On modularity of reducible residual Galois representations

# On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

Tobias Berger and Krzysztof Klosin School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK, email: tberger@cantab.net Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd, Queens, NY 11367, USA, email: kklosin@qc.cuny.edu
###### Abstract.

In this paper we study deformations of mod Galois representations (over an imaginary quadratic field ) of dimension whose semi-simplification is the direct sum of two characters and . As opposed to [BergerKlosin13] we do not impose any restrictions on the dimension of the crystalline Selmer group . We establish that there exists a basis of arising from automorphic representations over (Theorem LABEL:mainthm). Assuming among other things that the elements of admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if itself is modular then so is its every crystalline characteristic zero deformation (Theorems LABEL:mainthm2 and LABEL:version2).

###### Key words and phrases:
Galois representations, Galois deformations, automorphic forms, modularity
###### 2010 Mathematics Subject Classification:
11F80, 11F55
The work of the second author was partially supported by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

## 1. Introduction

Let be an odd prime. Let be a number field, a finite set of primes of (containing all primes of lying over ) and the Galois group of the maximal extension of unramified outside . Let be a finite extension of with ring of integers and residue field . Let be two absolutely irreducible non-isomorphic representations with , which we assume lift uniquely to crystalline representations .

The aim of this article is to study deformations of non-semi-simple continuous crystalline representations whose semi-simplification is in the case and is an imaginary quadratic field. We analyzed this deformation problem in [BergerKlosin13] under the additional assumption that is one-dimensional (which is equivalent to saying that there exists only one such up to isomorphism). Here denotes the subgroup of consisting of classes unramified outside and crystalline at all . In this paper we do not make any assumption on this dimension. Disposing of the “dim=1” assumption is more than a technicality as in the general case one can no longer expect to be able to identify the universal deformation ring with a Hecke algebra.

This question was studied by Skinner and Wiles for and totally real fields in the seminal paper [SkinnerWiles99]. In that paper the authors analyze primes of the (ordinary) universal deformation ring of and prove that they are ‘pro-modular’ in the sense that the trace of the deformation corresponding to occurs in the Hecke algebra . In particular no direct identification of and is made.

In this article we take a different approach and work with the reduced universal deformation ring and its ideal of reducibility. In the “dim=1”-case, the authors proved (as a consequence of an -theorem - Theorem 9.14 in [BergerKlosin13]) that is a finitely generated -module. In contrast, if , while there are only finitely many automorphic representations whose associated Galois representations are deformations of , the ring may potentially be infinite over (Remark 2.15). This is a direct consequence of the existence of linearly independent cohomology classes inside the Selmer group which can be used to construct non-trivial lifts to . The resulting (potentially large) characteristic components of do not arise from automorphic representations and in this paper we will ignore them by considering a certain torsion-free quotient of instead of itself. It is however possible that by doing so we are excluding some characteristic deformations whose traces may be modular in the sense of [CalegariMazur09] (i.e. arise from torsion Betti cohomology classes).

On the other hand, as opposed to the situation studied in [SkinnerWiles99], over an imaginary quadratic field there are no reducible deformations to characteristic zero which in turn is a consequence of the finiteness of the Bloch-Kato Selmer group (Lemma 2.19), where , are (unique) lifts to characteristic zero of and respectively.

While each may possess non-modular reducible characteristic deformations, the situation is complicated further by the fact that in general many ’s do not admit any modular deformations at all (this phenomenon does not arise in the “dim=1” case). Indeed, first note that two extensions in define isomorphic representation of if and only if they are (non-zero) scalar multiples of each other. In particular, if , then there is a unique non-semi-simple representation of with semi-simplification . (Similarly, if then there exists a unique crystalline such representation.) However, if , then there are non-isomorphic such representations where . This demonstrates that in general not all reducible representations can be modular (of a particular level and weight), as the number of such characteristic zero automorphic forms is fixed (in particular it is independent of making a residue field extension). Nevertheless, we are able to prove (see Corollary LABEL:surj1prime) that there exists an -basis of arising from modular forms. For this we combine a congruence ideal bound for a Hecke algebra with the upper bound on the Selmer group of predicted by the Bloch-Kato conjectures.

