Ordinary pseudorepresentations and modular forms

By Preston Wake and Carl Wang-Erickson

Abstract

In this note, we observe that the techniques of our paper “Pseudo-modularity and Iwasawa theory” can be used to provide a new proof of some of the residually reducible modularity lifting results of Skinner and Wiles. In these cases, we have found that a deformation ring of ordinary pseudorepresentations is equal to the Eisenstein local component of a Hida Hecke algebra. We also show that Vandiver’s conjecture implies Sharifi’s conjecture.

1. Introduction

The key technical innovation behind our previous work Reference WWE15 was our definition of an ordinary -dimensional pseudorepresentation of . Using this notion, we were able to study ordinary Galois deformations in the case where the residual representation is reducible. In particular, we constructed a universal ordinary pseudodeformation ring with residual pseudorepresentation . We also showed that the Galois action on the Eisenstein part of the cohomology of modular curves gives rise to an ordinary pseudorepresentation valued in the Eisenstein component of the ordinary Hecke algebra. Studying deformations of ordinary pseudorepresentations, we showed that if Greenberg’s conjecture holds, then certain characteristic localizations of are Gorenstein. Under the same assumption, we also proved an isomorphism .

In this note, we show that the methods of Reference WWE15 can be extended to the whole Eisenstein component , provided that we make stronger assumptions on class groups. Namely, we have to assume that the plus-part of the Iwasawa class group of the relevant cyclotomic field vanishes. When the tame level is , then this is known as Vandiver’s conjecture. When , there are examples where , but it is still often the case that . Assuming , we get an isomorphism (Theorem 4.2.8). As a consequence, we have a new technique to establish the residually reducible ordinary modularity theorem of Skinner and Wiles Reference SW99 over , in some cases (Theorem 5.2.4). We also derive new results on Gorensteinness of Hecke algebras (Corollary 5.1.1) and prove new results toward Sharifi’s conjecture (Corollary 5.1.2). In particular, we prove that is Gorenstein when , an implication that was known previously only after assuming Sharifi’s conjecture Reference Wak15b. Previous partial results in this direction by Skinner-Wiles Reference SW97 and Ohta Reference Oht05 require much stronger conditions on class groups.

As well as proving these new results, we review the most novel parts of Reference WWE15. In this way, this note may serve as an introduction to Reference WWE15.

1.1. Ordinary pseudorepresentations

A -dimensional pseudorepresentation of with values in a ring is the data of two functions that satisfy conditions as if they were the trace and determinant of a representation . The (fine) moduli of pseudorepresentations may be thought of as the coarse moduli of Galois representations produced by geometric invariant theory Reference WE15, Thm. A. In this respect, our results suggest that coarse moduli rings of Galois representations are most naturally comparable with Hecke algebras. Indeed, most previous theorems have been established where is a deformation ring for a residually irreducible Galois representation, in which case the fine and coarse moduli of Galois representations are identical.

The ordinary condition is somewhat subtle when applied to pseudorepresentations. For example, if one thinks about the case when is a field, a representation is defined to be ordinary when is reducible with a twist-unramified quotient. While knows nothing about which of the two Jordan-Hölder factors is the quotient, can often distinguish them. This allows for the definition of an ordinary pseudorepresentation of , which we extend to non-field coefficients. We overview this and other background from Reference WWE15 in §§2 and 3.

1.2. Outline of the proof

The étale cohomology of compactified modular curves defines a -module over the cuspidal quotient of . However, is a representation (i.e., locally free -module) if and only if is Gorenstein, which is not always true. Nonetheless, always induces an ordinary -valued pseudorepresentation deforming the residual pseudorepresentation . This pseudorepresentation extends to , resulting in a surjection .

This map is naturally a morphism of augmented -algebras, where is an Iwasawa algebra. The augmentation ideals

correspond to the Eisenstein family of -adic modular forms and the reducible locus of Galois representations, respectively. We can show that certain Iwasawa class groups surject onto , which is the cotangent module relative to the reducible family. The Vandiver-type assumption is used to show that one of the relevant Iwasawa class groups is cyclic. Using a version of Wiles’s numerical criterion Reference Wil95, Appendix, with the class groups playing the role of Wiles’s , we can show that this forces to be an isomorphism.

One aspect of this proof is that we are able to control a “cotangent space” of a pseudodeformation ring in terms of Galois cohomology. We use Galois cohomology groups with coefficients in . Such control is critical to proving theorems in the residually irreducible case, and theorems for pseudorepresentations were lacking because this control was not as available. In our situation, the relevant Galois cohomology is determined by class groups.

2. Background: Iwasawa theory and Hecke algebras

This section is a brief synopsis of §§2, 3, and 6 of Reference WWE15. We overview background information from Iwasawa theory and ordinary -adic Hecke algebras and modular forms.

2.1. Iwasawa algebra and Iwasawa modules

We review §2 of Reference WWE15.

Let be a prime number, and let be an integer such that . Let

be an even character. Let , where

is the Teichmüller character. Our assumption on implies that each of these characters is a Teichmüller lift of a character valued in a field extension of . By abuse of notation, we also use to refer to these characters.

We assume that satisfies the following conditions:

(a)

is primitive,

(b)

if , then , and

(c)

if , then .

A subscript or on a module refers to the eigenspace for an action of . A superscript will denote the -eigenspace for complex conjugation. Let denote the set of primes dividing along with the infinite place, and let be the unramified outside Galois group. We fix a decomposition group and let denote the inertia subgroup. Let denote the -adic cyclotomic character.

