IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONSOF SAITO-KUROKAWA TYPETOBIAS BERGER AND KRZYSZTOF KLOSIN

Abstract. We prove (under certain assumptions) the irreducibility of thelimit σ2 of a sequence of irreducible essentially self-dual Galois representations
σk : GQ → GL4 (Qp ) (as k approaches 2 in a p-adic sense) which mod p
reduce (after semi-simplifying) to 1 ⊕ ρ ⊕ χ with ρ irreducible, two-dimensionalof determinant χ, where χ is the mod p cyclotomic character. More precisely,
we assume that σk are crystalline (with a particular choice of weights) and
Siegel-ordinary at p. Such representations arise in the study of p-adic familiesof Siegel modular forms and properties of their limits as k → 2 appear to
be important in the context of the Paramodular Conjecture. The result isdeduced from the finiteness of two Selmer groups whose order is controlled byp-adic L-values of an elliptic modular form (giving rise to ρ) which we assume
are non-zero.

1. Introduction
In [BKar] the authors studied the modularity of abelian surfaces with rational
torsion. Let A be an abelian surface over Q, let p be a prime and suppose that A
has a rational point of order p, and a polarization of degree prime to p. Then the(semi-simplified) action of GQ := Gal(Q/Q) on A(Q)[p] is of the form 1 ⊕ ρ ⊕ χ, forχ the mod p cyclotomic character. Assuming that ρ is irreducible, Serre’s conjecture(Theorem of Khare-Wintenberger) implies that the mod p representation looks likethe reduction of that of a Saito-Kurakawa lift of an elliptic modular form f of weight
2. If End(A) = Z then the p-adic Tate module of A gives rise to an irreducible
p-adic Galois representation. The Paramodular Conjecture (formulated by Brumer
and Kramer [BK14]) predicts that this representation should be isomorphic to the
Galois representation attached to a weight 2 Siegel modular form of paramodular
level which is not in the space of Saito-Kurokawa lifts. Establishing the modularity
of A by a Siegel modular form therefore requires proving congruences between the
Saito-Kurokawa lift SK(f ) and “non-lifted type (G)” Siegel modular forms. The
latter are cuspforms staying cuspidal under the transfer to GL4 , and are expected
to be exactly the forms whose associated p-adic representation is irreducible.
Such congruences for Saito-Kurokawa lifts have been proven by Brown, Agarwal
and Li [Bro07], [AB14], [Brown-Li] for holomorphic Siegel modular forms of congruence level Γ20 (N ) and paramodular level Γpara (N ) for weights k larger than 6(see [Brown-Li] Corollary 6.15). With this new result [BKar] Theorem 10.2 can begeneralized to allow ramification at a squarefree level N , and establishes a so-called
The first author’s research was supported by the EPSRC Grant EP/R006563/1. The second
author was supported by a Collaboration for Mathematicians Grant #578231 from the Simons
Foundation and by a PSC-CUNY award jointly funded by the Professional Staff Congress and theCity University of New York.
1

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TOBIAS BERGER AND KRZYSZTOF KLOSIN

R = T result and the modularity of Fontaine-Laffaille representations that residually are of Saito-Kurokawa type (with an elliptic f of weight 2k − 2 for k ≥ 6). A
different type of congruences have also been constructed by Sorensen, see section
4.2.
The methods used to prove these congruences unfortunately do not extend toweight k = 2, the case of interest for the modularity of abelian surfaces. We propose
to use p-adic families to prove the relevant congruences in weight 2 (albeit a priori
only to a p-adic modular form - see below). For example, Skinner and Urban [SU06]
proved that for an ordinary elliptic form f the Γpara (N )-level holomorphic SaitoKurokawa lift SK(f ) can be p-adically interpolated by a semi-ordinary (also called
Siegel-ordinary) family. It is plausible that their arguments could be adapted forΓ20 (N )-level holomorphic Saito-Kurokawa lifts. Such p-adic families have also beenstudied by Kawamura [Kaw10] and Makiyama [Mak18].
As part of a work in progress we construct (under some assumptions) another
Siegel-ordinary p-adic family (of tame level either Γ20 (N ) or Γpara (N )) interpolating
the type of congruences constructed by Brown or Sorensen. At classical weightsk  0 its points would correspond to irreducible p-adic Galois representations thatare Siegel-ordinary (see Definition 2.3) and whose semi-simplified residual representation is the mod p representation associated to SK(f ).
One could then use this family to approach weight 2 via weights k  0, but
k → 2 p-adically. As points of weight 2 for such a family are critical (in the sensethat the Up = Up,1 Up,2 -slope is at least one and therefore does not satisfy the small
slope condition in Theorem 7.1.1 of [AIP15]; see section 4.1 for definitions of Up,1and Up,2 ) it is not clear whether this limit would correspond to a classical Siegel
modular form.
In fact, modularity by p-adic Siegel modular forms was proved for certain abeliansurfaces whose p-adic Galois representation is residually irreducible by Tilouine[Til06]. In a sense this paper provides a necessary ingredient to proving such p-adic
modularity for the residually reducible case as explained below. Let us also mention
that some strong potential modularity results in the residually irreducible situation
have recently been proven in [BCGP18].
One potential problem is that while the p-adic Galois representations attached tothe members of the family for k  0 are irreducible this is not a priori clear of thelimit. This property is on the one hand necessary for modularity purposes (as Tp A⊗
Qp is irreducible). On the other hand it allows one then to feed these ingredients
into a machinery similar to the one developed in [BKar] (modified appropriately
for representations that are Siegel-ordinary instead of Fontaine-Laffaille) and undersuitable conditions show that Tp A and the limit Galois representation are in factisomorphic, thus proving p-adic modularity of A.
In this paper we introduce a new way of proving that under certain assumptions
the limit of irreducible Galois representations is itself irreducible. This method isbased on finiteness of Selmer groups and while we only apply it here in our specificsituation (i.e., when the representations are residually of Saito-Kurokawa type, as
desired for proving the modularity of abelian surfaces with rational p-torsion) it
is not difficult to see how it can be modified to work in other contexts, cf. ourupcoming paper about a residually reducible R = T result for GL2 in weight 1.
In other words, while our overarching goal is to provide ingredients to prove
modularity of abelian surfaces as explained above, the theorems proven in this paper

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

3

could in principle be treated completely independently as a result on limits of Galois
representations. In particular, Siegel modular forms will be notably absent from ourstatements and their presence will manifest itself only through certain conditions
imposed on the Galois representations. We thus consider a family (which is partof a “refined” rigid analytic family in the sense of Ballaı̈che-Chenevier - see section3) of irreducible 4-dimensional p-adic Galois representations σk indexed by a set of
integers k > 2, k ≡ 2 (mod (p − 1)) which approach 2 in the p-adic sense. Supposethat tr σk converge p-adically to some pseudo-representation T when k → 2. We
require that for each k the representation σk reduces to some mod p representation
whose semi-simplification is isomorphic to 1 ⊕ χ ⊕ ρ for an irreducible 2-dimensional
representation ρ and that it is crystalline and Siegel-ordinary. We are interested in
conditions guaranteeing the irreducibility of T .
The basic idea is not difficult to explain. First we use the irreducibility of σk toconstruct Galois stable lattices in their representation spaces so that infinitely manyof the σk s reduce mod p to a non-semi-simple residual representation (whose semisimplification is 1⊕χ⊕ρ) with the same Jordan Holder factor as a subrepresentationand the same Jordan-Holder factor as a quotient. It is not possible to ensure that all
σk reduce to the same combination as σ k has three Jordan-Holder factors. Indeed,in general Ribet’s Lemma only tells us that there are enough (non-split) extensions
between different Jordan-Holder factors to guarantee connectivity of a certain graph- see section 5 - and absent any other assumptions (like for example lying in the
Fontaine-Laffaille range which was used in Corollary 4.3 of [BKar]) there is no way
to tell which extension will arise. However, as there are only finitely many suchextensions possible, we get an infinite subsequence T of σk with identical (non-split)
reduction.
Now, if T was reducible, there are several ways in which it can split into thesum of irreducible pseudo-representations. Let us discuss here the case of three
Jordan-Holder factors which can be regarded as the main result of this paper - see
Theorem 3.3. In that case as k ∈ T approaches 2 (p-adically) the representations
σk become reducible modulo pnk with nk tending to ∞. As the reduction of σkis non-split, we conclude that σk give rise to elements in a certain Selmer groupof arbitrary high order. Using symmetries built into the Galois representation one
shows that this Selmer group can only be one of two possibilities. Then the Main
Conjecture of Iwasawa Theory gives us that the orders of these Selmer groups arecontrolled by specializations to weight 2 (at two different points) of a certain p-adic
L-function. Hence to guarantee that these Selmer groups are finite (i.e., that T
cannot be reducible) we impose a non-vanishing condition on these L-values. As
we a priori do not know for which of the possible extensions we get the infinite
subsequence T we need to control both of the L-values as above. See section 5 fordetails.
A priori if T is reducible it could also split into 2 or 4 components and we deal
with them in sections 3 and 6. We are able to rule out all of them, albeit for the
reduction type dealt with in section 6, the so called Yoshida type, our theorems
require quite strong assumptions.
We would like to thank Adel Betina, Pol van Hoften, Chris Skinner, and Ariel
Weiss for helpful discussions related to the topics of this article and Andrew Sutherland for the example in section 4.2.