Let be the image in of the subalgebra generated by traces of for arising from a modular form. As pointed out we can extend the set consisting of to a modular basis of . Our ultimate goal is to show that it is possible to identify with the quotient of a Hecke algebra . Here the quotient corresponds to automorphic forms for which there exists a lattice in the associated Galois representation with respect to which the mod reduction equals .

To prove our main modularity lifting theorem (Theorem LABEL:mainthm2) we work under the following two assumptions. On the one hand we assume that the modular basis is unique in the sense that any other such consists of scalar multiples of the elements of . On the other hand we assume that all admit only finitely many characteristic zero deformations, which in particular implies that the quotient we define is a finitely generated -module. The first assumption can be replaced with the assumption that the Bloch-Kato Selmer group is annihilated by (Theorem LABEL:version2). This second result is in a sense ‘orthogonal’ to the main results of [BergerKlosin11] and [BergerKlosin13], where the same Selmer group is assumed to be cyclic, but of arbitrary finite order.

Our approach relies on simultaneously considering all the deformation problems for representations (). As in [BergerKlosin13] we first study “reducible” deformations via the quotients for the reducibility ideal of the trace of the universal deformation into as defined by Bellaïche and Chenevier. These ideals are the analogues of Eisenstein ideals on the Hecke algebra side. To relate to the order of a Bloch-Kato Selmer group we make use of a lattice construction of Urban (Theorem 1.1 of [Urban01], see Theorem 4.1 in this paper). In fact it is a repeated application of Urban’s theorem (on the Hecke side and on the deformation side) that allows us to prove a modularity lifting theorem. We show that when the upper bound on the Selmer group and the lower bound on the congruence ideal agree (which in many cases is a consequence of the Bloch-Kato conjecture), this implies that every reducible deformation which lifts to characteristic zero of every is modular (cf. section LABEL:urban). It is here that we make use of the assumption on the ‘uniqueness’ of to be able to use a result of Kenneth Kramer and the authors [BergerKlosinKramer14] on the distribution of Eisenstein-type congruences among various residual isomorphism classes of Galois representations (cf. Section LABEL:Tsection). Yet another application of Urban’s Theorem allows us to prove the existence of a deformation to and as a consequence to identify with (Theorem LABEL:deformtoRtr). Using the fact that the ideal of reducibility of is principal (Proposition LABEL:prin) and applying the commutative algebra criterion (Theorem 4.1 in [BergerKlosin13]) we are finally able to obtain an isomorphism and thus a modularity lifting theorem (Theorems LABEL:mainthm2 and LABEL:version2).

Throughout the paper we work in a slightly greater generality than necessary for the imaginary quadratic case to stress that our results apply in a more general context if one assumes some standard conjectures. However, in section LABEL:Main_result we gather all the assumptions in the imaginary quadratic case as well as the statements of the main theorems (Theorems LABEL:mainthm, LABEL:mainthm2 and LABEL:version2) in this context. Hence the reader may refer directly to that section for the precise (self-contained) statements of the main results of the paper in that case.

We would like to thank Gebhard Böckle and Jack Thorne for helpful comments and conversations related to the contents of this article. We would also like to express our gratitude to the anonymous referee for suggesting numerous improvements throughout the article. The second author was partially supported by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.

## 2. Deformation rings

Let be a number field and a prime with and unramified in . Let be a finite set of finite places of containing all the places lying over . Let denote the Galois group , where is the maximal extension of unramified outside . For every prime of we fix compatible embeddings and write and for the corresponding decomposition and inertia subgroups of (and also their images in by a slight abuse of notation). Let be a (sufficiently large) finite extension of with ring of integers and residue field . We fix a choice of a uniformizer .

### 2.1. Deformations

Denote the category of local complete Noetherian -algebras with residue field by . Let be any positive integer. Suppose

 r:GΣ→GLm(F)

is a continuous homomorphism.

