squarefree,p - N , k is even and Σ = {` | N } ∪ {p}. We also assume that ρ is absolutely
irreducible and ramified at every ` | N . By Proposition 2.5 (see Corollary 2.6)
1
1
we then have that Hf,min(Q, ρ(1 − k)) = Hf,ur(Q, ρ(1 − k)) which we will simply
1
abbreviate to Hf (Q, ρ(1 − k)). We assume in this section thatdimF Hf1 (Q, ρ(1 − k)) = 1.
The goal of this section is to prove the following proposition.
Proposition 5.1. Let σ, τ : GΣ → GL4 (F) be two Galois representations of the
form
 k−2

 k−2

χ
a
b
χ
a0
b0
ρ
c , τ = 
ρ
c0 
σ=
k−1
k−1
χ
χ
 k−2
  k−2
 



0
ρ
c
ρ
c0
χ
a
χ
a
with the extensions,
andall giving
,
ρ
χk−1
χk−1
ρ
rise to non-zero elements of Hf1 (Q, ρ(1 − k)) = F. Suppose also that both σ and τ
are short crystalline and semi-abelian. Then σ ∼
= τ.

18

TOBIAS BERGER, KRZYSZTOF KLOSIN

Proof. Denote the cohomology classes corresponding to the extensions in the statement of the Proposition by φ, φ0 , ψ, ψ 0 respectively. Note that they indeed giverise to elements in Hf1 by Corollary 3.3. Then there exist α, γ ∈ F× such thatφ0 = αφ and ψ 0 = γψ. The functions a, a0 , c, c0 are given by a(g) = φ(g)ρ(g),a0 (g) = φ0 (g)ρ(g), c(g) = ψ(g)χk−1 (g) and c0 (g) = ψ 0 (g)χk−1 (g), so in particulara0 = αa and c0 = γc. Hence 
−1  k−211
χa α1 γ −1 b

τ 
 =
αI2
αI2
ρ
c .
k−1
αγ
αγ
χ
Thus we may assume without loss of generality that a0 = a and c0 = c. We
will express σ as an iterated extension and apply Lemma 2.3 to show that suchextensions are a torsor under Hf1 (Q, F(−1)) = 0 from which the uniqueness of such
σ will follow.
Consider σ as in the statement of the Proposition. Its subrepresentation corresponding to the 3 × 3 upper-left block gives rise to the following exact sequence ofGΣ -representations:
(5.1)
 k−2

χ
a
0 → HomF (χk−1 , χk−2 ) → HomF (χk−1 ,) → HomF (χk−1 , ρ) → 0,ρand after restricting to the decomposition group at v, also to a corresponding short
exact sequence of GQv -representations.
We first consider the case v = p: Write M for the category of filtered Dieudonne
modules as defined in [CHT08] Section 2.4.1. Since M is closed under subquotients,
there exist objects M1 , M2 , M3 , M of M corresponding
 k−2
 to the restrictions to GQp ofχaχk−2 , ρ, χk−1 and the subrepresentation: GΣ → GL3 (F) respectively.
ρ
Since Hom(χk−1 , M ) = M (1 − k) we see that all the objects in (5.1) are in theessential image of G. Hence we can apply Lemma 5.3 in [BK13] to conclude thatit gives rise to an exact sequence of Selmer groups(5.2)
0

H (Qp , HomF (χk−1, ρ)) → Hf1 (Qp , HomF (χk−1 , χk−2 )) → Hf1 (Qp , HomF (χk−1 ,

 k−2
χ
0


a
))
ρ

→ Hf1 (Qp , HomF (χk−1 , ρ)) → 0.
It follows from (5.2) that (in the terminology of [Wes00], p.4) the finite/singularstructures on the first and last non-zero term in the GQp sequence (5.1) are induced
from the middle term.
Let us now show that this is also true for all primes v 6= p when we take Hf1 tobe the unramified structure defined in section 2. So, suppose that v | N . It suffices
to show that the sequence
(5.3)
1
1
H 0 (Qv , HomF (χk−1 , ρ)) → Hur(Qv , HomF (χk−1 , χk−2 )) → Hur(Qv , HomF (χk−1 ,

 k−2
χ
0


a
))
ρ

1
→ Hur
(Qv , HomF (χk−1 , ρ)) → 0

DEFORMATIONS OF SAITO-KUROKAWA TYPE

191
with Hur(Qv , M ) := ker(H 1 (Qv , M ) → H 1 (Iv , M )) is exact. Let us rewrite (5.1)
as
 −1

χ
a0
(5.4)
0 → F(−1) →) → ρ(1 − k) → 0
ρ(1 − k)and call the extension in the middle E.
By Corollary 3.3 we have E Iv = F(−1)Iv ⊕ρ(1−k)Iv , so in particular the sequence
0 → F(−1)Iv → E Iv → ρ(1 − k)Iv → 0
is exact. Since every module in that sequence has an action of G := Gal(Qurv /Qv ),
we get a long exact sequence in cohomology of that group (and note that (M Iv )G =M Gv , where Gv = GQv ):Iv(5.5) 0 → F(−1)Gv → E Gv → ρ(1 − k)Gv → H 1 (Qurv /Qv , F(−1) ) →
2
ur
Iv
1
ur
Iv
Iv
H 1 (Qurv /Qv , E ) → H (Qv /Qv , ρ(1 − k) ) → H (Qv /Qv , F(−1) )Note that the last group is zero since G has cohomological dimension one. For all
Iv
1
three modules M one has H 1 (Qurv /Qv , M ) = Hur (Qv , M ) by Lemma 3.2 (i) in[Rub00].
Since we now showed that the finite/singular structures on the first and last
non-zero term in the sequence 5.1 are induced from the middle term for all GQv ,Lemma 2.3 tells us that we have an exact sequence
(5.6)
 k−2

χ
a
Hf1 (Q, HomF (χk−1 , χk−2 )) → Hf1 (Q, HomF (χk−1 ,)) → Hf1 (Q, HomF (χk−1 , ρ)).
0
ρSince Hf1 (Q, HomF (χk−1 , χk−2 )) = Hf1 (Q, F(−1)) = 0 by Proposition 2.9, we seethat the middle Selmer group in (5.6) injects into the last one. Hence the cohomology class in Hf1 (Q, HomF (χk−1 , ρ)) corresponding to the quotient representation



 k−2
ρ
c
χ
a
k−1
of
σ
determinesashortcrystalline
extension
of
χ
by
0
ρ
0 χk−1
uniquely. Since both σ and τ are such extensions, we get σ ∼= τ.
6. The ideals of reducibility
Let ρf : GΣ → GL2 (E) be the Galois representation attached to a newformf ∈ S2k−2 (N ) for N a square-free integer with p - N and Σ = {` | N } ∪ {p}.
Suppose also that the residual representation ρ := ρf is absolutely irreducible and
1 ∗ramified at every prime ` | N . Then ρ|I` ∼(see (2.2)). Let
=
1
 k−2

χ
a
b
ρ
c  : GΣ → GL4 (F)σ=χk−1
be short crystalline at p and semi-abelian with a, c non-trivial classes in Hf1 (Q, ρ(1−k)).
The universal deformation σ red (resp. its trace) gives rise to an Rred -algebramorphism (which we denote by the same letter) σ red : Rred [GΣ ] → M4 (Rred ) (resp.
T = tr σ red : Rred [GΣ ] → Rred ). We define ker σ red (resp. ker T ) as in [BC09],
section 1.2.4.

20

TOBIAS BERGER, KRZYSZTOF KLOSIN

Since R, and hence also Rred , is Noetherian, both rings have a finite numberof minimal primes. Then the total ring of fractions Tot(Rred ) of Rred is a finite
product of fields. In fact we have an injectionY
red
Rred ,→ Tot(Rred ) =
RP
,
P
red

red
where P runs over minimal primes of R
and the localization RP
is a field (cf.
[BC09], Proposition 1.3.11).
To ease notation in this section we will write G for GΣ . Let us note that both
Rred [G]/ ker T and Rred [G]/ ker σ red are Cayley-Hamilton quotients of Rred [G] inthe sense of [BC09], section 1 and we have a canonical Rred -algebra map ϕ :Rred [G]/ ker σ red  Rred [G]/ ker T .

Theorem 6.1. One has the following:(i) Let ? ∈ {T, σ} and i, j ∈ {k−1, ρ, k−2}. Let S ? be Rred [G]/ ker T (when ? =T ) or Rred [G]/ ker σ red (when ? = σ). Then there are data E of idempotents
for which S ? is a GMA (Generalized Matrix Algebra) and there exist Rred submodules A?i,j of S ? which satisfy
A?i,j A?j,k ⊂ A?i,k ,∼T : A?i,i −→ Rred ,T (A?i,j A?j,i ) ⊂ mRred ,and
A?k−2,k−2
M1,2 (A?k−2,ρ )A?k−2,k−1
M2,1 (A?ρ,k−1 ) .
M2,2 (A?ρ,ρ )S? ∼= GMA(A? ) := M2,1 (A?ρ,k−2 )?
?
Ak−1,k−2
M1,2 (Ak−1,ρ )A?k−1,k−1The isomorphism S ? ∼= GMA(A? ) (and also S ? ∼= GMA(A? ) below) is anred
R -algebra morphism.
(ii) For E as in (i) one has
RredM1,2 (A?k−2,ρ )A?k−2,k−1
S? ∼= GMA(A? ) := M2,1 (A?ρ,k−2 ) M2,2 (Rred ) M2,1 (A?ρ,k−1 ) ⊂ M4 (Tot(Rred )),A?k−1,k−2
M1,2 (A?k−1,ρ )Rredwhere A?i,j are fractional ideals of Rred (and Aσi,j are ideals of Rred ) thatsatisfy
A?i,j A?j,i ⊂ A?i,k ,A?i,i = Rred ,A?i,j A?j,i ⊂ mRred .