4

TOBIAS BERGER AND KRZYSZTOF KLOSIN

2. Setup
Let p be an odd prime. Let E be a finite extension of Qp with integer ring O,
uniformizer $ and residue field F. We fix an embedding Qp ,→ C. Write  for thep-adic cyclotomic character and χ for its mod $ reduction. Let N be a square-free
positive integer with p - N . Let Σ be the set of primes of Q consisting of p andthe primes dividing N . We denote by GΣ the Galois group of the maximal Galois
extension of Q unramified outside of the set Σ.
Consider a Galois representation ρ : GΣ → GL2 (F) of which we assume that it
is odd and absolutely irreducible of determinant χ. Furthermore we assume that ρis ordinary and p-distinguished, i.e.,
 −1

η χ ∗
∼
ρ|Dp =,
η
where η is a non-trivial unramified character and that ρ|Ip is non-split. We further
assume that ρ is ramified at all primes dividing N and that ρ|I` has a fixed line forall ` | N (or equivalently that N is the prime-to-p-part of the conductor of ρ).
Let τ : G → GLn (O) be an n-dimensional representation of a group G or
τ : O[G] → O be an n-dimensional pseudo-representation of G. For a definition of a pseudo-representation, its dimension and basic properties we refer thereader to section 1.2.1 of [BC09]. However, let us only mention here that an ndimensional pseudo-representation τ is called reducible if τ = τ1 + τ2 for somepseudo-representations τ1 , τ2 (each necessarily of dimension smaller than n). A
pseudo-representation that is not reducible is called irreducible. In particular, if
τ : G → GLn (O) is a representation, then T := tr τ is an n-dimensional pseudorepresentation and T is reducible if and onlyPif τ is. Furthermore if τ is an nr
dimensional pseudo-representation and τ =i=1 τi with each τi an irreducible
pseudo-representation, then this decomposition as a sum of irreducible pseudorepresentations is unique (up to reordering of the summands).
Now let G = GΣ . By composing a representation or pseudo-representation τwith the reduction map O → F we obtain the reduction of τ which we will denoteby τ . If τ is an n-dimensional representation valued in GLn (E), one can always finda GΣ -stable O-lattice Λ such that when we choose a basis of E n to be a basis of Λ
we obtain a representation τΛ valued in GLn (O). The isomorphism class of τΛ andalso of its reduction τ Λ depends in general on the choice of Λ. However, the semisimplification τ ss
Λ (and hence also the pseudo-representation tr τ Λ ) is independentof Λ and so it makes sense to drop Λ from the notation.
Lemma 2.1. Let τ : GΣ → GLn (E) be a continuous representation and let V be therepresentation space of τ . Suppose that there exists a subspace L ⊂ V of dimensionr ≤ n with the following two properties: L is stable under GΣ and GΣ acts on L via
an irreducible representation ψ : GΣ → GLr (E) with values in GLr (O). Let Λ bea GΣ -stable O-lattice in V (Λ ⊗O E = V ). Then Λ has a rank r free O-submodule
which is stable under GΣ and on which GΣ acts via the representation ψ.
Proof. Fix a basis B = {e1 , e2 , . . . , en } for the O-module Λ. Then B is also abasis for the vector space for V . Let v = α1 e1 + · · · + αn en ∈ L be a non-zero
vector. Then there exists s ∈ Z≥0 such that $s αi ∈ O for all i = 1, 2, . . . , n.
Thus 0 6= v0 := $s v ∈ Λ ∩ L. Set Λ0 := O-span of {g · v0 | g ∈ GΣ } ⊂ L.
Clearly Λ0 is an O-module, but note that it is also stable under GΣ . Indeed, let

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

5v 0 := β1 g1 ·v0 +· · ·+βk gk ·v0 = β1 ψ(g1 )v0 +· · ·+βk ψ(gk )v0 ∈ Λ0 with β1 , . . . , βk ∈ O,
g1 , . . . gk ∈ GΣ . Let g ∈ GΣ . Then g · v 0 = ψ(g)β1 ψ(g1 )v0 · · · + ψ(g)βk ψ(gk )v0 =β1 ψ(gg1 )v0 · · ·+βk ψ(ggk )v0 ∈ Λ0 , the first equality being true since v 0 ∈ L, i.e., GΣ
acts on Λ0 via ψ. Furthermore, since ψ(g) has entries in O for every g ∈ GΣ andv0 ∈ Λ, we conclude that Λ0 is an O-submodule of Λ. Hence, as O is a PID, we getthat Λ0 is a free finitely generated O-module of rank r0 ≤ n. Finally, Λ0 ⊗O E ⊂ L
is a non-zero subspace of L which is stable under the action of GΣ , i.e., is a non-zero
subrepresentation of ψ. Since ψ is irreducible we must have Λ0 ⊗O E = L and so
r0 = r.

Lemma 2.2. Let τ : GΣ → GLn (E) an irreducible representation. Suppose thatwith respect to some
 GΣ -stable O-lattice Λ of the representation space V of τ oneτ
∗1for τi : GΣ → GLri (F), r1 + r2 = n. Then there exists a
has τ Λ ∼=
τ200GΣ -stable O-latticeΛ of the representation space V such that with respect to Λ we
τ1.
have τ Λ0 ∼
=
∗ τ2


a
bg
Proof. For g ∈ GΣ write τΛ (g) = g. Then cg is an r2 × r1 matrix whose
cg dg
entries we denote by cij (g). Let S = {g ∈ GΣ | cg 6= 0}. Irreducibility ofτ guarantees that S is non-empty. For g ∈ S set mg := min{val$ (cij (g)) |i, j such that cij (g) 6= 0}. Furthermore set m = ming∈S mg and note that m ≥ 1
as τ Λ is upper-triangular. Then



 

1
ag bg 1ag
$ m bg=.
$−m cg dg$m$−m cgdg
In this article we will be especially interested in 2-dimensional and 4-dimensional
Galois representations that are ordinary in a sense that we now define.
(1) A Galois
representation
τ : GΣ → GL2 (E) will be called −1 k−1ψ∗ordinary if τ |Dp ∼for some positive integer k and some=ψ
unramified character ψ.
(2) A Galois representation τ : GΣ → GL4 (E) will be called Siegel-ordinary if −1 2k−3

ψ 
∗ ∗ ∗

∗ ∗ ∗
,
τ |Dp ∼=∗ ∗ ∗
ψ

Definition 2.3.

for some positive integer k and some unramified Galois character ψ.
(3) A Galois representation τ : GΣ → GL4 (E) will be called Borel-ordinary if −1 2k−3
ψ 
∗
∗
∗

φ−1 k−1∗∗,τ |Dp ∼
=
φk−2 ∗ ψfor some positive integer k and some unramified Galois characters ψ and φ.