We recall from [ClozelHarrisTaylor08] p. 35 the definition of a crystalline representation: Let and be a complete Noetherian -algebra. A representation is crystalline if for each Artinian quotient of , lies in the essential image of the Fontaine-Lafaille functor (for its definition see e.g. [BergerKlosin13] Section 5.2.1). We also call a continuous finite-dimensional -representation over (short) crystalline if, for all primes , and for the filtered vector space defined by Fontaine (for details see [BergerKlosin13] Section 5.2.1).

Following Mazur we call two representations for such that strictly equivalent if there exists such that . A (crystalline) -deformation of is then a pair consisting of and a strict equivalence class of continuous representations that are crystalline at the primes dividing and such that , where is the maximal ideal of . (So, in particular we do not impose on our lifts any conditions at primes in .) Later we assume that if , then (mod ), which means that all deformations we consider will trivially be “-minimal”. As is customary we will denote a deformation by a single member of its strict equivalence class.

If has a scalar centralizer then the deformation functor is representable by since crystallinity is a deformation condition in the sense of [Mazur97]. We denote the universal crystalline -deformation by . Then for every there is a one-to-one correspondence between the set of -algebra maps and the set of crystalline deformations of .

For let be an absolutely irreducible continuous representation. Assume that . Consider the set of isomorphism classes of -dimensional residual (crystalline at all primes ) representations of the form:

 (2.1) τ=[τ1∗τ2]:GΣ→GLn(F),

which are non-semi-simple ().

From now on assume .

###### Lemma 2.1.

Every representation of the form (2.1) has scalar centralizer.

This is easy. ∎

### 2.2. Pseudo-representations and pseudo-deformations

We next recall the notion of a pseudo-representation (or pseudo-character) and pseudo-deformations (from [BellaicheChenevierbook] Section 1.2.1 and [Boeckle11] Definition 2.2.2).

###### Definition 2.2.

Let be a topological ring and a topological -algebra. A (continuous) -valued pseudo-representation on of dimension , for some , is a continuous function such that

1. and is a non-zero divisor of ;

2. is central, i.e. such that for all ;

3. is minimal such that , where, for every integer , is given by

 SN(T)(x)=∑σ∈SNϵ(σ)Tσ(x),

where for a cycle we define , and for a general permutation with cycle decomposition we let .

In the case when the pseudo-representation is determined by its restriction to (see [BellaicheChenevierbook] Section 1.2.1) and we will also call the restriction of to a pseudo-representation.

We note that if is a morphism of -algebras then is a pseudo-representation of dimension (see [BellaicheChenevierbook] Section 1.2.2).

According to [BellaicheChenevierbook] Section 1.2.1, if is a pseudo-representation of dimension and an -algebra, then is again a pseudo-representation of dimension .

Following [SkinnerWiles99] (see also [Boeckle11] Section 2.3) we define a pseudo-deformation of to be a pair consisting of and a continuous pseudo-representation such that , where is the maximal ideal of .

By the sentence following [SkinnerWiles99] Lemma 2.10 (see also [Boeckle11] Proposition 2.3.1) there exists a universal pseudo-deformation ring and we write for the universal pseudo-deformation. For every there is a one-to-one correspondence between the set of -algebra maps and the set of pseudo-deformations of . Any deformation of a representation as in (2.1) gives rise (via its trace) to a pseudo-deformation of , so there exists a unique -algebra map such that the trace of the deformation equals the composition of with .

We write for the quotient of by its nilradical and for the corresponding universal deformation, i.e. the composite of with . We further write for the closed -subalgebra of generated by the set

 S:={\textuptrρτ(Frobq)∣q∉Σ}.
###### Lemma 2.3.

The image of is and hence is an object in the category ..

###### Proof.

This is clear (cf. [ChoVatsal03] Theorem 3.11) since is topologically generated by ) (and is closed). ∎

### 2.3. Selmer groups

For a crystalline -adic -module (finitely generated or cofinitely generated over - for precise definitions cf. [BergerKlosin13], section 5) we define the Selmer group to be the subgroup of consisting of cohomology classes which are crystalline at all primes of dividing . Note that we place no restrictions at the primes in that do not lie over . For more details cf. [loc.cit.].