(iii) The data E can be adapted so that the Rred -algebra map ϕ0 : GMA(Aσ ) 
GMA(AT ) induced from ϕ has the property that ϕ0 (Aσi,j ) = ATi,j .
Proof. The statements (i), (ii) for ? = T and (i) for ? = σ are a direct consequenceof Theorem 1.4.4 in [BC09]. To show that we can also obtain (ii) for ? = σ weargue as in [BK13], Proposition 2.8 which uses Lemma 1.3.7 of [BC09]. Finally, we
proceed as in the proof of Lemma 2.5 in [BK13] to show that E can be adapted tosatisfy (iii).
Remark 6.2. The data E = {ei , ψi | i ∈ {χk−2 , ρ, χk−1 }} in Theorem 6.1 can∼be (and will be) chosen so that for i ∈ {χk−2 , ρ, χk−1 } the maps ψi : ei S σ ei −→red∼
Mdi ,di (R ) satisfy ψi ⊗ F = i. Here di = dim i. From now on fix E.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

21

Using Theorem 6.1 we get the following commutative diagram of Rred -algebras:
GMA(Aσ ) oι2∼Rred [GΣ ]/ ker σ red ∼
ι1

ϕ0


GMA(AT ) o

∼
ι3



/ GMA(Aσ )ϕRred [GΣ ]/ ker T

ι4∼/ GMA(AT )where the maps ιi are the ones given by Theorem 6.1. Using Lemma 1.3.8 (resp.
−1
Proposition 1.3.12) of [BC09] we get that the composite ι2 ◦ ι−1
1 (resp. ι4 ◦ ι3 ) give∼∼σTrise to isomorphisms of Rred -modules fi,j
: Aσi,j −→ Aσi,j (resp. fi,j: ATi,j −→ ATi,j ).
red
We define R -module surjections φi,j to make the following diagram commute
(6.1)

Aσi,j

σ
fi,j

/ Aσ

i,j

0

ϕ





ATi,j

T
fi,j

φi,j

/ AT

i,j

Definition 6.3 ([BC09] Definition 1.5.2). Let P = (P1 , . . . , Ps ) be a partition of
the set I = {χk−2 , ρ, χk−1 }. The ideal of reducibility I P (associated with partition P) is the smallest ideal I of Rred with the the property that there exist
pseudocharacters T1 , . . . , Ts : Rred [G]/IRred [G] → Rred /I such that
Ps(i) T ⊗ Rred /I = l=1 Tl ,P(ii) for each l ∈ {1, . . . , s}, Tl ⊗ F = ρ0 ∈Pl tr ρ0 .
To shorten notation we will sometimes write k − 1, k − 2, ρ for the elements of I
instead of χk−1 , χk−2 , ρ.
Proposition 6.4 (Proposition 1.5.1 in [BC09]). For every partition P the corresponding ideal of reducibility I P exists. Furthermore, let S be any Cayley-Hamilton
quotient of (Rred , T ) (we will only use (Rred , T ) and (Rred , σ red )) and choose dataof idempotents E as in Theorem 6.1 so that S ∼= GMA(A? ). Then I P is given by
the following formula (whose sides do not depend on the choice of S or E)X
IP =T (A?i,j A?j,i ).
(i,j)
i, j not in the same Pl

Corollary 6.5. For ? ∈ {T, σ red } let A?i,j be defined as in Theorem 6.1 (ii). Then
one hasX
IP =A?i,j A?j,i .
(i,j)
i, j not in the same Pl

Proof. Multiplication between elements of α ∈ A?i,j and β ∈ A?j,k corresponds tocorrect matrix multiplication, i.e., one puts α in the (i, j)th spot (which may be
a block) of a matrix and β in the (j, k)th spot and completes both matrices by
putting zeros elsewhere. Then αβ is the (i, k)th spot in the matrix obtained as a
product of the matrices above. The corollary follows from the commutativity of

22

TOBIAS BERGER, KRZYSZTOF KLOSIN

the following diagram

A?i,j A?j,i 
?
?
fi,j
⊗fj,i





/ Ai,i∼ T? 

A?i,j Aj,i
/ Rred?
since fi,i
= T : Ai,i → Rred .

In our situation we have 4 possible partitions of P and the following 4 corresponding ideals of reducibility.
(6.2)
P = {χk−2 , ρ} ∪ {χk−1 } =⇒ I P = I k−1 =Ak−1,k−2 Ak−2,k−1 + Aρ,k−1 Ak−1,ρ
P = {χk−1 , ρ} ∪ {χk−2 } =⇒ I P = I k−2 =Ak−1,k−2 Ak−2,k−1 + Aρ,k−2 Ak−2,ρ
P = {χk−2 , χk−1 } ∪ {ρ} =⇒ I P = I ρ =Ak−2,ρ Aρ,k−2 + Ak−1,ρ Aρ,k−1
P = {χk−2 } ∪ {ρ} ∪ {χk−1 } =⇒ I P = I tot =Ak−1,k−2 Ak−2,k−1 + Aρ,k−1 Ak−1,ρ
+Ak−1,k−2 Ak−2,k−1
We have the following analogue of [BC09], Lemma 9.3.1:
Theorem 6.6. One has(i) All the ideals of reducibility coincide with I tot ,(ii) I tot = ATk−2,ρ ATρ,k−2 ,(iii) If, in addition, dimF Hf1 (Q, ρ(2 − k)) = 1 and p - 1 + wf,` ` for all ` | N ,then I tot is a principal ideal of Rred .
Proof. We will prove Theorem 6.6 by a sequence of Lemmas some pertaining toATi,j and some to Aσi,j .
Lemma 6.7. One has
ATk−2,ρ ATρ,k−2 = ATk−1,ρ ATρ,k−1 .

(6.3)

Proof. This is proved exactly as Lemma 8.2.16 of [BC09] using the antiinvolutionτ.

We will need the following analogue of [BC09], Lemma 8.3.1:
Lemma 6.8. We haveAσk−1,k−2 = Aσk−1,ρ Aσρ,k−2 .
The strategy to prove Lemma 6.8 follows that of [BC09], Lemma 8.3.1, butinstead of using ExtT we use Lemma 6.9 below. For ρi , ρj ∈ {χk−1 , χk−2 , ρ},ρi ∼6 ρj , set A0i,j := Aσi,l Aσl,j , where ρl ∈ {χk−1 , χk−2 , ρ}, ρi ∼6 ρl ∼
6 ρj .
===
Lemma 6.9. There is an injection
HomRred (Aσi,j /A0i,j , F) ,→ HΣ1 (Q, Hom(ρj , ρi )).
If either (i) or (ii) hold, where(i) ρi = χk−2 and ρj = χk−1 ,(ii) ρi = χk−1 and ρj = χk−2 ,then the image of the injection is contained in Hf1 .

DEFORMATIONS OF SAITO-KUROKAWA TYPE

23

Proof. For simplicity in this proof only we write R for Rred . By [BC09], Theorem
1.5.5, taking J = m there is an injection
ιi,j : HomR (Aσi,j /A0i,j , F) ,→ Ext1R[G]/mR[G] (ρj , ρi ).
We havemR[G] = ker(R[G] → F[G]) = (mR)[G],
hence R[G]/mR[G] = (R/mR)[G] = F[G]. Thus we get an injection
ιi,j : HomR (Aσi,j /A0i,j , F) ,→ H 1 (Q, Hom(ρj , ρi )).
It remains to prove that the image is contained in the correct Selmer group. By[BC09], Theorem 1.5.6(1) the image consists precisely of the S/mS-extensions of ρiby ρj , whereS := (R[G])/(ker σ)(R[G]).
By [BC09], Theorem 1.5.6(2) any S/mS-extension is a quotient of Mj /mMj ⊕ ρi ,where Mj = SEj and Ej are defined as in [BC09], p. 21. Since p > 2k − 2,
the representations χk−2 , χk−1 and ρ are short crystalline. Since the category ofsuch representations is closed under taking subobjects, quotients and finite directsums it suffices therefore to prove that Mj is short crystalline. By [BC09], section
1.5.4, one has S = Mj ⊕ S(1 − Ej ), hence in particular Mj is an S-submodule, andhence also an O[G]-submodule of S. One has Mj ⊂ Mj ⊗ Tot(Rred ) and the latter
Galois module is isomorphic to the representation (Tot(Rred )4 , σ red ), which is short
crystalline. Thus Mj is short crystalline.
Let us now check that if ρi = χk−2 and ρj = χk−1 (or vice versa), then theextensions in the image of ιi,j are unramified away from p. Let ` | N be a prime. By
Remark 6.2 we can conjugate σ red so that it is adapted to the data of idempotents
E. Abusing notation we will in this proof denote this conjugate still by σ red . Thisimplies in particular that eρ (σ ⊗ F)eρ ∼= ρ and thatR
M1,2 (Aσk−2,ρ )Aσk−2,k−1
M2,2 (R)M2,1 (Aσρ,k−1 ) .
(6.4)
σ red (S σ ) = M2,1 (Aσρ,k−2 )σσ
Ak−1,k−2
M1,2 (Ak−1,ρ )R
Let X ∈ I` be such that σ red (X) topologically generates σ red (I` ). Write αk,l for the(k, l)-entry of Y := σ red (X) − I4 . Since ρ is ramified at ` at least one of the entries
α2,2 , α2,3 , α3,2 , α3,3 lies in R× . To fix attention let us assume that α2,3 ∈ R× . The
proof in the other three cases is identical.
The construction of the extensions in the image of ιi,j is given in [BC09], p.
37. Note that while there are in general three choices for i (corresponding toρi = χk−2 , χk−1 or ρ), and the same holds for j, there are four choices for k andfor l.
If ρi = ρj , let aii : S/mS → eρi (S/mS)eρi be the canonical projection definedin Lemma 1.5.4 of [BC09], where eρi is the idempotent corresponding to ρi . (Fornotation and terminology see [BC09].) We note that aii composed with the isomorphism eρi (S/mS)eρi ∼= F gives a representation isomorphic to ρi . If ρi 6= ρj , thenthe projection aij : S/mS → eρi (S/mS)eρj is defined in an analogous way and is
the mod m-reduction of the canonical projection ãij : S → eρi Seρj defined again
in the same way. The codomain of ãij is isomorphic to Aσij (via, say, a map φ)

24

TOBIAS BERGER, KRZYSZTOF KLOSIN

σσ
which then is isomorphic to Aσij via the map fi,j
. The composite fi,j◦ φ ◦ ãij equalsredi σ j , wherek−2diag(1, 0, 0, 0) if ρi = χi = diag(0, 1, 1, 0) if ρi = ρdiag(0, 0, 0, 1) if ρi = χk−1 .

Here we are only interested in the case when ρi = χk−1 and ρj = χk−2 or vice versa(but the proof in the second case is identical to the first, so we omit it).
Let f ∈ Hom(Aσχ,1 /A0χ,1 , F). Recall that we have A0χ,1 = Aσχ,ρ Aσρ,1 . Since χ and
1 are unramified to show that the extension
aχχ f ◦ aχ1
a11
is unramified at ` it suffices to show that aχ1 (I` ) ⊂ A0χ,1 . Note that aχ1 (X) = α4,1 .