6

TOBIAS BERGER AND KRZYSZTOF KLOSIN

For later it will be useful to introduce the following notation. If α ∈ E × , thenthe unramified character from Dp to E × that takes the arithmetic Frobenius to α
will be denoted by φα .
3. Irreducibility
3.1. Main assumptions. Assume we have a p-adic family of Galois representations in the sense of [BC09], i.e. we have a rigid analytic space X over Qp
and a 4-dimensional pseudo-representation T : GΣ → O(X). We denote by
σx : GΣ → GL4 (E(x)) (for some finite extension E(x) of Qp ) the semi-simple
representation of GΣ whose trace is the evaluation Tx of T at x ∈ X (for existencesee [Tay91], Theorem 1). We are interested in the case when the family satisfies
nice p-adic Hodge properties for all points in a Zariski dense set Z ⊂ X and want
to deduce properties at a point x0 ∈ X\Z, in particular control the ramification at
p of the corresponding Galois representation. The reader should think of X as (anaffinoid subdomain of) an eigenvariety parametrizing Siegel modular forms. We
therefore also assume the existence of a weight morphism w : X → W, where W is2
×the rigid analytic space over Qp such that W(Cp ) = Homcts ((Z×p ) , Cp ).
More precisely, assume that we have data (X, T, {κn }, {Fn }, Z), a refined family
in the sense of [BC09] Definition 4.2.3, where n = 1, . . . 4 and κn and Fn are analytic
functions in O(X). For z ∈ Z we have 0 = κ1 (z) < κ2 (z) < κ3 (z) < κ4 (z) are
the Hodge-Tate weights of σz . The case of interest to us is where for a point z ofweight w(z) = (w1 , w2 ) with w1 ≥ w2 we have κ2 (z) = w2 − 2, κ3 (z) = w1 − 1and κ4 (z) = w1 + w2 + 3. We assume σz is crystalline and the eigenvalues of ϕ on
Dcris (σz ) are given by (pκ1 (z) F1 (z), . . . , pκ4 (z) F4 (z)). Furthermore, suppose there
exists an involution τ : O(X)[GΣ ] → O(X)[GΣ ] given by τ (g) = Φ(g)g −1 for somecharacter Φ : GΣ → O(X)× with Φ|Dp = κ4 (z) such that T ◦ τ = T.
We also assume that for z ∈ Z the representation σz |Dp is Siegel-ordinary, i.e.
that
 −1 κ (z)
ψ  4
∗ ∗ ∗

∗ ∗ ∗
.
σz |Dp ∼=∗ ∗ ∗
ψThis is equivalent to demanding that |F1 (z)| = 1 and then ψ = φF1 (z) . The
existence of τ then implies that F4 (z) = F1 (z)−1 . In addition we assume that σz is
p-distinguished, i.e., ψ 6= 1.
Fix x0 ∈ X \ Z of weight w(x0 ) = (2, 2) and from now we reserve the notation
E for the field E(x0 ) and denote by O the ring of integers in E with uniformizer $and residue field F. Put T = Tx0 and σ2 := σx0 . We assume that T ≡ 1 + tr (ρ) + χ
mod $ for ρ as in section 2 and that F2 (x0 ) 6= 0.
Let S be a sequence of integers k ≡ 2 (mod pmk −1 (p − 1)) with mk → ∞as k → ∞. We single out a sequence of points zk ∈ Z converging to x0 withw(zk ) = (k, k) for k ∈ S. Denote the corresponding family of Galois representations
σk := σzk : GΣ → GL4 (Ek ), where we set Ek := E(zk ). Extending Ek if necessary
we may assume that O ⊂ Ok , where Ok is the ring of integers of Ek with uniformizer$k . Then we define nk ∈ Z≥0 to be the largest integer n such that tr σk ≡ T mod$n . Note the convergence zk → x0 implies nk → ∞ as k → ∞ but approaches 2
p-adically.

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

7

We assume that for each k ∈ S the representations σk have the following properties (of which (2), (3) and (5) follow from the assumption made on T and so does(4) for k  0, but we record them here again for the ease of reference):(1) σk is irreducible,(2) det σk = 4k−6 ,(3) σk∨ ∼= σk (3 − 2k),∼(4) σ ss
k = 1 ⊕ ρ ⊕ χ,(5) σk |Dp is crystalline with weights 2k − 3, k − 1, k − 2, 0 and σk is Siegelordinary at p, i.e., −1 2k−3
φβ k 
∗ ∗
∗

∗ ∗
∗ 
∼
,
σk |Dp =

∗ ∗
∗ 
φβk
for βk ∈ Ok× and we assume that βk 6≡ 1 mod $k , i.e., σ k is p-distinguished;(6) If ` ∈ Σ − {p} then ` 6≡ 1 (mod p) or σk |I` is unipotent.
We refer the reader to Remark 4.1 for a relation between these properties of σk andSiegel modular forms.
Lemma 3.1. We have(i) T |Dp = φ−1
β  + φβ + tr γ for β = F1 (x0 ) and a continuous representation
γ : Dp → GL2 (O).
(ii) The pseudo-representation T (or rather σ2 ) has Hodge-Tate-Sen weights 0,0,1,1.
(iii) Furthermore, if Ψ is any character that occurs in the decomposition of T |Dp
into pseudo-representations then we must have Ψ|Ip =  or Ψ|Ip = 1
Proof. For (i) we use the Siegel-ordinarity of the σz for z ∈ Z and continuity.
For (ii) we apply [BC09] Lemma 7.5.12 and deduce that the Hodge-Tate-Sen
weights in weight 2 are 0,0,1,1.
For (iii) first note that the statement is clear if Ψ = φβ or Ψ = φ−1β . So we00
now consider the case whenγ|ss
=
Ψ
⊕
Ψ
for
some
characterΨ.
Part(ii) tells
Dp
us that Ψ is Hodge-Tate of weight 0 or 1, so equal to a finite order character (notnecessarily unramified) or the product of such a character and . We want to use
the crystallinity of σz for z ∈ Z to deduce that Ψ is crystalline. Results of Kisin andBellaı̈che-Chenevier allow to continue crystalline periods for the smallest HodgeTate weight. Note that either φβ or φ−1β  has the same Hodge-Tate weight as Ψ.
To be able to attribute the crystalline period to Ψ (rather than φβ or φ−1β ) we use
the Siegel-ordinary and p-distinguishedness assumptions we made on σz for z ∈ Z:
As in [BB19] proof of Theorem 5.3 (which uses geometric Frobenius convention,
so considers representations dual to the ones we have here) we can quotient out thesheaf M corresponding to O(X)[Dp ]/ ker T (cf. [BC09] Lemma 4.3.7) by a subsheaf
L corresponding to the line stabilised by Ip on which Frobp acts by F4 pκ4 . The^ is generically of rank 3 and its semi-simplification specializes
quotient sheaf M/L
at x0 to Ψ ⊕ Ψ0 ⊕ φβ . As in the proof of [BB19] Proposition 8.2 Siegel-ordinarity^ has a torsion-free subsheaf M0 of generic rank 2 such thatfurther tells us that M/Lthe specialisations σz0 at z ∈ Z are 2-dimensional crystalline representations with
Hodge-Tate weights κ2 (z), κ3 (z) and with crystalline period for the appropriate
κi (z)
Hodge-Tate weight, i.e. Dcris (σz0 )ϕ=Fi (z)p6= 0 for i = 2 or 3. (Note that for

8

TOBIAS BERGER AND KRZYSZTOF KLOSINk ∈ S we have κ2 (zk ) = k − 2 and κ3 (zk ) = k − 1.) The semi-simplification of thesheaf M0 specializes at x0 to Ψ ⊕ Ψ0 .
Applying (the torsion-free analogue of) [BC09] Theorem 3.3.3(i) to M0 then givesκi (x0 )that Dcris (M0 )ϕ=Fi (x0 )p6= 0. Since by assumption F2 (x0 ) 6= 0 (and so also
F3 (x0 ) 6= 0) this means that one of the characters Ψ or Ψ0 is crystalline, so equal toa power of the cyclotomic character times a finite order unramified character. As
discussed before this power must be 0 or 1. As T |Dp = T |Dp ◦τ with τ (g) = (g)g −1
we get ΨΨ0 = . So we are done.