We are now going to state our assumptions. The role of the first one is to rigidify the problem of deforming the representations appearing on the diagonal of the residual representations. The role of the second is to rule out characteristic zero upper triangular deformations.

###### Assumption 2.4.

Assume that and denote by the unique lifts of to .

###### Assumption 2.5 (“Bloch-Kato conjecture”).

One has the following bound:

 #H1Σ(F,HomO(~τ2,~τ1)⊗OE/O)≤#O/L,

for some non-zero .

###### Remark 2.6.

In applications the constant will be the special -value at zero of the Galois representation divided by an appropriate period.

For the remainder of this section we will work under the above two assumptions.

### 2.4. Ideal of reducibility

Let be a Noetherian Henselian local (commutative) ring with maximal ideal and residue field and let be an -algebra. We recall from [BellaicheChenevierbook] Proposition 1.5.1 the definition of the ideal of reducibility of a (residually multiplicity free) pseudo-representation of dimension , for which we assume that

 T=\textuptrτ1+\textuptrτ2modmA
###### Definition 2.7 ([BellaicheChenevierbook] Proposition 1.5.1 and Definition 1.5.2).

There exists a smallest ideal of such that mod is the sum of two pseudo-characters with mod . We call this smallest ideal the ideal of reducibility of and denote it by .

###### Definition 2.8.

We will write for the ideal of reducibility of the universal pseudo-deformation , for the ideal of reducibility of , for the ideal of reducibility of and for the ideal of reducibility of .

###### Lemma 2.9.

Let be the smallest closed ideal of containing the set

 {\textuptrρredτ(Frobv)−\textuptr~τ1(Frobv)−\textuptr~τ2(Frobv)∣v∉Σ}.

Then equals the ideal of reducibility .

###### Proof.

By the Chebotarev density theorem we get (mod ), hence . Conversely, we know from the definition of the ideal of reducibility that (mod ) is given by the sum of two pseudo-characters reducing to . By Assumption 2.4 and Theorems 7.6 and 7.7 of [BergerKlosin13] (see also [Boeckle11] Theorem 2.4.1) these two pseudo-characters must equal (mod ). This shows that . ∎

###### Corollary 2.10.

The quotient is cyclic. ∎

###### Remark 2.11.

Combined with Lemma 7.11 of [BergerKlosin13] this shows that for any pseudo-deformation of with ideal of reducibility for which there is a surjection , the quotient is cyclic.

###### Proposition 2.12.

The module is a torsion -module.

###### Proof.

Fix and set . Suppose that is not torsion. Let be the canonical surjection (of -algebras). Let . We first claim that . Clearly in , so we just need to prove that . Suppose on the contrary that . Then there exists such that

 (2.2) ϖσ−1(1−ϖs)=0\textupin$S$.

Since the residue field of is , we see that is not a unit in , and hence is a unit in . Thus (2.2) implies that in , which leads to a contradiction and hence we have proved that .

We now use the following lemma.

###### Lemma 2.13.

There exists an -submodule such that

 R=A⊕B

as -modules.

###### Proof.

This follows from the following result.

###### Lemma 2.14 (Lemma 6.8(ii), p.222 in [Hungerford]).

Let be a module over a PID such that and for some prime and a positive integer . Let be an element of of order . Then there is a submodule of such that .

Apply Lemma 2.14 for , , , , . Then . ∎

We now finish the proof of Proposition 2.12. Let be an -module generator of . Write for the deformation corresponding to the canonical map . Then we can write

 ρI=[~τ1αe+β~τ2],

where and are maps (here we identify with its composition with ). Define

 ρ+I:G→GLn(A)g↦[~τ1(g)α(g)e~τ2(g)].

We must check that is a homomorphism. This follows easily from the fact that is a homomorphism and the fact that is a direct summand of . Moreover, note that the image of is not contained in because reduces to which is not semi-simple.