0 0 0 0
0 0 1 0σ
Since Y is a conjugate of 
0 0 0 0 and Ak−1,ρ ⊂ R (by Theorem 6.1 (ii)) while0 0 0 0×
α2,3 ∈ R we see that the fourth row of Y is a scalar multiple of the second row
and that this scalar α ∈ Tot(R) in fact lies in R. If α = 0, then aχ1 (X) = α4,1 = 0
and we are done, otherwise let s be the largest integer such that α ∈ ms . Then
α4,1 = αα2,1 . If α2,1 = 0, we are done, otherwise let r be the largest integer suchthat α2,1 ∈ mr . Then α4,1 = αα2,1 ∈ mr+s . Finally α4,3 = αα2,3 ∈ ms − ms+1 since
α2,3 is a unit. Comparing the matrix


∗
∗
∗
∗
 α2,1 ∗ α2.,3 ∗

Y =
 ∗
∗
∗
∗
αα2,1 ∗ αα2,3 ∗with (6.4) we see that by the definition of r we get that Aσρ,1 ⊃ mr (since Aσρ,1 is
the ideal of R generated by the entries of σ falling in the corresponding spots andα2,1 ∈ mr −mr+1 ). Similarly by the definition of s and the fact that α4,3 ∈ ms −ms+1
we get that Aσχ,ρ ⊃ ms and so, A0χ,1 = Aσχ,ρ Aσρ,1 ⊃ mr+s . Thus, since α4,1 ∈ mr+s
we get α4,1 ∈ A0χ,1 . This completes the proof of the lemma.
Proof of Lemma 6.8. We have by Proposition 2.9 that Hf1 (Q, Hom(χk−2 , χk−1 )) =0. Thus by Lemma 6.9 and Nakayama’s Lemma we get that Aσk−1,k−2 = A0k−1,k−2 .
Lemma 6.10. We haveATk−1,k−2 = ATk−1,ρ ATρ,k−2 .
Proof. This follows by applying φk−1,k−2 to both sides of the equality in Lemma
6.8 and noting that φk−1,k−2 preserves multiplication after identifying the elementsα ∈ Aσk−1,ρ and β ∈ Aσρ,k−2 as matrices with non-zero entries in the correct spots,i.e., that one has φk−1,k−2 (αβ) = φk−1,ρ (α)φρ,k−2 (β).
We can now finish the proof of parts (i) and (ii) of Theorem 6.6. First we apply
the Chasles relation to see that one has ATk−2,k−1 ATk−1,ρ ⊂ ATk−2,ρ (cf. Theorem

DEFORMATIONS OF SAITO-KUROKAWA TYPE

25

6.1). So, using Lemma 6.8 one gets ATk−2,k−1 ATk−1,k−2 ⊂ ATk−2,ρ ATρ,k−2 . This gives
I tot = ATk−2,k−1 ATk−1,k−2 + ATk−2,ρ ATρ,k−2 + ATk−1,ρ ATρ,k−1 = ATk−2,ρ ATρ,k−2 .
This proves (i) and (ii). To prove (iii) we begin with the following lemma.
Lemma 6.11. One has Aσk−2,ρ = Rred and Aσρ,k−2 is a principal ideal of Rred .
Proof. We have
Rred
red
red
σ (R [GΣ ]) ∼= M2,1 (Aσρ,k−2 )Aσk−1,k−2
M1,2 (Aσk−2,ρ )Aσk−2,k−1
M2,2 (Rred ) M2,1 (Aσρ,k−1 ) .
M1,2 (Aσk−1,ρ )Rred

Recall that
 k−2
χ
σ=

a
ρ


b
c ,

χk−1

with a and c both non-trivial classes. Thus a conjugate of σ red which is adapted tothe data of idempotents E will have the corresponding properties. More precisely,
σ red ⊗ F will have a subquotient which is a non-split extension of ρ by χk−2 and one
which is a non-split extension of χk−1 by ρ. Thus we get that Aσk−2,ρ = Aσρ,k−1 =Rred . Then by Chasles relations we get that also Aσk−2,k−1 = Rred . Finally, by
Theorem 6.1(ii) we have that Aσij Aσji ⊂ m for each pair i 6= j. Thus we get thateach of Aσk−1,k−2 , Aσk−1,ρ and Aσρ,k−2 is contained in m.
By Lemma 6.8 we have Aσk−1,k−2 = Aσk−1,ρ Aσρ,k−2 ⊂ mAσρ,k−2 . This gives us
red
R -module maps (note that Aσρ,k−1 = Rred )!
!
Aσρ,k−2Aσρ,k−2Aσρ,k−2Aσρ,k−2=⇒ HomRred, F ,→ HomRred,F .
Aσρ,k−1 Aσk−1,k−2mAσρ,k−2mAσρ,k−2
Aσρ,k−1 Aσk−1,k−2
We have (the first isomorphism following from our assumptions in part (iii) of
Theorem 6.6 and from Propositions 2.4(1) and 2.7 while the first injection following
from Lemma 6.9)
(6.5)

F∼
= HΣ1 (Q, Hom(χk−2 , ρ)) ←- HomRred (Aσρ,k−2 /A0ρ,k−2 , F) =!
!
Aσρ,k−2Aσρ,k−2, F ←- HomRred,F ,= HomRredAσρ,k−1 Aσk−1,k−2mAσρ,k−2
Aσso, the (clearly non-trivial) F-vector space mAρ,k−2
is one-dimensional.
σ
ρ,k−2
σ
Aρ,k−2 is generated over R by one element by Nakayama’s lemma.

Thus,


Lemma 6.12. One has ATk−2,ρ = Rred and ATρ,k−2 is a principal ideal of Rred .
Proof. As explained in the proof of Lemma 6.10 this follows by applying φk−2,ρ
(resp. φρ,k−2 ) to Aσk−2,ρ (resp. Aσρ,k−2 ) and using Lemma 6.11 along with the factthat both φ maps are R-linear.
Lemma 6.12 along with part (ii) of the Theorem imply part (iii).



26

TOBIAS BERGER, KRZYSZTOF KLOSIN

7. Sufficient conditions for Rred to be a discrete valuation ringLet ρ : GΣ → GL2 (F) be a continuous absolutely irreducible and odd Galois
representation of determinant χ2k−3
crystalline at p and such that for every
 , short1∗prime ` ∈ Σ\{p} one has ρ|I` ∼6= I2 . Let Rρ be the universal deformation=
0 1
0ring for deformations ρ : GΣ → GL2 (A) (here A is an object of LCN(E)) of ρwhich have determinant equal to 2k−3 , are short crystalline at p and minimal in
the sense that they satisfy


1 ∗
(7.1)
ρ0 |I` ∼.
=0 1
We will assume that Rρ ∼= O, i.e., is a discrete valuation ring (cf. [BK13], section
6.1 for discussion of this assumption as well as section 9 in the current paper).
This implies that the set of strict equivalence classes of short crystalline, minimal
deformations of ρ to GL2 (O) with determinant equal to 2k−3 contains a single
element. We will write ρ̃ for a fixed representative of this strict equivalence class.
Consider short crystalline, semi-abelian σ : GΣ → GL4 (F) of the form
 k−2

χ
a
b
ρ
c 
σ=
χk−1
with a, c giving rise to non-zero elements of Hf1 (Q, ρ(1 − k)) (cf. Corollary 3.3).
We assume in this section thatdimF Hf1 (Q, ρ(1 − k)) = 1.
For A an object of LCN(E) we will call a matrix M ∈ GL4 (A) upper-triangular if∗ ∗ ∗ ∗
0 ∗ ∗ ∗M =0 ∗ ∗ ∗ .
0 0 0 ∗
We will say that a homomorphism σ 0 : GΣ → GL4 (A) is upper-triangular if for
all g ∈ GΣ , the matrix σ 0 (g) ∈ GL4 (A) is upper-triangular. Finally, we say that a
deformation σ : GΣ → GL4 (A) of σ is upper-triangular if there exists a member σ 0of the strict equivalence class of σ which is upper-triangular.
Lemma 7.1. Every short crystalline minimal deformation σ of σ to GL4 (A) suchthat tr σ = T1 + T2 + T3 with T1 , T2 , T3 pseudocharacters such that T1 ≡ χk−2 ,T2 ≡ tr ρ, T3 ≡ χk−1 mod mA is upper-triangular.
Proof. This is Theorem 1.1 in [Bro08]. Note that the Artinian assumption in [Bro08]
is unnecessary. See also remarks to Theorem 1 in [Urb99].
Lemma 7.2. Consider an upper-triangular deformation of σ to GL4 (F[X]/X 2 ).
Then every upper-triangular member σ 0 of its strict equivalence class has the form
 k−2

∗
∗
χ
ρ
∗ .
σ0 = 
χk−1

DEFORMATIONS OF SAITO-KUROKAWA TYPE

27

Proof. We need to show that the only deformations of χk−2 , ρ, χk−1 are the trivial
ones. Suppose that χk−2 +αX is a deformation of χk−2 to (F[X]/X 2 )× . Then sinceχ2−k is also crystalline, we get that 1 + χ2−k αX is crystalline. Then α0 := χ2−k α
is a homomorphism from GΣ to the additive group F. Let ` | N be a prime.
Note that the total ring of fractions of F[X]/X 2 is F[X]/X 2 . If x ∈ GΣ is suchthat σ 0 (I` ) is generated by σ 0 (x)then by minimality
we know that σ 0 (x) − I4

0 0 0 0

0 0 f 0×
is conjugate (over F[X]/X 2 ) to 
0 0 0 0 for f ∈ F . Using the upper0 0 0 0
triangular form of σ 0 this implies that α must be unramified at `. Hence thecharacter 1 + α0 X : GΣ → (F[X]/X 2 )× can only be ramified at p. However,crystallinity forces the character to be trivial (by a simple modification of the proof
of Lemma 9.6 in [BK13]), so α = 0. Similarly one proves the only deformation ofχk−1 is the trivial one. Finally, the claim for ρ follows from the assumption thatRρ is a dvr.
Proposition 7.3. There do not exist any non-trivial upper-triangular deformationsof σ to F[X]/X 2 .

 k−2


ρ
c
χ
a
is a subrepresentation of σ and that
Proof. First note that
is
ρ
χk−1
a quotient of σ hence they are both short crystalline since σ is. Thus (by Corollary3.3) a gives rise to a (non-zero) element of Hf1 (Q, Hom(ρ, χk−2 )) = Hf1 (Q, ρ(1 − k))and c gives rise to a (non-zero) element of Hf1 (Q, Hom(χk−1 , ρ)) = Hf1 (Q, ρ(1 − k))(cf. Remark 3.1). Since this last Selmer group is assumed to be one-dimensional,
a and c give rise to linearly dependent cohomology classes.
By Lemma 7.2 any upper-triangular deformation % of σ to GL4 (F[X]/X 2 ) has
the form
 k−2χa + Xa0 b + Xb0
ρ
c + Xc0 %=χk−1for a0 , b0 , c0 valued in F.
Let us now show that all the off-block-diagonal entries are unramified outside
p. Suppose ` | N . By Corollary  3.3 we see that a, b, c are all unramified at `.