3.2. Possible splitting types of T . Now suppose that T is reducible. Then T isin one of the following cases:(i) T = T1 + T2 + T3 + T4 , where each Ti is a character;(ii) T = T1 + T2 + T3 , where T1 and T3 are characters and T2 is an irreducible
pseudo-representation of dimension 2 (we refer to this type of splitting as
the Saito-Kurokawa type);(iii) T = T1 + T2 , where T1 , T2 are both irreducible pseudo-representations ofdimension 2 (we refer to this type of splitting as the Yoshida type);(iv) T = T1 + T2 , where T1 is an irreducible pseudo-representation of dimension
3 and T2 is a character.
Proposition 3.2. Cases (i) and (iv) cannot occur.
∼
Proof. Case (i) cannot occur because σ ss
k = 1 ⊕ ρ ⊕ χ for every k ∈ S, so also
T = 1 + tr ρ + χ and ρ is irreducible (so also tr ρ is irreducible as a pseudorepresentation).
Let us now show that T is not as in case (iv). Suppose T is as in case (iv). Then
T = ξ+tr ρ0 , where ξ : GΣ → O× is a character and ρ0 is a 3-dimensional irreducible
representation. As T = T ◦ τ , we must have ξ|Ip = ξ|−1
Ip . This contradicts Lemma
3.1(iii).
P∞n
For an ordinary newform g =of weight 2 let L(g, s) denote
n=1 an (g)q
the standard L-function of g and let Lp (g, 2) be the p-adic L-value denoted by−1Lan, T = p) in section 2 of [BKar]. The proof of the following theorem willp (g, ωbe given in the next section.
p
Theorem 3.3. Assume that ρ|GK is absolutely irreducible for K = Q( (−1)(p−1)/2 p).
Suppose that L(g, 1)Lp (g, 2) 6= 0 for all p-ordinary newforms g of weight 2 and level
dividing N 2 p such that a` (g) ≡ tr ρ(Frob` ) mod $ for all primes ` - N p. Then T
is not of Saito-Kurokawa type.
Note that there are only finitely many (possibly none) forms g as in Theorem
3.3.
In section 6 we discuss some conditions that guarantee that T is not of Yoshidatype either. All these results combined would guarantee that T is in fact irreducible,
however, the assumptions allowing us to rule out the Yoshida type are quite strong(cf. Remark 6.2).
4. Siegel modular forms and Paramodular Conjecture
4.1. Siegel modular forms. We recall some facts about Siegel modular forms andtheir associated Galois representations to motivate our discussion. By Arthur’sclassification (see [Art04] and [GT19]) cuspidal automorphic representations forIRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

9

GSp4 (AQ ) fall into different types. Cuspidal automorphic representations π whosetransfer to GL4 stays cuspidal are called of “general type” or type (G). Such type(G) representations are expected to have irreducible p-adic Galois representations(see [Wei19] for a summary of what’s known and results in the low weight case).
Other types in the classification are known to be associated to reducible p-adic
Galois representations, see [BCGP18] Lemma 2.9.1. Particular examples of suchtypes are the Saito-Kurokawa lifts and Yoshida lifts of elliptic modular forms, whose
associated Galois representations have trace of Saito-Kurokawa or Yoshida typerespectively. Schmidt [Sch18] proved that holomorphic Siegel modular forms ofparamodular level are either of type (G) or Saito-Kurokawa lifts, while other CAP
types or Yoshida lifts do not occur.
We denote by Up,1 (resp. Up,2 ) the Hecke operators associated to diag(1, 1, p, p)(resp. diag(1, p, p2 , p)). For π of sufficiently high weight (i.e. corresponding toclassical Siegel eigenforms of weights k1 ≥ k2 ≥ 3) we have the following result aboutproperties of the associated Galois representations (for a more detailed statementsee [BCGP18] Theorem 2.7.1):
Theorem 4.1 (Laumon, Weissauer, Sorensen, Mok, Faltings-Chai, Urban). Suppose π is a cuspidal automorphic representation for GSp4 (AQ ) of weight k1 ≥ k2 ≥3. Then there is a continuous semi-simple representation ρπ : GQ → GSp4 (Qp )
with
∼
ρ∨
π = ρπ (3 − k1 − k2 )satisfying the following properties:(1) For each prime ` 6= p we have local-global compatibility up to semi-simplification
with the local Langlands correspondence proved by Gan-Takeda. In particular, if π is unramified at ` then so is ρπ and if π is of Iwahori level at `then ρπ |I` is unipotent.
(2) If ρπ is irreducible then for each prime ` 6= p one has local-global compatibility up to Frobenius semi-simplification.
(3) ρπ |Dp is de Rham with Hodge-Tate weights k1 + k2 − 3, k1 − 1, k2 − 2, 0.
(4) Assume that π is Siegel-ordinary at p (i.e λp,1 is a p-adic unit, λp,2 has
finite p-valuation, where λp,i is the Up,i -eigenvalue of π for i = 1, 2), thenρπ |Dp is Siegel-ordinary in the sense of Definition 2.4 with the unramified
character having λp,1 as value at Frobp .
(5) If π is unramified at p then the p-adic representation ρπ is crystalline at
p. If π is also Siegel-ordinary then the characteristic polynomial of Frobenius acting on Dcris (ρπ |Dp ) equals the Hecke polynomial. In particular, theeigenvalues are
k2 −2
k1 −1
k1 +k2 −3λp,1 , λ−1, λp,1 λ−1, λ−1.
p,1 λp,2 p
p,2 p
p,1 p

Suppose now that ρ as in section 2 equals ρf for f ∈ S2 (N p). If f is ordinary
it lies in a Hida family of eigenforms fk . Brown et al. [Bro07], [AB14], [BL19]
then prove that there exists a family of holomorphic Siegel modular eigenforms Fk(2)
for k ∈ S with S as in section 3 of Iwahori level N (level Γ0 (N ) or Γpara (N )) that are congruent to the Saito-Kurokawa lifts SK(fk ) modulo p and σFk isirreducible (see e.g. [AB14] Corollary 7.5). The theorem above then shows that theassociated Galois representations σFk satisfy the conditions (1)-(6) in section 3.1
except possibly for Siegel-ordinarity (note that we assume that ρ is p-distinguished).
To establish that the tr σFk interpolate p-adically is work in progress.

10

TOBIAS BERGER AND KRZYSZTOF KLOSIN

The pseudo-representation of the (Siegel-ordinary, tame level N ) eigenvariety
then gives rise to T : GΣ → O(X) for an affinoid X containing the limit point
x0 of weight (2, 2). One obtains a Zariski dense subset Z ⊂ X of classical points
that are old at p such that (X, T, {κn }, {Fn }, Z) is a refined family in the sense of
Bellaı̈che-Chenevier. By the above theorem the function F1 = F4−1 interpolates theUp,1 -eigenvalue λp,1 , F2 = F3−1 interpolates λ−1p,1 λp,2 , so our assumption F2 (x0 ) 6= 0
corresponds to the Up,2 -slope of the limit form being finite.
4.2. Discussion of applicability to the Paramodular Conjecture. For an
elliptic modular form f of weight 2k − 2 a holomorphic Saito-Kurokawa lift exists
under the following conditions on f and k: for Γ20 (N )-level k has to be even, forΓpara (N )-level the sign of the functional equation of f has to be −1 (see [Sch07]).
Suppose ρ = ρf for an ordinary newform f of level N . For Theorem 3.3 weneed to assume that L(f, 1) 6= 0. Continuing our discussion from the introduction
about Saito-Kurokawa congruences, we note that in the case that L(f, 1) 6= 0
we would therefore need to consider congruences with holomorphic Γ20 (N )-level
Saito-Kurokawa lifts. However, a different method to the one used by Brown etal. (pointed out to us by Pol van Hoften) could be used to prove the required
congruences for paramodular level: Using the arguments from the proof of [Sor09]
Theorem D one should be able to prove congruences for the generic (as opposed tothe holomorphic) Saito-Kurokawa lift, for which the conditions on k and the root
number are reversed.
Once the congruence between the generic Saito-Kurokawa lift and a type (G)form has been proved, one could then switch to the holomorphic element of thesame packet. If such a congruence could be proved in weight 2 this would also
explain the example of the abelian surface of conductor 997 mentioned in [BKar](which involves an elliptic modular form f with root number  = 1 and L(f, 1) = 0).
To demonstrate that examples with L(f, 1) 6= 0 occur when studying the modularity of abelian surfaces we thank Andrew Sutherland for providing us with thefollowing abelian surface: Let A be the Jacobian of the genus 2 curve
C : y 2 + (x + 1)y = −2x6 + x5 − x4 + 9x3 − 2x2 + 2x − 9(see [LMF20, Genus 2 Curve 1870.a] and [BSS+ 16]). Then A has conductor 1870 =2 ∗ 5 ∗ 11 ∗ 17 and comparing values on Frob` for ` < 106 indicates that
A(Q)[3] ∼=1⊕ρ ⊕χf

for f the unique weight 2 newform of level Γ0 (17) corresponding to the isogeny
class of rank 0 elliptic curves over Q with conductor 17.
5. Ruling out Saito-Kurokawa type
We keep the notation and assumptions of section 2, 3.1 and Theorem 3.3. In
this section we will prove Theorem 3.3. Recall that by assumption (4) we haveσ ss
k = 1 ⊕ ρ ⊕ χ for every k ∈ S.
Lemma 5.1. Let k ∈ S. Then there exists a GΣ -stable lattice Λ in the representation space of σk such that(i)