Note that is an upper-triangular deformation into . Moreover, since reduces to , it gives rise to an element in which generates an -submodule isomorphic to . Since was arbitrary we conclude that must be infinite which contradicts Assumption 2.5. This concludes the proof of Proposition 2.12. ∎

###### Remark 2.15.

If then and are cyclic -modules by Corollary 7.12 in [BergerKlosin13] which combined with Proposition 2.12 implies finiteness of and . On the other hand given that it is easy to construct an upper-triangular (not necessarily crystalline) lift of to which would suggest that in general , and even (since is reduced), may have positive Krull dimension. Indeed, to see this, let be a cohomology class corresponding to and let be a cohomology class linearly independent from . Then the representation

 ρ=[τ1τ2(f+gX)0τ2]

is a non-trivial lift of to . In particular there is no guarantee that is a finitely generated -module. Since our method of proving modularity relies on that property we will restrict in the following section to the ‘characteristic zero’ part of of which we will demand that it is finite over .

### 2.5. The ring R0τ

Set . For the rest of this article we assume the following:

###### Assumption 2.16.

Assume that is finite.

We then define to be the image of in . It is clear that is a finitely generated -module and an object in . Note that the canonical surjection factors through . Write for the composition of with the map . Write for the ideal of reducibility of . By [BergerKlosin13], Lemma 7.11, we have (in fact equality holds since the opposite inclusion is obvious) and thus induces a surjection .

###### Lemma 2.17.

The quotient is finite.

###### Proof.

This follows immediately from Proposition 2.12 and the surjectivity of .∎

Define to be the closed -subalgebra generated by the set

 S:={\textuptrρ0τ(Frobq)∣q∉Σ}.
###### Lemma 2.18.

The image of under is . Thus is an object in the category LCN(E).

###### Proof.

It is clear that . On the other hand , so the equality holds because is dense in . ∎

We will write for the ideal of reducibility of . By Lemma 2.18 and Lemma 7.11 in [BergerKlosin13] we get that (in fact equality holds) and thus induces a surjection . By Remark 2.11 the quotient is a cyclic -module.

### 2.6. Generic irreducibility of ρ0τ

###### Lemma 2.19.

For any as in (2.1) is irreducible. Here is any of the fields in , where is the total ring of fractions of .

###### Proof.

First note that since is a finitely generated -module and since is assumed to be sufficiently large we may assume that all of the fields are equal to . If any of the representations is reducible write for its semi-simplification with each irreducible, . Then by compactness of for each there exists a -stable -lattice inside the representation space of . This implies that for all and all . Hence splits over into the sum of traces of . Since is a deformation of we easily conclude that with (with respect to some lattice) being a deformation of (). Using the fact that is a deformation of we now deduce that there is an -lattice inside the space of with respect to which is block-upper-triangular (with correct dimensions) and non-semi-simple. When we reduce it modulo , the upper-right shoulder will give rise to an element of order in . Since is arbitrary this contradicts Assumption 2.5. ∎

## 3. The rings Tτ

Let us now define the rings that will correspond to on the Hecke side.

###### Proposition 3.1.

If is irreducible and satisfies

 (3.1) ¯¯¯ρss≅τ1⊕τ2

then there exists a lattice inside so that with respect to that lattice the mod reduction of has the form

 ¯¯¯ρ=[τ1∗0τ2]

and is non-semi-simple.

###### Proof.

This is a special case of [Urban01], Theorem 1.1, where the ring in [loc.cit.] is a discrete valuation ring . ∎

For each representation as in (2.1) let be the set of (inequivalent) characteristic zero deformations of , i.e. crystalline at Galois representations whose reduction equals . Also, let be the set of (inequivalent) crystalline at Galois representations such that there exists a -stable lattice in the space of so that the mod -reduction of equals .

The following is a higher-dimensional analogue of Lemma 2.13(ii) from [SkinnerWiles99]:

One has if .