 0
c
0
Write a = a1 a2 , c = 1 with ai , ci : GΣ → F. Let x ∈ GΣ be suchc2
that %(x)%(I` ). After conjugating ρ (if necessary), we may assume that
 generates

1 1
ρ(x) =. As the (2,3)-entry of %(x) − I4 lies in F× minimality of % forces1
the first row of %(x) − I4 to be a scalar multiple of the second row and the fourthcolumn of %(x) − I4 to beThis gives a1 (x) = a scalar multiple of the third column.
A
1 −a2 (x)X1 c1 (x)Xc2 (x) = 0. Set E0 :=with A =and D =.
D11
0 0 0 b0 (x)X

0 0 ∗
0 
 . Note that E0 ≡ I4 (mod $),One has E0 (%(x) − I4 )E0−1 = 0 0 00 
0 0 0
0so the conjugate E0 %E0−1 is strictly equivalent to %. Hence we may assume that

28

TOBIAS BERGER, KRZYSZTOF KLOSIN

a2 (x) = c1 (x) = 0. Since again the first row must be a scalar multiple of the second
row we also getb0 (x) = 0.
 k−2

χ
a + Xa0
Note thatis a subrepresentation of % (with a + Xa0 unramifiedρaway from p). Then using that dimF Hf1 (Q, ρ(1 − k)) = 1 we obtain by Proposition
7.2 of [BK13] (where the assumption 6.1(ii) is satisfied for ρ and for χk−2 it can be
replaced by the lack of a non-trivialof χk−2 to (F[X]/X 2 )× - cf. Proof k−2 deformation
0χ
a + Xaof Lemma 7.2 above) thatcan be conjugated (over F[X]/X 2 ) to
ρ
 k−2

χ
a
the representation, i.e., there exists A ∈ I3 + XM3 (F[X]/X 2 ) such that
ρ
 k−2

 k−2

χ
a + Xa0 −1
χ
a
A
A =
. Then we have
ρ
ρ
 k−2


 
−1
χ
a b00 + Xb000
A 0
A 0ρ c00 + Xc000  .
%
=
0 1
0 1
χk−1
Since the image of A in M3 (F) is the identity, we see that b00 = b and c00 = c, so inparticular k−2 this is still 0a deformation of σ. Hence we conclude that % is equivalentχa b + Xb
ρ c + Xc0  which we will still denote by %.
to 
χk−1
Let V be the representation space of % (i.e., the free F[X]/X 2 -module of rank 4
on which GΣ acts via %). Let W be the GΣ -invariant free F[X]/X 2 -submodule ofV spanned by the vector t [1, 0, 0, 0]. Then GΣ acts on the quotient V /W viaρ c + Xc0
%0 :=
.
χk−1
Hence we can again conclude by Proposition
 7.2 in [BK13] that there exists A ∈
ρ
c
2
0 −1
. Then we haveI3 + XM3 (F[X]/X ) such that A% A =
χk−1
 k−2


 
−1
χ
a + Xa00 b + Xb00
1 0
1 0
.
ρ
c
%
=
0 A
0 A
k−1
χ
Denote the right-hand side of the above equation by %0 . Now, let us re-write %0 in
the F-basis of (F[X]/X 2 )4 given by               
1
0
0
0
X
0
0
0 


               

0
1
0
0
0
X
0
, , , , , , ,0 .
B 
0 0 1 0  0   0  X   0 





0
0
0
1
0
0
0
X
We get that %0 in the basis B has the form
 k−2
χ
a
b

ρ
c


χk−1

00

a
b00
χk−2





a
ρ




.
b 

c 

χk−1

DEFORMATIONS OF SAITO-KUROKAWA TYPE

Taking a sequence of submodules and quotients we get

 k−2

χ
a
b
ρ
c


ρ
c
k−1



χ
→


χk−1
a00
b00
00
00
k−2
a
b
χ


I2 0 0which if we conjugate it by  0 0 1 will give us
0 1 0


ρ
c
a00 χk−2
b00  ,
(7.2)
χk−1

29


χk−2

,

from which we can extract a short crystalline 2-dimensional subrepresentation of
the formρ
,a00 χk−2
and again (by crystallinity of the above representation) we see that a00 (which isunramified away from p by an argument as before) gives rise to an element of
Hf1 (Q, Hom(ρ, χk−2 )) = Hf1 (Q, ρ(1 − k)). So, by one-dimensionality of the latter
there exists a constant α ∈ F such that a00 = αa.
Hence we get that k−2
  k−2χ
a b + Xb0
χ
a + Xαa b + Xb00
 = %0 ,ρ c + Xc0  ∼
ρ
c
(7.3)
%=
=
k−1
k−1
χ
χ
and the isomorphism is by conjugation by an element of I4 +XM4 (F). Furthermore
we note that

 k−2
χ
a b + Xb000
,ρc
M0 %0 M0−1 = 
χk−1
where


1
 ∈ 1 + XM4 (F).
M0 =  (1 + αX)I2
1 + αX
Hence we conclude that
can take a00 = 0.
 we
χk−2
b00
Then we see thatis a quotient of the representation in (7.2).
χk−1
00
Thus, b gives rise to an element of HΣ1 (Q, F(−1)). By the argument from the
beginning of the proof we see that b00 is unramified at all primes ` | N , so it in factgives rise to an element of Hf1 (Q, F(−1)) which is zero by Proposition 2.9.
Let IR0 ⊂ R0 denote the total ideal of reducibility corresponding to the universal
deformation (σ 0 )univ and IR ⊂ R the total ideal of reducibility corresponding to theuniversal deformation σ univ .
Corollary 7.4. The structure maps O → R0 /IR0 , O → R/IR and O → Rred /I tot
are all surjective.
Proof. By Lemma 7.11 of [BK13] it is enough to show that the structure map
O → R0 /IR0 is surjective. We need the following lemma.

30

TOBIAS BERGER, KRZYSZTOF KLOSIN

0
Lemma 7.5. Let I ⊂ R0 be an ideal such that R0 /I ∈ LCN(E). Then I ⊃ IRif
0
0 univand only if (σ )mod I is an upper-triangular deformation of σ to GL4 (R /I).

Proof. The proof of Corollary 7.8 in [BK13] carries over to three Jordan-Hölder
factors with the application of Theorem 7.7 in [BK13] replaced by an applicationof Lemma 7.1 above.
The rest of the proof of Corollary 7.4 is the same as the proof of Proposition
7.10 in [BK13] with the application of Corollary 7.8 in [BK13] now replaced with
an application of Lemma 7.5.
Theorem 7.6. Let ρ̃ be a representative of the unique deformation of ρ to characteristic zero. Suppose that #Hf1 (Q, ρ̃(1 − k) ⊗ E/O) ≤ #O/LO for some L ∈ O.
Then #Rred /I tot ≤ #O/LO.
Proof. Again by Lemma 7.11 of [BK13] it is enough to show that #R0 /IR0 ≤#O/LO. First let us note that it follows from the proof of Lemma 7.2 that there
do not exist any non-trivial short crystalline deformations of χk−2 and of χk−1
to (F[X]/X 2 )× . Let Rk−2 (resp. Rk−1 ) denote the universal short crystalline
deformation ring of χk−2 (resp. χk−1 ). Then we can adapt the proof of Proposition
7.10 in [BK13] with S there replaced by Rk−2 to show that Rk−2 /$Rk−2 = F andhence by Nakayama’s lemma the structure map O → Rk−2 is onto. However, sincek−2 is a deformation of χk−2 to O× , we must in fact have Rk−2 = O and so k−2
is a unique such deformation. The same conclusion holds for Rk−1 . Assume that
#R0 /IR0 = #O/$s O > #O/LO (we allow s = ∞ here). Then by Lemma 7.5
there exists an upper-triangular deformation σ : GΣ → GL4 (O/$s O) of σ. By theabove argument and using the fact that ρ̃ is the unique deformation of ρ to GL2 (O)the deformation σ must have the form
 k−2


∗1
∗2

ρ̃
∗3  ,
k−1where the diagonal pieces are understood to be taken mod $s .
Arguing as in the proof of Proposition 7.3 we see that minimality guarantees thatthe classes of ∗1 and ∗3 are unramified away from p and thus give rise to classes
in Hf1 (Q, ρ̃(1 − k) ⊗ E/O[$s ]) = Hf1 (Q, ρ̃(1 − k) ⊗ E/O)[$s ] (cf. Proposition
2.4(1)). Since the reduction σ of σ has the property that its entry correspondingto ∗3 above gives rise to a non-trivial element in Hf1 (Q, ρ(1 − k)), we conclude thatthe image of ∗3 is not contained in $O/$s O, hence gives rise to an element of
Hf1 (Q, ρ̃(1 − k) ⊗ E/O) of order #O/$s , which is a contradiction.

Corollary 7.7. The rings R0 , R and Rred are all topologically generated as an
O-algebra by the set {tr σ(Frobl ) | l - N p}, where σ stands for (σ 0 )univ , σ univ andσ red respectively.
Proof. It is enough to prove this for R0 as the other rings are quotients of it. For
this one can use the same proof as the one of Proposition 7.13 in [BK13] replacing
again the application of Corollary 7.8 with Lemma 7.5.
The upshot of this section is the following theorem identifying conditions when
the universal deformation ring is a discrete valuation ring. To make the statement
self-contained we will include all the assumptions made so far.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

31

Theorem 7.8. Let ρf : GΣ → GL2 (E) be the Galois representation attached toa newform f ∈ S2k−2 (N ) for N a square-free integer with p - N and Σ = {` |N } ∪ {p} such that p - 1 + wf,` ` for all primes ` | N . Suppose also that the residual
representation ρ := ρf is absolutely irreducible and ramified at every prime ` | N .
Assume that dimF Hf1 (Q, ρ(2 − k)) = 1 and that #Hf1 (Q, ρf (1 − k) ⊗ E/O) ≤ #F.
Furthermore assume that Rρ is a dvr. Let σ : GΣ → GL4 (F) be a continuous
representation short crystalline at p and semi-abelian of the form
 k−2

χ
a
b
ρ
c 
σ=
k−1
χ
with a, c giving rise to non-zero elements of Hf1 (Q, ρ(1 − k)). Suppose σ admits adeformation σ to GL4 (O) which is τ -self-dual.
σ→ O.
Then σ gives rise to an O-algebra isomorphism Rσred = Rred −Proof. The O-algebra map R  O induced by σ factors through Rred since O is
torsion-free. Theorem 6.6 combined with Theorem 7.6 imply that the maximal idealof Rred is principal. Because of the surjection to O its generator is not nilpotent.
As in [Cal06] Lemma 3.4 we can therefore deduce that Rred is a discrete valuation
ring and the surjection an isomorphism Rred ∼
= O.
8. Application to the Paramodular Conjecture
The following conjecture is due to Brumer and Kramer (cf. Conjecture 1.1 in[BK14]) and is often referred to as the Paramodular Conjecture.
Conjecture 8.1. There is a one-to-one correspondence between isogeny classesof abelian surfaces A/Q of conductor N with EndQ A = Z and weight 2 Siegel
modular forms F , which are not in the space spanned by the Saito-Kurokawa lifts,
have level K(N ) and rational eigenvalues, up to scalar multiplication, where K(N )is the paramodular group of level N defined by K(N ) = γM4 (Z)γ −1 ∩ Sp4 (Q) with
γ = diag[1, 1, 1, N ]. Moreover, the L-series of A and F should agree and the p-adic
representation of Vp (A) := Tp (A) ⊗ Qp should be isomorphic to those associated to
F for any p prime to N where Tp (A) is the p-adic Tate module.
In this section we will show how our results can be used to verify this conjecture
in some cases when A has rational p-torsion. More precisely, our method will only
allow for the verification of the second claim, i.e., the existence of the isomorphismof the Tate module with the representation associated to F .
Let p > 2 be a prime and let A be an abelian surface of square-free conductor N
as in Conjecture 8.1. Suppose that p - N and that A has a polarization of degree
prime to p and a Q-rational point of order p. Then the p-adic Tate module gives
rise to a Galois representation
σA := Vp (A) : GΣ → GL4 (Qp ),where Σ = {p} ∪ {` | N }, short crystalline at p (by [ST68] and [Fon82]), which is
absolutely irreducible since it is semisimple and EndQp [GQ ] (Vp (A)) = EndQ (A) ⊗Qp = Qp by [Fal83]. Furthermore, under our assumptions the semisimple residual
representation σ ss
A : GΣ → GL4 (Fp ) has the formss
σ ss
A = 1 ⊕ ρ ⊕ χ,