1 ak bk
σ k,Λ = dk ρ ck 
ek f k χ

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

11

and for at least one xk ∈ {ak , ck , dk , f k }, the representation σ k,Λ has
a subquotientisomorphic
to a non-semisimple representation of the form


ρ1 xk
, where ρ1 , ρ2 are distinct elements of {1, ρ, χ}, one of
σxk :=
ρ2
which equals ρ, and are determined by xk .
(ii) σ k,Λ is indecomposable.
Proof. Consider the graph G whose vertices are elements of the set V = {1, ρ, χ}and where we draw a directed edge from ρ0 ∈ V to ρ00 ∈ V if there exists a GΣ stable lattice Λ0 such that σk,Λ0 hasa subquotient isomorphic to a non-semi-simple
ρ0 x
representation of the form
. Then by a theorem of Bellaı̈che for any two
ρ00ρ0 , ρ00 ∈ V, there exists a directed path from ρ0 to ρ00 (see Corollaire 1 in [Bel03]).
In particular there must be at least one edge originating at ρ and at least oneedge ending at ρ. One of these edges gives rise to the desired σxk . This proves(i). If for such a Λ0 the representation σ k,Λ0 is already indecomposable then weare done. Otherwise σ k,Λ0 is the direct sum of σxk and the remaining element v of
V. Hence we can apply Theorem 4.1 in [BKar] with σxk as a quotient and v as a(non-semi-simple) subrepresentation. This gives us a new lattice Λ so we have σ k,Λ
is a non-split extension of σxk by v, hence indecomposable.

Corollary 5.2. Under the same assumptions as in Lemma 5.1 the subquotient σxk
can be taken to be a subrepresentation or a quotient, i.e., there exists at least one
lattice for each option.
Proof. This follows from Lemma 2.2.

For the rest of the section assume that T = T1 + T2 + T3 with T1 , T2 , T3 whereΨ1 := T1 and Ψ2 := T3 are characters and T2 is two-dimensional and irreducible.
We assume that Ψ1 = 1, Ψ2 = χ and T 2 = tr ρ. Our goal is to show that theseassumptions lead to a contradiction, and thus prove Theorem 3.3. Since T2 isirreducible we get by [Tay91] Theorem 1 that T2 = tr ρ̃ for some irreducible 2dimensional representation ρ̃ : GΣ → GL2 (E) reducing to ρ.
Lemma 5.3. The representation ρ̃ is ordinary.
−1
Proof. By Lemma 3.1 we have σ2 |ss
Dp = φβ  ⊕ φβ ⊕ γ, where γ is two-dimensional.
Since β 6≡ 1 mod $ by our assumption (5), we cannot have Ψ1 |Dp , Ψ2 |Dp ∈
∼ −1
∼
{φ−1
, φβ }. Hence it must be the case that ρ̃|ss
Dp = φβ  ⊕ φβ . Suppose ρ̃|Dp =
 β

φβ
∗
. Note that ρ̃ ∼= ρ is irreducible, so in particular well-defined and we
0 φ−1
β 
have by assumption that ρ|Dp does not have an unramified subrepresentation of −1∼ φβ  ∗ asdimension 1. Thus neither can ρ̃|Dp . Hence we get that ρ̃|Dp =0φβdesired.


Recall that for every k ∈ S we write nk for the largest integer such that tr σk ≡ T(mod $nk ). Note that under the assumptions from section 3.1 one clearly has
nk → ∞ as k approaches 2 p-adically.

12

TOBIAS BERGER AND KRZYSZTOF KLOSIN

Lemma 5.4. Let k ∈ S, J = {Ψ1 , ρ̃, Ψ2 } and let Λ be a lattice from Lemma 5.1
so that
ρ3 ∗∗ρ1 xk σ k := σ k,Λ = 
ρ2


ρ1 xk
is indecomposable with non-semi-simple 3-dimensional quotient(cf. Corolρ2
lary 5.2). Then
ρ3 yk zk
ρ1 xk  (mod $nk ).
σk ∼=O k 
ρ2
Here ρi are distinct elements of J and ρi = ρi mod $ and xk = xk mod $. In
particular the class [xk ] ∈ H 1 (Q, Hom(ρ2 , ρ1 ) ⊗ Ok /$nk ) has the property that$nk −1 [xk ] 6= 0.
Proof. This follows from Remarks (a) and (d) in [Urb99] (cf. also Theorem 1.1
in
The last statement follows directly from the fact that the quotient [Bro08]).

ρ1 xk
is not semi-simple.
ρ2
Lemma 5.5. There exists an ordinary newform g of weight 2 and level dividing
N 2 p such that ρ̃ = ρg .
 −1
φβ  ∗∼
Proof. By Lemma 5.3 we have that ρ̃|Dp =, i.e., ρ̃ is an ordinary de0
φβformation of ρ. In particular, its Hodge-Tate weights are 1 and 0. Furthermore, theassumption that ρ|GK be absolutely irreducible (with K as in Theorem 3.3) guarantees that ρ̃ is modular by some ordinary newform g of weight 2 by a generalizationof a theorem of Wiles due to Diamond - see Theorem 5.3 in [Dia96]. The p-part ofthe level of g is p or 1 (see e.g., Lemma 3.26 in [DDT97]). For primes ` | N suchthat ` 6≡ 1 mod p the discussion in section 10.2 of [BK13] explains how a result ofLivne implies that the level of g is at most `2 . If ` | N and ` ≡ 1 (mod p), then thelevel is, in fact, at most ` due to our unipotency assumption. For this note that if
V denotes the space corresponding to ρ̃ this means that V I` is 1-dimensional. As
we are also assuming that the residual reduction VρI` is 1-dimensional and that ρ
has Artin conductor `, the Artin conductors of ρ and ρ̃ agree and equal ` (as theirvaluations are given by dim V − dim V I` + sw(ρ̃) and dim Vρ − dim VρI` + sw(ρ),
respectively, and sw(ρ) = sw(ρ̃) by Serre).
Remark 5.6.
(1) The reader may note that if no g as in the statement ofTheorem 3.3 exists then Lemma 5.5 already gives a contradiction to theassumption that T is of Saito-Kurokawa type.
(2) Similar analyses of reducibility ideals for families approximating holomorphic paramodular Saito-Kurokawa lifts were carried out in [SU06] and[BB19] in characteristic zero (necessarily under different assumptions, inparticular for L(g; 1) = 0). In the following we present arguments working in characteristic p. However, it is possible that a characteristic zero
approach would also yield our result.
Write Vg for the representation space of ρg and let Vg+ ⊂ Vg be the onedimensional subspace on which Ip acts via . Let Tg ⊂ Vg be any GΣ -stable latticeIRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