###### Proof.

Let be a representation such that and let equal its trace. Suppose there exist two lattices in the representation space of such that the reductions of the corresponding representations are given by and with as in (2.1). We now consider the classes of the cocycles corresponding to in . Using Assumption 2.5 above and Corollary 7.8 in [BergerKlosin13] we conclude that the quotient is finite. Thus arguing as in the proof of Proposition 1.7.4 in [BellaicheChenevierbook] but using Proposition 3.1 in [BergerKlosin13] instead of generic irreducibility of to conclude that (see [BellaicheChenevierbook], Proof of Proposition 1.7.2, on how this equality - which follows from Proposition 1.6.4 in [loc.cit.] in the generically irreducible case - is used) we obtain that the existence of with trace and non-split reduction as in (2.1) implies that is 1-dimensional, where .

First note that . Secondly one clearly has that . These two facts imply that the map factors through and that the kernel of the resulting surjection equals . Thus we have , so by the above we conclude that is one-dimensional. This means the corresponding representations of are isomorphic. Since (as noted above) the reductions both factor through this quotient of , and so they are isomorphic as representation of , in contradiction to our assumption. ∎

The following notation will remain in force throughout the paper.

###### Notation 3.3.

Write for the set of isomorphism classes of residual representations of the form (2.1). Set .

###### Remark 3.4.

Assumption 2.16 that is finite is equivalent to assuming that the set is a finite set.

We now fix subsets and of deformations. In our later application these will be taken to correspond to all the modular deformations corresponding to cuspforms of a particular weight and level which are congruent to a fixed Eisenstein series. In particular may be empty.

Whenever we obtain an -algebra map This induces a map

 (3.2) Rtrτ→∏ρ∈ΠτO.
###### Definition 3.5.

We (suggestively) write for the image of the map (3.2) - note that this also depends on the choice of the set - and denote the resulting surjective -algebra map by . Also we will write for the image of , where is induced from the traces of the deformations . Finally we will write for the ideal of reducibility of the pseudo-representation and for the ideal of reducibility of the pseudo-representation .

###### Lemma 3.6.

The maps and factor through and the image of inside respectively.

###### Proof.

Clearly the kernel of contains . Thus the map factors through . Then the claim follows since by Lemma 2.18. ∎

###### Lemma 3.7.

The quotient is cyclic and one has .

###### Proof.

The first part is a consequence of Lemma 2.9 and was already mentioned in Remark 2.11.

By Lemma 7.11 in [BergerKlosin13] we know that . For the opposite inclusion we argue as follows. We need to show that for pseudo-representations.

Put , and write for and for . Let . Since is surjective there exists such that . Then by definition of we have for some pseudo-representations and . Now set for . ∎

One has .

###### Proof.

By Lemma 3.6 the map factors through . By abuse of notation we will denote the induced map also by as in the statement of the Corollary. Then since (with as in section 2.5) we get the corollary. ∎

###### Lemma 3.9.

The quotient is cyclic.

###### Proof.

For this we prove as in Lemma 2.9 that is equal to the smallest closed ideal of generated by the set We note that Assumption 2.4 can again be applied as Definition 3.5 tells us that is induced by the traces of the crystalline deformations . ∎

## 4. The lattice L and modular extensions

We will make a frequent use of the following result that is due to Urban [Urban01]. Let be a Henselian and reduced local commutative algebra that is a finitely generated -module. Since is assumed to be sufficiently large and is reduced we have

 B⊂^B=s∏i=1O⊂s∏i=1E=FB,

where stands for the normalization of and for its total ring of fractions. Write for the maximal ideal of . For any finitely generated free -module , any -submodule which is finitely generated as a -module and has the property that will be called a -lattice.

###### Theorem 4.1 ([Urban01] Theorem 1.1).

Let be a -algebra, and let be an absolutely irreducible representation of on (i.e., composed with each of the projections is absolutely irreducible) such that there exist two representations for in and a proper ideal of such that

• the coefficients of the characteristic polynomial of belong to ;

• the characteristic polynomials of and are congruent modulo ;

• mod and mod are absolutely irreducible;

• .

Then there exist an -stable -lattice in and a -lattice of such that we have the following exact sequence of -modules:

 0→ρ