32

TOBIAS BERGER, KRZYSZTOF KLOSIN

where, as before, χ is the mod p cyclotomic character and ρ is a two-dimensional
representation. The representation σA has determinant 2 and hence det ρ = χ, so
in particular, ρ is odd.
We will assume that ρ is absolutely irreducible, ramified at all ` | N . It follows
from Serre’s conjecture [KW09] (see Proposition 4 of [Ser87] for the determinationof the Serre weight) that ρ ∼= ρf where ρf : GΣ → GL2 (Qp ) is the Galois representation attached to a newform f ∈ S2 (N ) and ρf denotes its mod p reduction.
We now assume that A has semi-abelian reduction at all ` | N , i.e. the reduction
is an extension of an elliptic curve by a torus and that p does not divide theTamagawa number at `, i.e. the number of connected components of the special
fiber of a Neron model of A at `. Under these assumptions [LD98] Proposition 1.3
proves that σ A |I` ∼= exp(t` N1 ). The proof generalizes to A[pn ] as [ST68] Lemma 1∼ exp(t` N1 ).
and 2 hold in this generality, so we deduce that σA |I` =For a positive integer N and an integer k ≥ 2 we will denote by Sk (N )para thespace of Siegel modular forms of genus 2, (parallel) weight k and paramodular level
N . For a prime ` - N , write T`,2 for the Hecke operator given by the double coset
K(N ) diag(1, 1, `, `)K(N ), T`,1 for the Hecke operator given by the double coset
K(N ) diag(1, `, `2 , `)K(N ), and T`,0 for the Hecke operator given by the double
coset `K(N ) (see e.g. p. 18 of [Pil17]). We will say that F ∈ Sk (N )para is aneigenform if F is an eigenform for T`,i for all ` - N and i = 0, 1, 2. If F is aneigenform we will write λ`,i (F ) for the eigenvalue of T`,i corresponding to F .
Theorem 8.2 (Taylor, Weissauer, Laumon). For k ≥ 2 let F ∈ Sk (N )para be an
eigenform. There exists a finite extension E of Qp and a continuous semisimple
representation
σF : GΣ → GL4 (E)∨ ∼3−2kwith σF = σF ⊗such that for all primes ` - N p one has det(I4 −XσF (Frob` ))−1 =Q` (X) for the Hecke polynomialQ` (X) = 1−λ`,2 (F )X+`(λ`,1 (F )+(1+`2 )λ`,0 (F ))X 2 −`3 λ`,2 (F )λ`,0 (F )X 3 +`6 λ`,0 (F )2 X 4 ,where λ`,0 (F ) = `2k−6 .
Remark 8.3. This result is well-known, but for the benefit of the reader we explain the relationship to results in the literature. The eigenform F generates anautomorphic representation π that decomposes into a direct sum π1 ⊕ . . . ⊕ πnof irreducible cuspidal automorphic representations of GSp4 whose local representations agree away from N . Laumon [Lau05] and Weissauer [Wei05] constructed
the Galois representations for irreducible automorphic representations π with π∞holomorphic discrete series (and hence for eigenforms F as above for weight k > 2
as all πi give rise to the same semi-simple Galois representation). The result fork = 2 can be deduced from this using Taylor’s argument in Example 1 of section
1.3 in [Tay91]. [Jor10] and [Mok14] Proposition 4.14 provide an alternative proof(for [Mok14] under the restriction that F be of type (G) in the sense of [Sch17]
section 2.1, i.e. such that all πi have a cuspidal transfer to GL4 , but see Remark[Mok14] 4.15).
Theorem 8.4 ([FC90], [Jor12] Theorem 3.1). Let F be as in Theorem 8.2, k ≥ 2
and p - N . If k = 2 assume that F has pairwise distinct roots for the Hecke
polynomial Qp (F ). Then σF is crystalline at p and short crystalline if p > 2k − 2.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

33

Proposition 8.5. For k ≥ 2 and N squarefree let F ∈ Sk (N )para be an eigenformof type (G) in the sense of [Sch17] section 2.1. If k = 2 we further assume that σF
is absolutely irreducible and that σ ss
F is ramified at ` for all ` | N . Then we have

σF |I` ∼= exp(tp N1 ) for all ` | N.
Proof. For k > 2 this is proved by [Mok14] Theorem 3.5 (local-global compatibility
up to Frobenius semisimplification) and [Sor10] Corollary 1 (monodromy rank 1).
(We know by [Sch17] Proposition 1.2.1 that type (G) paramodular eigenforms giverise to generic local representations, so that Sorenson’s result applies.)
As for k = 2 [Mok14] Theorem 4.14 only proves local-global compatibility up
to semisimplification we argue as follows: Theorem 1.1 of [MT15] allows us top-adically approximate the Hecke eigenvalues of a weight 2 form F by those of cohomological Hecke eigenforms Fn . This means that the corresponding Galois representations σFn are trace convergent to σF in the sense of [BCKL05], i.e. tr (σFn )(g)converges to tr (σF )(g) in Cp for all g ∈ GQ . As we assume that σF is absolutely
irreducible we can invoke Theorem 1.2 of [BCKL05] to deduce that σFN are physically convergent to σF , i.e. such that there exist Cp -bases for σFn and ρ suchthat the matrix entries of σFn converge to the corresponding entries of σF . This
result also tells us that the σFn are absolutely irreducible for n  0 so we knowthat the Fn are eventually of type (G). As these are of weight k > 2 we know that
σFn |I` ∼= exp(tp N1 ) and the rank of the monodromy remains 1 in the limit by ourassumption that σ ss
F is ramified.

Theorem 8.6. Let A and ρ = ρf be as above and assume that p - 1 + wf,` ` for all` | N , Rρ = O, #Hf1 (Q, ρf ) = #F and #Hf1 (Q, ρf (−1) ⊗ E/O) ≤ #F.
Furthermore suppose that there exists F ∈ S2 (N )para such that λ`,2 (F ) ≡ 1 +` + a` (f ) (mod p) for all primes ` - N p and such that its Hecke polynomial at someprime ` - N p vanishes neither at 1 nor at 1/`. We also assume that F has pairwise
distinct roots for the Hecke polynomial Qp (F ). Then σA ∼= σF . In particular, A isparamodular of level N .
Proof. The congruence, Tchebotarev’s density Theorem and the Brauer-Nesbitt
Theorem imply that the semisimplification σ ss
F of the mod p reduction of σF has
ss
the form σ ss
=
1
⊕
ρ
⊕
χ
=
σ
.
f
F
A
We will show below that σF is absolutely irreducible. Using Corollary 4.4 (andthe fact that σF is short crystalline at p - cf. Theorem 8.4) we can then chooseGΣ -stable lattices in σA and in the representation space of σF so that with respect
to these lattices σ A and σ F have the form as in Corollary 4.4. Then by Proposition
5.1 we get that in fact σ A ∼= σ F . By adjusting the basis of σF if necessary we canassume that σ A = σ F , hence we obtain that σF is a deformation of σ A . Let R be
the quotient of the universal deformation ring of σ A as in Proposition 3.6. Thenby Theorem 7.8 we see that σA and σF both give rise to an isomorphism Rred ∼=O

34

TOBIAS BERGER, KRZYSZTOF KLOSIN

of O-algebras. Hence we get a commutative diagram
O
id

!
id

Rred

∼
σA

(/

O

σF

 
O

from which we see that σA = σF as maps from Rred to O, which implies that the
representations σF and σA are isomorphic since they are both composites of theuniversal deformation with the map Rred → O.
Let us now show that σF is absolutely irreducible. Indeed, note that σF cannot
be the sum of 4 characters because its reduction has an absolutely irreducible
two-dimensional component ρf . Suppose that σFss splits as a direct sum of two
irreducible 2-dimensional representations σ1 ⊕ σ2 . Note that σ1∨ ∼6 σ2 ⊗ −1 since=sswithout loss of generality σ 1 = ρ 6= σ 2 = 1 ⊕ χ. We therefore have σ1∨ ∼= σ1 ⊗ −1 ,which implies det(σ1 ) = det(σ2 ) = . Indeed we must have det(σ1 ) = φ for somequadratic character φ reducing to the identity mod p since det(σ 1 ) = χ. Sincep > 2 the character φ has to be trivial.
We can therefore argue as in case (v) on p. 46 of [SU06]. Note that σ1 has tobe odd since ρ is, so σ2 must be odd as well. Since the automorphic representation
π corresponding to F can be transferred to an isobaric automorphic representationon GL4 the assumptions in Theorem C of [Ram13] are satisfied, which tells usthat LS (σ1∨ ⊗ σ2 , 1) 6= 0 for some finite set of places S. This means that thestandard L-function LS (s, π, std) = ζ S (s)LS (σ1∨ ⊗ σ2 , s) has a pole at s = 1, which
by [KRS92] implies that π is endoscopic, i.e. corresponds to type (Y ) in [Sch17].
Since F is paramodular this contradicts [Sch17] Lemma 2.2.1. This excludes thecase σA = σ1 ⊕ σ2 .
It remains to show that σA cannot split as σ1 ⊕ χ1 with σ1 a 3-dimensional
representation and χ1 a character. If it did, then the spin L-function of A would
have a linear factor. Since the only short crystalline, minimal deformations of 1
(resp. χ) are 1 (resp. ) - cf. the beginning of the proof of Lemma 7.2 - we musthave that χ1 is either 1 or . Thus the linear factor of the local spin L-function
must be 1 − `−s (if χ1 = 1) or 1 − (`)`−s = 1 − ``−s . But the local spin L-factor
is the local Hecke polynomial at ` with X replaced by `−s , so we’re done by ourassumption.

9. Examples
In this section we work out in detail how our result proves paramodularity of
the abelian surface of conductor 731. We also discuss other cases where the result
may be applicable without going into details.
9.1. Conductor N = 731. We will show that for an abelian surface of this conductor (using a congruence result proved in the Appendix) the conditions for Theorem
8.6 are satisfied. This establishes a new case of the Paramodular Conjecture.