13

in Vg . The following Lemma follows from the fact that any two GΣ -stable lattices
are homothetic.
Lemma 5.7. Let τ : GΣ → GL2 (E) be residually irreducible. Let Λ, Λ0 be two GΣ stable lattices in the representation space of τ . Then τΛ ∼= τΛ0 (over O). In otherwords, Λ and Λ0 are isomorphic as O[GΣ ]-modules, i.e., there exists M ∈ GL2 (O)such that τΛ0 = M τΛ M −1 .
In particular, the action of GΣ on Tg /$Tg (which we denote by ρg,Tg ) is isomorphic to ρg ∼= ρ as the latter representation is irreducible. Furthermore, by Lemma
5.7 we get that the isomorphism class of the restriction of the action of GΣ to Ip on
Tg is independent of the choice of Tg inside the representation space of ρg . Moreprecisely, we have the following result.
 ∗Lemma 5.8. One has ρg,Tg |Ip ∼.
=O
1
Proof. By Lemma 5.7
 it isenough to show that there exists a GΣ -stable lattice Λ0 xsuch that ρg,Λ0 |Ip =. For this see proof of Proposition 6 of [Gha05].
1
Write Wg for Vg /Tg ∼= ρg,Tg ⊗E/O. By Lemma 5.8 we know that there exist rank
one free O-submodules Tg+ and Tg− of Tg such that Tg = Tg+ ⊕ Tg− as O-modules
and that if e1 ∈ Tg+ and e2 ∈ Tg− form a basis of Tg then in the basis {e1 , e2 } one xhas ρg,Tg |Ip =with x 6≡ 0 mod $ (as ρg |Ip = ρ|Ip is non-split). One clearly1++
has Tg ⊗O E = Vg . Set Wg+ := Vg+ /Tg+ ∼= Tg+ ⊗O E/O.
Following [SU06] 3.1.3 we define Greenberg-style Selmer groups


resIp
−1
1
+
Seli := ker H 1 (GΣ , W ⊗ Ψ−1
)
−
−
−
→
H
(I
,
(W
/W
)
⊗
Ψ
)
, i = 1, 2.
p
g
g
i
i
Lemma 5.9. One has Ψ1 = 1 and Ψ2 = .
Proof. By assumption (6) we know that Ψ1 and Ψ2 are unramified away from p.
Since Ψ1 = 1 and Ψ2 = χ we know by Lemma 3.1(iii) that Ψ1 is unramified
everywhere, hence trivial. As Ψ1 Ψ2 =  we get Ψ2 = .
Proposition 5.10. The groups Seli are finite for i = 1, 2.
Proof. Recall
Y
Y
L(g, s) =(1 − a` (g)`−s + `−2s+1 )−1 (1 − a` (g)`−s )−1`-N

for Re(s)  0.

`|NLet LN (g, s) be defined in the same way but omitting the Euler factors at primes` | N . By Theorem 4.6.17 in [Miy89] we get that the `-eigenvalue a` (g) of g equals 0
or ±1, hence 1−a` (g)`−i 6= 0 for i = 1, 2. This implies that L(g, i) 6= 0 if and only if
LN (g, i) 6= 0 for i ∈ {1, 2}. By [SU14] Theorem 3.36 we have #Sel1 ≤ #O/LN (g, 1).
In the notation of [SU14] we are in the case m = 0 and ap (g) − 1 ∈ O× due toour p-distinguishedness assumption on ρ. For i = 2 we use the argument from theproof of [BKar] Proposition 2.10. In particular, by the Main Conjecture of Iwasawa
theory and the control theorem ([BKar] Theorem 2.11) we deduce#Sel2 ≤ #O/LN
p (g, 2).


14

TOBIAS BERGER AND KRZYSZTOF KLOSINAs the representations σk,Λ are valued in Ok , rather than O we need to introduce
some auxiliary Selmer groups. For k ∈ S and r ∈ Z+ we set


resp
+
−1
1
Seli,k,r := ker H 1 (GΣ , Tg,k,r ⊗ Ψ−1i ) −−→ H (Ip , (Tg,k,r /Tg,k,r ) ⊗ Ψi ) , i = 1, 2,?
where Tg,k,r= Tg? ⊗ Ok /$r Ok for ? ∈ {+, ∅}. Note that for k = 2 we have an
inclusion Seli,2,r ,→ Seli [$r ].

Proposition 5.11. If xk ∈ {dk , f k }, then [xk ] ∈ Sel1,k,nk . If xk ∈ {ak , ck }, then[xk ] ∈ Sel2,k,nk . In either case $nk −1 [xk ] 6= 0.
Proof. Write
Ψ1 a bσk =  d ρ̃ c  (mod $nk )e f Ψ2
t


 1


t

as before with ak = ak a2k , dk = d1k d2k , ck = c1k c2k and fk = fk1 fk2 .
By Siegel-ordinarity we have −1
φβ  ∗ ∗ ∗∗ ∗ ∗
.
σk |Dp ∼=Ek ∗ ∗ ∗
φβ −1
φβ  hFurthermore, by Lemma 5.3 we have ρ̃|Dp =. Thus in particularφβ(σk |Dp(mod $nk ))ss = Ψ1 ⊕ Ψ2 ⊕ φ−1β  ⊕ φβ(mod $nk ).

Conjugating σk by a permutation matrix we see that −1
φβ  d1k c1k h
 a1
Ψ1 bk a2k k (mod $nk ).
σk |Dp ∼=O k 
 f1
ek Ψ2 fk2 k0
d2k c2k φβ
To complete the proof of Proposition 5.11 we need several lemmas.
Lemma 5.12. One has• If xk = ak , then a1k gives rise to an extension of Dp -modules
mod $nk , which splits, i.e., [a1k ] = 0.
•

If xk = ck , then c2k gives rise to an extension of Dp -modules


Ψ1


φβ

$nk , which splits, i.e., [c2k ] = 0.
•

If xk = dk , then d2k

gives rise to an extension of Dp -modules


c2k
mod
Ψ2

φβ

mod $nk , which splits, i.e., [d2k ] = 0.
• If xk = f k , then fk1 gives rise to an extension of Dp -modules
mod $nk , which splits, i.e., [fk1 ] = 0.


a1k
φ−1
β 


Ψ2

d2k
Ψ1




fk1
φ−1
β 

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONSΨ2
Proof. To fix attention assume that xk = ak , i.e., that σk = 

15


zk
ak  modρ̃$nk as in Lemma 5.4. First note that (after possibly changing to an appropriate
basis for the ρ̃-piece and using Lemma 5.8) Siegel-ordinarity implies that


Ψ2 yk
zk1
zk2

Ψ1
a1k
a2k 

(5.1)
σk |Dp = (mod $nk ).
−1
φβ  h 
φβ
yk
Ψ1

nk

 Hence1 we
 see that there indeed is a rank 2 free Ok /$ [Dp ]-subquotient S =Ψ1ak
as claimed in the Lemma. It remains to show that S splits. Assume
φ−1β it does not. Let V be the representation space for σk . By Siegel-ordinarity it has
a Dp -stable line L on which Dp acts via φ−1β . Let Λ be a GΣ -stable lattice givingnk
σk such that σk |Dp mod $ has the form (5.1). Then we see by Lemma 2.1 that
this Λ must have a Dp -stable rank one submodule with Dp action by φ−1β , hence
finally Λk := Λ mod $nk must have a free Ok /$nk -submodule Λ0 of rank one on
which Dp acts by φ−1β .
We now claim that the subquotient S also has a free Ok /$nk -submodule which isstabilized by Dp and on which Dp acts via φ−1β . Indeed, write B = {e1 , . . . , e4 } fornk
an Ok /$ -basis of Λk such that with respect to that basis we have σk |Dp in form
(5.1). Write Λ0 = (Ok /$nk )e1 ⊕(Ok /$nk )e2 ⊕(Ok /$nk )e3 and Λ00 := (Ok /$nk )e4 .
We note that Λ0 is stable under the action of Dp . We first want to show that
Λ0 ⊂ Λ0 . Let v0 ∈ Λ0 be an Ok /$nk -module generator. Using the fact that
B is a basis we can decompose v0 uniquely as v0 = v00 + v000 with v00 ∈ Λ0 and
v000 ∈ Λ00 . We want to show that v000 = 0. Let g ∈ Ip be such that χ(g) 6= 1. Then
000
g · v0 = φ−1β (g)v0 = (g)v0 . On the other hand g · v0 = g · v0 + g · v0 . We have that
0
0
00
00
0
00
0
0
0
g · v0 ∈ Λ and g · v0 = φβ (g)v0 + v = v0 + v for some v ∈ Λ . So we have(g)v00 + (g)v000 = (g)v0 = g · v0 = g · v00 + v000 + v 0 =⇒ (g)v000 − v000 ∈ Λ0 ∩ Λ00 = 0.
Since χ(g) 6= 1, we see that (g) − 1 ∈ (Ok /$nk )× , which implies that v000 = 0. So
Λ0 ⊂ Λ0 .
Now set Λ00 = (Ok /$nk )e1 . This is a Dp -stable submodule of Λ0 on which Dp
acts via Ψ2 . Notice that we have S = Λ0 /Λ00 as Dp -modules. Clearly the image ofΛ0 ⊂ Λ0 in S is the desired Dp -stable Ok /$nk -submodule of S on which Dp acts via
φ−1
β . We just need to show that this image is non-zero. Suppose to the contrary
that it is zero, i.e., that Λ0 ⊂ Λ00 . Let d ∈ Dp be such that Ψ1 (d) 6≡ φ−1
β (d) mod $.
Then we get φ−1
(d)v
=
d
·
v
=
Ψ
(d)v
,
which
impliesv=
0,acontradiction.
0
0
1
0
0
β
This now proves the claim about S.
a b
In other words there must exist a matrix A =∈ GL2 (Ok ) such that
c d