DEFORMATIONS OF SAITO-KUROKAWA TYPE

35

For N = 731 = 17 ∗ 43 there exists an abelian surface A of conductor N , whicharises as the Jacobian of the hyperelliptic curve with equationy 2 + (x3 + x2 )y = x5 + 2x4 − x − 3.
According to LMFDB [LMF17b] it has semi-abelian reduction at 17 and 43 andhas a rational point of order p = 5. We first note that 5 - 1 ± ` for ` = 17, 43.
Since A has rational 5-torsion and is principally polarised we know that σ ssA,p =1 ⊕ ρ ⊕ χ. Furthermore A has semi-abelian reduction and has Tamagawa numbers
at 17 and 43 not divisible by 5, which tells us that if ρ were reducible over any finite
extension of Qp its semisimplification would be unramified away from 5, i.e. of theform χi ⊕ χ1−i mod 5 for i = 1 or 2. We check on specific Frobenius elements thatthe trace of ρ is not equal to χi + χ1−i for i = 1, 2 so that ρ has to be absolutely
irreducible. Serre’s conjecture implies that ρ is modular by a cuspidal weight 2eigenform of level 17, 43 or 731 (see section 8). Using MAGMA [BCP97] we check(by comparing the linear term of the Euler factor of A at ` with 1 + ` + a` (f )) thatthe only newform giving rise to ρ is the modular form corresponding to the elliptic
curve E (Cremona label 731a1) of conductor 731 given by
y 2 + xy + y = x3 − 539x + 4765,
which has rank 1 and trivial rational torsion. This also shows that 5 is not acongruence prime for the corresponding weight 2 modular form, i.e. that Rρ = Zp .
By consulting LMFDB [LMF17a] we know that ρE,p is ramified at 17 and 43 (since
E has non-split reduction at both primes). We also note that E has good ordinary
reduction at 5.
We can check the Selmer group conditions as follows: We have Hf1 (Q, ρE,p ) =1
Hf (Q, ρE,p ⊗ Qp /Zp )[p] = Selp (E)[p] where the first equality follows from Proposition 2.4 and for the second one see e.g. [Rub00] Proposition 1.6.8. Here Selp (E)is defined by
0 → E(Q) ⊗ Qp /Zp → Selp (E) → X(E)[p∞ ] → 0.

(9.1)

Using snake lemma on the diagram:0/ E(Q) ⊗ Qp /Zp0/ E(Q) ⊗ Qp /Zp/ Selp (E)

·p

·p



/ Selp (E)/ X(E)[p∞ ]

/0

·p


/ X(E)[p∞ ]/0

we get an exact sequence
0 → E(Q) ⊗ Qp /Zp [p] → Selp (E)[p] → X(E)[p] → 0.
Since ρE,p is absolutely irreducible, we conclude that E has no rational p-torsion
and hence (E(Q) ⊗ Qp /Zp )[p] = E(Q)/pE(Q). This implies that (9.1) gives rise
to a short exact sequence
0 → E(Q)/pE(Q) → Hf1 (Q, ρE,p ) → X(E)[p] → 0,
Since the rank of E is 1 we therefore deduce that #Hf1 (Q, ρE,p ) = 5 if X(E)[5] =0. According to LMFDB the analytic order of X(E) is 1. By Skinner et al. [JSW17]
the p-part of the BSD formula is satisfied, so X(E)[5] = 0.
It now remains to bound the order of #Hf1 (Q, ρE,p (−1) ⊗ Qp /Zp ) (which weknow to be non-trivial) from above by 5. Setting f ∈ S2 (Γ0 (731)) in Proposition

36

TOBIAS BERGER, KRZYSZTOF KLOSIN

2.10 to be the newform corresponding to E it is enough to calculate p-valuationof the p-adic L-series L(E, ω −1 ) ∈ Zp [[T ]] specialized at T = p. Using SAGE we
confirm that this p-valuation is indeed 1.
As proved in the Appendix there is a unique non-lift paramodular eigenform F
new at level 731 (non-vanishing modulo 5 and with Hecke eigenvalue λ5,2 (F ) = 0)
that is congruent modulo 5 to a Gritsenko lift (paramodular Saito-Kurokawa lift)of a modular form of level dividing 731. This congruence of Fourier expansions
implies a congruence of Hecke eigenvalues modulo 5.
Using MAGMA we can rule out that F is congruent to all but the SaitoKurokawa lift of the rational modular form f corresponding to E. Since λ5,2 (F ) = 0
its Hecke polynomial at 5 is of the form 1 + cT 2 + 25T 4 for some c ∈ Z, which either
has distinct roots or more than one repeated root. As a5 (f ) ≡ −1 mod 5 the Hecke
polynomial of the Saito-Kurokawa lift of f (given by (1−αT )(1−βT )(1−T )(1−5T )with α + β = a5 (f )) is congruent to 1 − T 2 modulo 5, so we deduce that the rootsof the Hecke polynomial of F are distinct modulo 5.
Theorem A.1(4) in the appendix also shows that the Hecke polynomial at 2 isgiven by 4T 4 + 2T 3 + 2T 2 + T + 1 = (1 − T + 2T 2 )(1 + 2T + 2T 2 ). Its complex
roots do not include 1, 2 or 1/2.
9.2. Other examples. Poor and Yuen have found candidate paramodular forms
for some other examples of abelian surfaces. For the seven conductors in Table
5 of [PY15] one can check that only the abelian surface of conductor N = 277
involves a congruence (for the torsion prime p = 3, but not for p = 5) with theparamodular Saito-Kurokawa lift of a modular form corresponding to an elliptic
curve. In this case the p-valuation of the p-adic L-series gives us a bound of 9 on
#Hf1 (Q, ρf (−1) ⊗ E/O). It may still be possible that the Bloch-Kato Selmer group
has order 3, but we were not able to confirm this. However, this abelian surface has
been proved to be paramodular by Brumer et al. using the Faltings-Serre method,
but a proof has not yet appeared in print.
When the conjectured congruence is with the Saito-Kurokawa lift of a modular
form with non-rational Fourier coefficients it is sometimes possible to check thatRρf is a discrete valuation ring. This is the case for the examples of conductor
N = 349, 353 and 389 on the list in [PY15]. These surfaces have rational p-torsion
for p = 13, 11 and 5, respectively. We have, however, not tried to check the Selmer
group assumptions for these cases.
Another example we want to highlight is that of conductor N = 997, where
there exists an abelian surface with 3-torsion. In this case the expected congruenceis between a paramodular non-lift and a Saito-Kurokawa lift of a modular form
corresponding to a rank 2 elliptic curve. As the root number of the modular form is+1 this Saito-Kurokawa lift has to be of congruence level, rather than paramodularas the other cases. It would be interesting to confirm the existence of a matchingparamodular non-lift Siegel modular form in this case, but Theorem 8.6 does not
apply because p = 3 and the elliptic curve rank >1 should imply #Hf1 (Q, ρE,p ) > 3.
10. Modularity theorem for cohomological weights k
In this section we will prove a modularity theorem in cases k > 2 and N = 1. The
sole reason for these two restrictions is that at present only in this case we havea result (which is due to Brown) providing us with enough congruences among

DEFORMATIONS OF SAITO-KUROKAWA TYPE

37

Siegel modular forms [Bro11]. This result also allows us to replace the assumption that #Hf1 (Q, ρf (1 − k) ⊗ E/O) ≤ #F with the weaker assumption thatdimF Hf1 (Q, ρf (1 − k)) = 1. In particular, the proof does not proceed via provingthat Rred is a discrete valuation ring - in fact the latter property is not impliedby our R = T theorem. We again include all the assumptions to make the statement self-contained, however to make their use more transparent we will separate
the assumptions that are necessary for Brown’s congruence result (collecting thembelow as Assumption 10.1 - cf. Theorem 5.4 and Corollary 5.6 in [Bro11]) fromthe assumptions required on the deformation side (which will be spelled out in thestatement of Theorem 10.2).
Let k be a positive integer and p a prime such that p > 2k − 2.
Assumption 10.1. Suppose k > 9 is even. Assume also that there exists N ∈Z>1 and a fundamental discriminant D < 0 such that χD (−1) = −1 and p (2)(2)
N D[Sp4 (Z) : Γ0 (N )]. Here Γ0 (N ) is the congruence subgroup of Sp4 (Z) of level
N . Let f ∈ S2k−2 (1) be a p-ordinary newform. Also assume that there exists a
Dirichlet character ψ of conductor N such thatval$ (LΣ (3 − k, ψ)Lalg (k − 1, f, χD )Lalg (1, f, ψ)Lalg (2, f, ψ)) = 0.
For precise definitions of the L-factors, cf. [Bro11], section 5.
Theorem 10.2. Assume Assumption 10.1. Let ρf : GΣ → GL2 (E) be the Galois
representation attached to the newform f and write ρ for the residual representation
ρf , which we assume to be absolutely irreducible. Here Σ = {p}. Assume thatdimF Hf1 (Q, ρ(1 − k)) = dimF Hf1 (Q, ρ(2 − k)) = 1.
Furthermore assume that Rρ is a dvr. Let σ : GΣ → GL4 (E) be an absolutely
irreducible short crystalline Galois representation with det σ = 4k−6 , satisfyingσ∨ ∼= σ(3 − 2k) and such that its residual representation σ (defined only up tosemisimplification) satisfies
σ ss ∼= χk−2 ⊕ ρ ⊕ χk−1 .
Then σ is modular, i.e., there exists a cuspidal non-lift Siegel modular eigenform
F of weight k, level one such that Lspin (s, F ) = L(s, σ).
Proof. Let σ be as above. Then by Corollary 4.3, there is a GΣ -stable lattice Λ inthe space of σ such that
 k−2

χ
a
b
ρ
c 
σ=
χk−1
with a and c non-trivial classes in Hf1 (Q, ρ(1 − k)). Let Rred denote, as before, the(reduced, self-dual) universal deformation ring of σ.
Let Φ0 denote the set of linearly independent Siegel modular eigenforms of weightk, level 1 which are orthogonal to the subspace spanned by Saito-Kurokawa lifts.
By Theorems 8.2 and 8.4 we get that for every F ∈ Φ0 there exists a Galois representation
σF : GΣ → GL4 (E)which is short crystalline, satisfies det σF = 4k−6 and is τ -self-dual. Let Φ ⊂ Φ0 be∼ k−1 ⊕ ρ ⊕ χk−2 . For F ∈ Φ thethe subset consisting of forms F such that σ ss
F = χ
representation σF is absolutely irreducible which can be proven as in [Bro07] (cf.

38

TOBIAS BERGER, KRZYSZTOF KLOSIN

the arguments following the proof of Proposition 8.3). Write T for the quotient of
T0m acting on the space spanned by Φ. Here T0 denotes the Hecke algebra generated
over O by the operators T`,0 , T`,1 and T`,2 for all primes ` 6= p and T0m denotes thecompletion of T0 at the maximal ideal corresponding to Φ.
Thanks to Kato’s result (Theorem 17.14 in [Kat04]) we can use an argument
similar to that in the proof of Proposition 2.10 to deduce that
#Hf1 (Q, ρf (1 − k) ⊗ E/O) ≤ #O/Lalg (k, f ).