 −1

Ψ1
a1k
φβ  ∗A=
A(mod $nk ).
φ−1

Ψ
1
β
Suppose that [a1k ] 6= 0, i.e., that there exists g ∈ Dp such that Ψ1 (g) = φ−1β (g) = 1
but a1k (g) 6= 0. Then comparing the upper left entries of both sides evaluated at g
we get a + a1k (g)c = a, from which we get that c ≡ 0 mod $. For the same entry,

16

TOBIAS BERGER AND KRZYSZTOF KLOSIN

00
but for a general element g 0 ∈ Dp such that φ−1β (g ) 6≡ Ψ1 (g ) (mod $), we get−1Ψ1 (g 0 )a + ca1k (g 0 ) = aφβ (g 0 ). Reducing this equation mod $ we thus conclude
that a ≡ 0 (mod $). This is a contradiction since A is invertible.
The other cases, i.e., where xk = ck , dk , f k are handled similarly using the factthat Ψ1 |Dp , Ψ2 |Dp , φ−1β , φβ are all pairwise distinct mod $. This finishes theproof of Lemma 5.12.


We are now ready to complete the proof of Proposition
5.11.
Suppose first that


ρ̃ ∗
nk
xk = dk . Then σk mod $ has a subquotient τ =, i.e., σk mod $nk has
Ψ1


Ψ1
a submodule τ =which is non-split mod $ as [xk ] 6= 0. Thus dk givesdk ρ̃
rise to a class in H 1 (GΣ , Hom(Ψ1 , ρ̃) ⊗ Ok /$n ) such that $nk −1 [dk ] 6= 0.
Ψ1
0
0
−1
Furthermore, by Lemma 5.12 we must have τ |Dp ∼=  d1k φβ  h  . (Note00φβthat while d2k as in Lemma 5.12 gives rise to an (in fact split) extension of Ψ1 by
φβ , it is not necessarily true that d1k gives rise to an extension of Ψ1 by φ−1β  because
h may be non-zero. However, it is still true that d gives rise to an extension of Ψ1by ρ̃.) Recall that ρ̃ = ρg .
Furthermore, in thebasis giving rise to τ as above, the module Tg,k,nk cor 
0+responds to vectors α while the submodule Tg,k,nof Tg,k,nk corresponds to
k
β
 
0
vectors of the form α ∈ Tg,k,nk , as on these vectors Ip acts via . Since in the
0
 1 
d (γ)
+
same basis for every γ ∈ Ip we have dk (γ) = k, we see that dk (γ)(α) ∈ Tg,k,nk
0
for every α in the representation space of Ψ1 .
Let A be an Ok -algebra (we will only use A = Ok or A = Ok /$s Ok ). Given twoψfree A-modules V1 , V2 we have a canonical isomorphism V2 ⊗A V1∨ −→ HomA (V1 , V2 )given by v ⊗ φ 7→ (v 0 7→ φ(v 0 )v). If W2 ⊂ V2 is an A-direct summand (i.e.,
V2 = W2 ⊕ W20 for some A-submodule W20 of V2 ) then the isomorphism ψ carries
W2 ⊗ V1∨ ⊂ V2 ⊗ V1∨ onto the (direct summand of HomA (V1 , V2 ) consisting of)homomorphisms whose image is contained in W2 . Similarly, if W1 ⊂ V1 is an Adirect summand and we denote by W10 ⊂ V1∨ the submodule (which is a direct
summand) of linear functionals which kill W1 , then ψ carries W10 onto the direct
summand of Hom(V1 , V2 ) consisting of homomorphisms that kill W1 . All of thesefollow immediately from the fact that tensor product as well as the Hom-functor
commute with direct sums in both coordinates.
Even though by Lemma 5.9 we have Ψ1 = 1, in the argument below we keep
the notation Ψ1 to convince the reader that analogous calculations hold for any
character, hence in particular can be applied in the cases when xk = ck or f k , whereΨ1 is replaced by Ψ2 which is a non-trivial character. By the above we see that for+every γ ∈ Ip we get that dk (γ) ∈ Hom(Ψ1 , Tg,k,nk ) has image contained in Tg,k,n,k−1
so the image of dk (γ) under the inverse of the isomorphism ψ : Tg,k,nk ⊗ Ψ1 →

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

17

+
−1
HomOk /$n Ok (Ψ1 , Tg,k,nk ) is an element of Tg,k,n⊗ Ψ−1
1 ⊂ Tg,k,nk ⊗ Ψ1 . Thus,kdk gives rise to an element of Sel1,k,nk as desired.


Ψ1 ak
nk
Suppose now that xk = ak . Then σk mod $ has a submodule τ =ρ̃which is non-split mod $ as [xk ] 6= 0. Thus ak gives rise to a class in

H 1 (GΣ , Hom(Tg,k,nk , Ok /$nk Ok (Ψ1 )))


Ψ1
0
a2k
h .
such that $nk −1 [ak ] 6= 0. Again by Lemma 5.12 we must have τ |Dp =  0 φ−1
β 
0
0
φβ
+
We will now show that for every γ ∈ Ip the homomorphism ak (γ) kills Tg,k,n.
In k deed, note that in the basis which gives the above formof τ we have ak = 0 a2k ,0+α ∈ Tg,k,nk .
while Tg,k,n
is
given
again
by
the
vectors
of
the
form
k
0
By the discussion above we conclude that the inverse of the isomorphism ψ :∨Ok /$nk (Ψ1 )⊗Tg,k,n→ Hom(Tg,k,nk , Ok /$nk (Ψ1 )) carries ak (γ) into the subspacek++∨
Ok /$nk (Ψ1 )⊗(Tg,k,n)0 ⊂ Ok /$nk (Ψ1 )⊗Tg,k,n, where as above (Tg,k,n)0 denotes
k
k
k
+
∨
the submodule of Tg,k,nk consisting of functionals which kill Tg,k,nk .
−1
∨
∨ ∼
∼ −1
Note that since Ψ1 Ψ2 = , we get Ψ1 ⊗ ρ∨
g = Ψ2  ⊗ ρg = Ψ2 ⊗ ρg (1).
+nk
0Under these isomorphisms the module Ok /$ (Ψ1 ) ⊗ (Tg,k,nk ) gets mapped to
+
−1
+
0
nk
0
Ok /$nk (Ψ−12 ) ⊗ (Tg,k,nk ) and finally to Ok /$ (Ψ2 ) ⊗ (Tg,k,nk ) (1). Finally (byessential self-duality of ρg ) there is an isomorphism of GΣ -modules ψ 0 : ρg → ρ∨
g (1).
+
We note that Tg,k,n
istheuniquedirectsummand
of
T
which
is
stableunder
g,k,nkk
Ip and such that Ip acts on it by . Hence ψ 0 (as it is GΣ -equivariant) must carry+∨
Tg,k,nonto the unique direct summand of Tg,k,n(1) with the same property, i.e.,
k
k
+
0
∨
on which Ip
ψ (Tg,k,nk ) = X ⊗  where X is the unique direct summand of Tg,k,nk
acts trivially.
 
v1
+
0
Let φ ∈ (Tg,k,n
)
.
Let
γ
∈
I
,
v
=
∈ Tg,k,nk . (We suppress the 0 from
p
k
v2
 