(10.1)

Here Lalg (k, f ) is the algebraic part of the L-function of f (for precise normalizationsee [Bro11]). For this we apply an analogue of the control theorem 2.11 for ξ = −mwith m = k − 1 and the fact that the p-adic L-function interpolates the classicalL-value at critical points.
We will now demonstrate that Φ is non-empty by showing that T 6= 0. Since theSelmer group on the left of (10.1) is non-trivial, we get that val$ (Lalg (k, f )) > 0.
Let SK(f ) denote the Saito-Kurokawa lift of f . It is a Siegel modular eigenformof weight k and level 1. Let J be the ideal of T generated by the image of theannihilator AnnT0m (SK(f )) of SK(f ) under the map T0m  T. Then it follows from
Corollary 5.6 in [Bro11] that #T/J ≥ #O/Lalg (k, f ). In particular Φ 6= ∅.
every F ∈Φ, we get by Corollary 4.3 that we may assume that σ F = Forχk−2 ∗1∗2ρ
∗3 . Since σF are short crystalline (by our assumption that p >χk−1
2k − 2) Proposition 5.1 then shows that we have σ F ∼= σ. We further get an
O-algebra map
Y
φ : Rred →O ⊃ T.
F ∈Φ
redClearly φ(R ) ⊃ T, but we in fact get that φ(Rred ) = T by Corollary 7.7. The
map φ descends to a map φ : Rred /I tot  T/J. Now combining Corollary 7.4 with
Theorem 7.6, inequality (10.1) and the fact #T/J ≥ #O/Lalg (k, f ), we conclude
that φ is an isomorphism. Since I tot is principal by Theorem 6.6, φ is also an
isomorphism by the commutative algebra criterion [BK13], Theorem 4.1.

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40

Maz97.

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Mok14.
MT15.
Och00.
Pil12.
Pil17.

PW11.
PY15.
Ram93.
Ram13.

Rib76.
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Ski16.
Sor10.
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Appendix A. by Cris Poor3 , Jerry Shurman4 , and David S. Yuen5
We prove the following theorem.
Theorem A.1. The space S2 (K(731)) has dimension 19, and its subspace of Gritsenko lifts has dimension 18. There exists a new Hecke eigenform f731 ∈ S2 (K(731))
such that:(1) f731 has integer Fourier coefficients with content 1.
(2) f731 is not a Gritsenko lift.
(3) f731 is congruent modulo 5 to a Gritsenko lift Hecke eigenform that has
integer Fourier coefficients.
(4) the first few Hecke eigenvalues of f731 are
λ2 = −1, λ3 = 0, λ4 = −2, λ5 = 0,
and its spin 2-Euler factor is
1 + x + 2x2 + 2x3 + 4x4 .
The proof will take several steps. The existence argument is motivated by techniques in [8, 6, 11], while the congruence is shown using a new technique.
A.1. Notation and Jacobi Restriction method. Let Sk (K(N )) denote thespace of weight k Siegel cusp forms for the level N paramodular group, or, more
briefly, the space of weight k, level N paramodular cusp forms. We use the notation “[d]” for projection onto the first d Jacobi coefficients, and “(d)” for thesubspace where the first d − 1 Jacobi coefficients vanish; because all Jacobi coefficients have index divisible by N , this condition is that the Jacobi coefficients ofindex N, 2N, . . . , (d − 1)N are 0. Thus
Sk (K(N ))[d] = Sk (K(N )) elements truncated to the first d Jacobi coefficients,
Sk (K(N ))(d) = Sk (K(N )) elements with vanishing first d − 1 Jacobi coefficients,
and there is an exact sequence
0 −→ Sk (K(N ))(d + 1) −→ Sk (K(N )) −→ Sk (K(N ))[d] −→ 0.
Analogous notation is understood to hold with either Fricke eigenspace Sk (K(N ))±of Sk (K(N )), or any Atkin–Lehner eigenspace, and the exact sequence persists.
Let Jcusp
k,m denote the space of weight k, index m Jacobi cusp forms. Using
the Fourier–Jacobi expansions of paramodular cusp forms, we view two maps as
containments for simplicity,∞
M∞
Sk (K(N )) ⊂Jcusp
k,jN ⊂ C .
j=1

The latter containment entails some chosen ordering of the index sets for Fourier
expansions of Jacobi forms. For any prime p, let a subscript p denote the map from
subsets of C∞ to subsets of Fp∞ that reduces integral elements modulo p,
Vp = (V ∩ Z∞ ) mod p,

V ⊂ C∞ .

42

TOBIAS BERGER, KRZYSZTOF KLOSIN

Thus
Sk (K(N ))p ⊂∞M∞(Jcuspk,jN )p ⊂ Fp .

j=1∞
The spaces Sk (K(N )) and Jcusp
by results of [16] and [3], rek,m have bases in Z
spectively. Therefore dimFp Sk (K(N ))p = dimC Sk (K(N )) and dimFp (Jcuspk,m )p =dimC Jcusp.
Extendingthenotationfromabove,k,m

Sk (K(N ))[d]p = reduction modulo p of elements of Sk (K(N ))[d] ∩ Z∞ ,Sk (K(N ))p = reduction modulo p of elements in Sk (K(N )) ∩ Z∞ ,Sk (K(N ))p (d) = elements of Sk (K(N ))p with first d − 1 Jacobi coefficients 0,
and in consequence of an integral basis of Sk (K(N )),dim Sk (K(N ))[d] = dim Sk (K(N ))[d]p ,dim Sk (K(N ))(d) ≤ dim Sk (K(N ))p (d).
Again, analogous notation is understood to hold with either Fricke eigenspace or
any Atkin–Lehner eigenspace of Sk (K(N )), and the dimension relations persist.
For either Fricke eigenvalue  = ±1, the Jacobi Restriction method [2, 5] runs
with d as a parameter, returning a basis of a finite-dimensional complex vector
space Jd of truncated formal Fourier–Jacobi expansions such that
Sk (K(N )) [d] ⊂ Jd ⊂

d
M
Jcusp
k,jN .
j=1

An additional parameter, detmax , bounds the Fourier coefficient index determinantsin the calculation, thus making the method an algorithm; this parameter must be
chosen so that each space Jcusp
k,jN with j ≤ d is determined by the Fourier coefficients
whose indices satisfy the determinant bound. For simplicity we suppress detmax
from the notation Jd , along with the weight k and the level N . We can also run
the Jacobi Restriction method modulo a prime p. In this case, the algorithm returns
over Fp such thata basis of a finite dimensional vector space Jd,p
Sk (K(N )) [d]p ⊂ Jd,p⊂d
M

(Jcuspk,jN )p .

j=1

The Jacobi Restriction method can also be run on any Atkin–Lehner eigenspace.
For a vector of signs describing an Atkin–Lehner eigenspace, the Jacobi Restriction
signs
method returns a basis of a space Jdsigns or Jd,pthat contains Sk (K(N ))signs [d]or Sk (K(N ))signs [d]p .
The methods of [11] let us attempt to show that S2 (K(N ))(d) or S2 (K(N ))p (d)is 0 for a certain d. The method we will use here requires us to span enoughof S4 (K(N )), namely a subspace of codimension less than dim Jcusp
2,N . A formula=
18 from [3].
from [4] gives dim S4 (K(731)) = 972, and we also have dim Jcusp
2,731
A.2. Spanning S4 (K(731)). For  = (−1)k , let Grit : Jcusp
k,N −→ Sk (K(N )) be thelinear map sending a Jacobi cusp form to its Gritsenko lift. Let g1 , . . . , g18 be abasis of the space of Gritsenko lifts in the Fricke plus space S2 (K(731))+ . The set

S = {T` (gi gj ) : 1 ≤ i ≤ j ≤ 18, 1 ≤ ` ≤ 4}

DEFORMATIONS OF SAITO-KUROKAWA TYPE

43

turns out to span at least a 719-dimensional space in S4 (K(731))+ , as shownby its Fourier coefficients for indices out to determinant 206. Establishing thisrequired expanding the gi initially out to indices of determinant 3296. Thus
dim S4 (K(731))+ ≥ 719, and we may select a basis of a 719-dimensional subspace.
For ` - N , the Hecke operator T` employed here is the T`,2 of the main article.
Note that 731 = 17 · 43. Let S4 (K(731))+,− denote the subspace of S4 (K(731))
where the Atkin–Lehner operators AL17 and AL43 have respective eigenvalues ±1,
and similarly for S4 (K(731))−,+ . Jacobi Restriction with parameters d = 6, p = 5,+,−
and detmax = 18275 on these Atkin–Lehner subspaces returns spaces J6,5
and−,+
J6,5 such that+,−
S4 (K(731))+,− [6]5 ⊂J6,5,+,−= 125,
dimF5 J6,5−,+
S4 (K(731))−,+ [6]5 ⊂J6,5,−,+
dimF5 J6,5
= 128.

Even though 719 + 125 + 128 = 972, it is not mathematically certain that 719,
125, and 128 are the dimensions of the Fricke plus space and the two Atkin–Lehnersubspaces of the Fricke minus space in S4 (K(731)); however, in practice we areconfident that they are, and this guided our search for spanning elements.
We construct a Borcherds product f4,731 ∈ S4 (K(731))− . The website [17] gives
the details of the construction, but here we note that the Borcherds product’s
leading theta block was located among more than 354, 000 candidate theta blocks
in Jcusp
4,731 of q-vanishing order 2. Its symmetrizations
h+,− , h−,+ = f4,731 ± AL17 (f4,731 ) ∈ S4 (K(731))+,− , S4 (K(731))−,+
are both nonzero. To generate many more elements of these Atkin–Lehner subspaces, we use a technique called bootstrapping [10], created to overcome the obstacle
that, computationally, a Hecke operator returns shorter truncations of paramodular
Fourier expansions than it receives. This shortening allows only a few iterations ofHecke operators, even starting with very long vectors that are expensive to compute. A brief summary of bootstrapping in the present context is as follows.
(1) We have a sufficiently long expansion of h+,− to identify it modulo 5 in+,−
J6,5
. The vector that we identify it as modulo 5 is longer.
+,−(2) Using the longer vector in J6,5
, we have sufficiently many Fourier coefficients to apply T2 , obtaining a shorter expansion of T2 h+,− modulo 5. But+,−
this vector is sufficiently long to identify in J6,5
, again giving us a longenough expansion to apply T2 again, and so on.
This technique shows that T2i h+,− for i = 0, 1, . . . , 124 are linearly independent
modulo 5, and so dimF5 S4 (K(731))+,− [6]5 ≥ 125. Similarly, bootstrapping withh−,+ gives dimF5 S4 (K(731))−,+ [6]5 ≥ 124. Thus altogether we have shown thatdimF5 S4 (K(731))− [6]5 ≥ 125 + 124 = 249, and we may select a basis modulo 5of a 249-dimensional subspace of S4 (K(731))−5 . So, because 719 + 249 = 968,
altogether we have reductions modulo 5 of linearly independent elements that spana codimension 4 subspace of S4 (K(731)). This is comfortably beyond the needed
codimension dim Jcusp
2,731 − 1 = 17.
A.3. Determining S2 (K(731))− . The H4 (N, 2, 1)− method of Proposition 4.6 in[11] uses a low-codimension subspace in S4 (K(N ))− to show that S2 (K(N ))−p (2) =
0. Briefly, the idea is as follows. Suppose that some h in S2 (K(731))−(2)
isnonzero.
5
Then the space W = h Grit(J2,731 )5 ⊆ S4 (K(731))−
5 has dimension 18. But certain