0
v1 .) Then
v2




(γ)−1 h(γ −1 )(γ)−1 v1 + h(γ −1 )v2−1(γ·φ)(v) = φ(ρg (γ )v) = φ(v) = φ= v2 = φ(v).
1v2++Hence Ip acts trivially on (Tg,k,n)0 , i.e., we must have X = (Tg,k,n)0 . In other
k
k
+
+
0
0
words ψ carries Tg,k,nk onto (Tg,k,nk ) (1). This proves that for γ ∈ Ip we have+
)0 ∼that ak (γ) is mapped under ψ −1 into Ok /$nk (Ψ1 ) ⊗ (Tg,k,n= Ok /$nk (Ψ−1
2 )⊗
k
+
0
0 −1
(Tg,k,nk ) (1) and further mapped under (ψ ) into the the direct summand Ok /$nk (Ψ−1
2 )⊗
+
−1
nk
Tg,k,nk ⊂ Ok /$ (Ψ2 ) ⊗ Tg,k,nk . Hence we get [ak ] ∈ Sel2,k,nk .
The cases xk = ck and xk = f k are handled in an analogous way. Finally thefact that $nk −1 [xk ] 6= 0 follows from Lemma 5.4.
Corollary 5.13. If xk ∈ {dk , f k }, then there exists an element x0k ∈ Sel1 such that$nk −1 x0k 6= 0. If, on the other hand, xk ∈ {ak , ck }, then there exists an elementx0k ∈ Sel2 such that $nk −1 x0k 6= 0.

18

TOBIAS BERGER AND KRZYSZTOF KLOSIN

Proof. First note that as the formation of Selmer groups commutes with direct sumsof Galois modules and Ok /$r = (O/$r )s where s = [Ok : O] one has Seli,k,nk =s(Seli,2,nk ) . If xk ∈ {dk , f k } then by Proposition 5.11 we get that [xk ] ∈ Sel1,k,nk
is such that $nk −1 [xk ] 6= 0. Thus there must exist an element x0k ∈ Sel1,2,nk which
is not annihilated by $nk −1 . As we have an inclusion Sel1,2,nk ,→ Sel1 [$nk ], we
can regard x0k as an element of Sel1 which is not killed by $nk −1 . The other caseis analogous.
We are now ready to finish the proof of Theorem 3.3, i.e., that the pseudorepresentation T is not of Saito-Kurokawa type. Indeed, we will now arrive at acontradiction. Since by Lemma 5.1 for every k ∈ S there exists xk ∈ {ak , ck , dk , f k }such that [xk ] gives rise to a non-split extension of the corresponding Jordan-Holder
blocks of 1 ⊕ ρ ⊕ χ, there exists A ∈ {a, c, d, f } and an infinite subsequence T ⊂ Ssuch that for all k ∈ T we have that [xk ] = [Ak ] is such a non-split extension. Fix
such an A. Then Proposition 5.11 gives us an extension [Ak ] ∈ Seli,k,nk for i = 1
or 2 such that $nk −1 [Ak ] 6= 0. Set i(A) = 1 if the extension [Ak ] lies in Sel1,k,nk
and i(A) = 2 if the extension [Ak ] lies in Sel2,k,nk . Then by Corollary 5.13 we get
an element A0k ∈ Seli(A) not annihilated by $nk −1 . As nk tends to ∞ for k ∈ T ,we see that Seli(A) must be infinite. Thus we obtain a contradiction to Proposition
5.10.

6. Ruling out Yoshida typeIn this section we show that σ2 is not the direct sum of two irreducible twodimensional representations under some assumptions.
(2)
For a positive integer N we will write S2 (Γpara (N )) for weight 2 genus 2 Siegel
modular forms of paramodular level N .
Proposition 6.1. Suppose at least one of the following holds:(I) One has ` 6≡ ±1 mod p for all ` | N and σ2 is Borel-ordinary at p.
(II) One has ` 6≡ ±1 mod p for all ` | N and σ2 is crystalline at p.
(III) One has p > 3 and σ2 = σF for some classical Siegel modular form F ∈(2)S2 (Γpara (N )) which has distinct roots for its Hecke polynomial at p.
Then σ2 is not of Yoshida type.
Proof. Assume that in fact σ2 = ρ1 ⊕ ρ2 with ρ1 , ρ2 irreducible and ρ1 = ρ and
ss
ρss
= φ−1
2 = 1 ⊕ χ. By Lemma 3.1(i) we have (σ2 |Dp )β ⊕ φβ ⊕ γ , which as inφ−1  ∗
Lemma 5.3 implies that ρ1 is ordinary, i.e., that ρ1 |Dp ∼. By Lemma=E βφβ3.1(ii) the Hodge-Tate-Sen weights of σ2 are 0,0,1,1.
Proof of (I): As σ2 is Borel-ordinary, this forces ρ2 |Dp to be ordinary, i.e., ρ2 |Dp ∼= −1
φα  ∗
×for some α ∈ O . On the other hand since ρ2 is irreducible there exists
φαa GΣ -stable lattice Λ in the space of ρ2 such that with respect to that lattice we
have


1 a ∼
(6.1)
ρ2,Λ =
6 1 ⊕ χ.
=χ

IRREDUCIBILITY OF LIMITS OF GALOIS REPRESENTATIONS

19−1
By Lemma 2.1,
" the lattice# Λ must have a Dp -stable line on which Dp acts via φα ,−1∗. By comparing with the form (6.1) and using that χ isφα∼ χ ∗ . Thus ρ |D =∼ 1⊕χ.
ramified we conclude that φα = 1, so in fact ρ2,Λ |Dp =2 p
1This in particular implies that ρ2 splits when restricted to Ip . Hence a gives rise
to a class inso ρ2,Λ |Dp ∼
=

φα χ

resp

HΣ1 (Q, F(−1)) := ker(H 1 (GΣ , F(−1)) → H 1 (Ip , F(−1))).
Since ` 6≡ ±1 mod p for all ` | N we use Q
Lemma 6.3 in [BK19] to conclude thatHΣ1 (Q, F(−1)) = ker(H 1 (GΣ , F(−1)) → `∈Σ H 1 (I` , F(−1))). This part of theclass group of Q(µp ) is zero by Proposition 6.16 in [Was97]. This implies that ρ2,Λis split which leads to a contradiction.
Proof of (II): As before there existsa GΣ -stable lattice Λ such that with respect
1 a ∼6 1 ⊕ χ. Since σ2 is crystalline and its
to that lattice we have ρ2,Λ ==χ
Hodge-Tate-Sen weights are 0,0,1,1, it is in the Fontaine-Laffaille range. Hence so
is ρ2 . This implies (see e.g. [BK19] Lemma 6.1) that the extension given by a gives
rise to a non-zero element in HΣ1 (Q, F(−1)), which again gives a contradiction asHΣ1 (Q, F(−1)) = 0.
Proof of (III): We have σ2 = σF for some classical Siegel modular form F ∈(2)S2 (Γpara (N )). We can assume that F is not a Saito-Kurokawa lift (as then tr σF
would not be of Yoshida type). By [Sch18] this means that F is of type (G). The
assumption on the roots of the Hecke polynomial implies by [Jor10] Theorem 4.3.4or [Mok14] Proposition 4.16 that σ2 is crystalline at p. If ` 6≡ ±1 mod p for all` | N then we get a contradiction as in (I) and (II). Without this assumption weargue as in the proof of [BKar] Theorem 8.6, i.e. apply [Ram13] Theorem C and[KRS92] Theorem 7.1 to deduce that F would have to be of Yoshida type, i.e. notof type (G), a contradiction.
Remark 6.2. Note that the key issue in the Yoshida case is ruling out that σ2 isthe sum of an (ordinary) 2-dimensional Galois representation associated to a classical form (with associated mod p-representation ρ) and a 2-dimensional Galois
representation that is a priori not de Rham.
It is worth noting that whilst we are able to rule out that σ2 is of Saito-Kurokawa
type only using properties of the representations σk for k ∈ S the Yoshida type caserequires additional information about σ2 . In particular, while for both the SaitoKurokawa and the Yoshida type we assume crystallinity of the representations σk ,in case (II) of Proposition 6.1 we also need to assume that σ2 itself is crystalline.
On the other hand, work in progress by Ariel Weiss shows that a classical Siegelordinary type (G) eigenform has irreducible Galois representation. This would
allow us to drop the assumption in (III) on the distinctness of the roots of the
Hecke polynomial.
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