44

TOBIAS BERGER, KRZYSZTOF KLOSIN

Fourier coefficients of every element w of W must be zero modulo 5, because such
w are a particular kind of product. We make a 249 × r matrix M of coefficientsfrom a basis of the known 249-dimensional subspace of S4 (K(731))−
5 where the rcoefficients of every w ∈ W must be zero, and we compute the rank of M modulo 5
to be 249. Because the 249-dimensional subspace has codimension at most 4 in
S4 (K(731))−
5 , we conclude that dim W ≤ 4, contradicting dim W = 18. This proves−that S2 (K(731))−
5 (2) = 0, which in turn proves that S2 (K(731)) (2) = 0. Further,−
Jacobi Restriction with parameters d = 1, p = 5, and detmax = 1462 gives J1,5
= 0,−−
and so S2 (K(731)) [1] = 0, and the exact sequence gives S2 (K(731)) = 0.
A.4. Determining S2 (K(731))+ . The H4 (N, 6, 1)+ method of Proposition 4.6 in[11] shows that S2 (K(731))+
5 (6) = 0, using very similar reasoning. This implies
that S2 (K(731))+ (6) = 0. These critical vanishing results are ultimately whatallow us to prove equalities and congruences in S2 (K(731)). The exact sequencesays that the projection map S2 (K(731))+ −→ S2 (K(731))+ [5] is an isomorphism.
Further, Jacobi Restriction with parameters d = 5, p = 5, and detmax = 1462 gives+
dim J5,5= 19, so dim S2 (K(731))+ [5] ≤ 19. Thus dim S2 (K(731))+ ≤ 19, which is
to say that there is at most one dimension of nonlift eigenforms. The existence of anonlift, given in section A.6, will complete the proof that dim S2 (K(731))+ = 19.
A.5. Borcherds products. Our current best method for finding a paramodular
form that is not a Gritsenko lift in the low weight 2 and the fairly high level 731
is to try constructing Borcherds products (see [8, 9, 11]). However, the algorithm
from [9] shows that there are no nonlift Borcherds products in S2 (K(731)). The
method from [6, 8] is to go to twice the level and search for a Borcherds product inthe minus Fricke subspace S2 (K(1462))− . Indeed, we find two Borcherds products
f2,1462,a , f2,1462,b ∈ S2 (K(1462))− .
See the website [17] for their constructions.
A.6. Tracing down from S2 (K(1462))− to S2 (K(731))+ . We prove the existenceof a nonlift in S2 (K(731))+ by using the trace down map from [11],TrDn : S2 (K(1462)) → S2 (K(731))+ .
Here the codomain is the Fricke plus space because S2 (K(731))− = 0, as shown
above. Tracing down the Borcherds products f2,1462,a and f2,1462,b requires many
more Fourier coefficients than we can directly compute. So we run Jacobi Restriction on S2 (K(1462))− with parameters d = 4 and detmax = 5848, and get
dim J4− = 2. After identifying the projections of the Borcherds products in J4− ,we can compute two Fourier coefficients of TrDn(f2,1462,a ) and TrDn(f2,1462,b ), but
this is insufficient for our purposes. To remedy this, we use the fact that the
Borcherds products are Fricke minus forms to compute further Fourier coefficientsfor indices that are beyond the fourth Jacobi coefficient but still within the detmax
bound. This permits the computation of 46 Fourier coefficients of TrDn(f2,1462,a )and TrDn(f2,1462,b ). The computed coefficients of TrDn(f2,1462,a ) show that it
is linearly independent of the Gritsenko lifts in S2 (K(731))+ . This proves thatdim S2 (K(731))+ = 19.
Because there is only one nonlift eigenform in S2 (K(731))+ , it must scale to haverational coefficients, and then rescale to a nonlift eigenform f731 that has integer
Fourier coefficients with content 1. Jacobi Restriction with parameters d = 5 andDEFORMATIONS OF SAITO-KUROKAWA TYPE

45

detmax = 1462 returns dim J5+ = 19. Hence S2 (K(731))+ [5] = J5+ and we have
have enough coefficients to compute eigenvalues on the Hecke operators through T5 .
Especially, the nonlift eigenform f731 has eigenvalues
λ2 = −1, λ3 = 0, λ4 = −2, λ5 = 0.
We refer to [13, RS07] for the theory of global paramodular newforms, and notehere that the nonlift eigenform f731 must be a newform because by [12] the spaces
S2 (K(1)), S2 (K(17)), and S2 (K(43)) have no nonlifts.
A.7. Proving the mod 5 congruence. First we compute the action of T2 on
the 18-dimensional space of Gritsenko lifts in S2 (K(731))+ . The characteristic
polynomial of T2 on this space, factored into irreducibles over Q, is(x − 4)(x − 2)2 (x − 1)
· (x6 − 17x5 + 112x4 − 358x3 + 566x2 − 400x + 101)
· (x8 − 25x7 + 264x6 − 1530x5 + 5286x4 − 11046x3 + 13465x2 − 8612x + 2174).
Thus there is one Gritsenko lift eigenform with λ2 = 4, which we denote g andscale to have integer coefficients with content 1. The first few eigenvalues of g canbe computed,
λ2 = 4, λ3 = 5, λ4 = 8, λ5 = 5.
We now prove that the Fourier expansion of f731 is congruent to an integer multipleof the Fourier expansion of g modulo 5. For any h ∈ S2 (K(731))+ with integer
coefficients, introduce correspondingly
h[5] ∈ S2 (K(731))+ [5],h5 ∈ S2 (K(731))+5,h[5]5 ∈ S2 (K(731))+ [5]5 .

For any v ∈ S2 (K(731))+ [5]5 , there are enough computed Fourier coefficients of
T2 v to determine T2 v as an element of S2 (K(731))+ [5]5 , and so we can compute amatrix representative of the map
T2 [5]5 : S2 (K(731))+ [5]5 −→ S2 (K(731))+ [5]5 .
Basic linear algebra on the matrix representative shows that the T2 [5]5 -eigenspacewith eigenvalue 4,
V = {h ∈ S2 (K(731))+ [5]5 : T2 [5]5 h = 4h},is 2-dimensional. Although we did not have enough Fourier coefficients to compute the matrix representative of T4 [5]5 on S2 (K(731))+ [5]5 , we do have enoughto compute its matrix representative on the space V , which has a smaller set ofdetermining Fourier coefficients, and then basic linear algebra shows that the T4 [5]5 eigenspace with eigenvalue 3 in V is 1-dimensional. Both f731 [5]5 and g[5]5 lie in V ,and both have T4 [5]5 -eigenvalue 3. They cannot be linearly independent, because
then they would span V , giving the contradiction that T4 [5]5 |V = 3 IdV . Thus
f731 [5]5 and g[5]5 are linearly dependent, and so there exist an integer β 6= 0 mod 5
such that f731 [5]5 + βg[5]5 = 0 mod 5. This implies that
(f731 + βg)5 ∈ S2 (K(731))+
5 (6).
We earlier proved the critical result that S2 (K(731))+
5 (6) = 0, and this implies(f731 + βg)5 = 0. This shows that f731 is congruent to a multiple of g modulo 5.
The multiple of g is still a Gritsenko lift, so f731 is congruent modulo 5 to a Gritsenko
lift. This completes the proof of Theorem A.1.

46

TOBIAS BERGER, KRZYSZTOF KLOSIN

We thank Ralf Schmidt for helpful discussions.
Appendix References
1. James Arthur. The endoscopic classification of representations, volume 61 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI,
2013. Orthogonal and symplectic groups.
2. Jeffery Breeding, II, Cris Poor, and David S. Yuen. Computations of spaces of paramodular
forms of general level. J. Korean Math. Soc., 53(3):645–689, 2016.
3. Martin Eichler and Don Zagier. The theory of Jacobi forms, volume 55 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1985.
4. Tomoyoshi Ibukiyama and Hidetaka Kitayama. Dimension formulas of paramodular forms ofsquarefree level and comparison with inner twist. To appear in Journal of the MathematicalSociety of Japan.
5. Tomoyoshi Ibukiyama, Cris Poor, and David S. Yuen. Jacobi forms that characterize paramodular forms. Abh. Math. Semin. Univ. Hambg., 83(1):111–128, 2013.
6. Cris Poor, Ralf Schmidt, and David S. Yuen. Paramodular forms of level 16 and supercuspidal
representations. In preparation.
7. Cris Poor, Ralf Schmidt, and David S. Yuen. Paramodular forms of level 8 and weights 10
and 12. To appear in Int. J. Num. Th.
8. Cris Poor, Jerry Shurman, and David S. Yuen. Constructing paramodular nonlift newforms
via Borcherds products. In preparation.
9. Cris Poor, Jerry Shurman, and David S. Yuen. Finding all Borcherds lift paramodular cusp
forms of a given weight and level. preprint.
10. Cris Poor, Jerry Shurman, and David S. Yuen. Weight three paramodular forms and applications. In preparation.
11. Cris Poor, Jerry Shurman, and David S. Yuen. Siegel paramodular forms of weight 2 andsquarefree level. Int. J. Number Theory, 13(10):2627–2652, 2017.
12. Cris Poor and David S. Yuen. Paramodular cusp forms. Math. Comp., 84(293):1401–1438,
2015.
13. Brooks Roberts and Ralf Schmidt. On modular forms for the paramodular groups. In Automorphic forms and zeta functions, pages 334–364. World Sci. Publ., Hackensack, NJ, 2006.
14. Brooks Roberts and Ralf Schmidt. Local Newforms for GSp(4), volume 1918 of Lecture Notes
in Mathematics. Springer, Berlin, 2007.
15. Ralf Schmidt. Packet structure and paramodular forms. Transactions of the AMS, (17), 2017.
16. Goro Shimura. On the Fourier coefficients of modular forms of several variables. Nachr. Akad.
Wiss. Göttingen Math.-Phys. Kl. II, (17):261–268, 1975.
17. David S. Yuen. Degree 2 Siegel paramodular forms of weight 2 and level 731, 2017. http://siegelmodularforms.org/pages/degree2/paramodular-wt2-lev731/.
1 School of Mathematics and Statistics, University of Sheffield, Hicks Building,Hounsfield Road, Sheffield S3 7RH, UK.
E-mail address: tberger@cantab.net
2 Princeton University, Department of Mathematics, Fine Hall, Washington Rd, Princeton, NJ 08544, USA
E-mail address: krzysztof.klosin@qc.cuny.edu
3 Dept. of Mathematics, Fordham University, Bronx, NY 10458, USA

E-mail address: poor@fordham.edu
4 Reed College, Portland, OR 97202 USA

E-mail address: jerry@reed.edu
5 Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu,
HI, 96822 USA
E-mail address: yuen@math.hawaii.edu