Fix a system of primitive -th roots of unity such that for all . Let and let .

Let be the class group, and let

There is action of on . By class field theory, where is the maximal pro-, abelian, unramified extension of . A closely related object is , where is the maximal pro- abelian extension unramified outside .

Let . Let . We write for when is implicit. This is a local component of the semilocal ring and is abstractly isomorphic to , where is the extension of generated by the values of . Notice that the action on gives an isomorphism .

Let and be the functors on -modules as defined in Reference Wak15a, §2.1.3. Namely, as -modules, but acts on (resp. ) as (resp. ) acts on . We sometimes, especially when using duality, are forced to consider -modules with characters other than , but we use these functors to make the actions factor through so we can treat all modules uniformly.

We define to be a generator of the principal ideal . By the Iwasawa Main Conjecture, it may be chosen to be a power series associated to a Kubota-Leopoldt -adic -function.

Consider the -valued character , where is the quotient map. We define -modules and to be with acting by and , respectively.

2.2. Duality and consequences

We review some relevant parts of §6 of Reference WWE15. To compare conditions on various class groups, we use the following -adic version of Poitou-Tate duality. It is a generalization of Reference WWE15, Prop. 6.2.1.

Here, is a number field, and is an open dense subset of . The compactly supported cohomology is defined to be the cohomology of

where and are the standard complexes that compute Galois cohomology.

Proposition 2.2.1.

Let be a finitely generated projective -module equipped with a continuous action of , unramified at places outside , and let be a finitely generated -module. Then there is a quasi-isomorphism

that is functorial in . There is a similar quasi-isomorphism when and are swapped, i.e.,

Here is the dual representation .

Proof.

We prove the first quasi-isomorphism. The proof of the second is similar.

For the case where , see Reference Nek06, Prop. 5.4.3, p. 99 or Reference FK12, §1.6.12. Then we have quasi-isomorphisms

where the first comes from the case, and the second is standard (for example Reference Wei94, Exer. 10.8.3). To prove the proposition, we are reduced to producing a quasi-isomorphism

This follows from Reference LS13, Prop. 3.1.3 (and its compactly supported analog, which, as remarked in the proof of Proposition 4.1.1 of Reference LS13, can be established similarly).

The proposition yields spectral sequences with second pages

These spectral sequences are functorial in .

We record the influence of the assumption that . In the proof, we make use of the following lemma on the structure of -modules.

Lemma 2.2.5.

Let be a finitely generated -module. Say that is type if is free, type if is torsion and has projective dimension , and type if is finite. Then is type if and only if for all . Moreover, is type if and only if is torsion and has no non-zero finite submodule.

Proof.

See Reference Jan89, §3.

The following is well-known to experts.

Proposition 2.2.6.

if and only if is a free -module of rank .

Proof.

As in Reference WWE15, Cor. 6.3.1, we have . Since is the Pontryagin dual of , (classical) Poitou-Tate duality implies that

We can then deduce that , as in Reference WWE15, Cor. 6.1.3.

Analyzing spectral sequence Equation 2.2.4 above with and , we see that for cohomological dimension reasons unless . We find that

Then is a free -module if and only if by Lemma 2.2.5.

The fact that the rank is then 1 follows from class field theory and Iwasawa’s theorem. Indeed, class field theory implies that there is an exact sequence

where is an Iwasawa local unit group at , and Iwasawa’s theorem implies that is free of rank 1 over (see Reference WWE15, §2.1 and the references given there; note that there is no contribution from the local units at primes dividing because is primitive). Since is -torsion, this implies that has rank .

2.3. Hecke algebras

We review §3 of Reference WWE15. Let

where the subscript denotes the eigenspace for the diamond operators. Let and denote the Hida Hecke algebras acting on and , respectively. There is a unique maximal ideal of containing the Eisenstein ideal for ; let and be the localizations of and at the Eisenstein maximal ideal, and let and . Let be the Eisenstein ideal, and let be the image of .

By Hida theory, each of , , , and is finite and flat over . There are also canonical isomorphisms of -modules , , and (see Reference WWE15, Prop. 3.2.5), making an augmented -algebra.

3. Ordinary pseudorepresentations

We define ordinary pseudorepresentations and show that they are representable by an ordinary pseudodeformation ring , recapitulating results of Reference WWE15. In particular, we will review background on pseudorepresentations, Cayley-Hamilton algebras, and generalized matrix algebras from §5 of Reference WWE15.

We highlight the following important points:

The definition is “not local”, in the sense that it does not have the form is ordinary if is ordinary”.

When is a field, we can say that is ordinary if there exists an ordinary -representation such that is induced by .

While not every pseudorepresentation comes from a representation, we fix this problem by broadening the category of representations to include Cayley-Hamilton representations. We first define ordinary Cayley-Hamilton representations, and then say a pseudorepresentation is ordinary when there exists an ordinary Cayley-Hamilton representation inducing it.

We fix some notation. We use the letter to denote the functor that associates to a representation its induced pseudorepresentation. Let , which is the -valued residual pseudorepresentation induced by the Galois action on . Write for the pseudodeformation ring for Reference WWE15, §5.4 with universal object . In this section, will denote a Noetherian local -algebra with residue field . If , then denotes the image of .

3.1. Representations valued in generalized matrix algebras

As in Reference BC09, §1.3, we say that a generalized matrix algebra over is an associative -algebra equipped with an -algebra isomorphism

which we call a GMA structure. That is, we have as -modules for some -modules and , and there is an -linear map such that the multiplication in is given by 2-by-2 matrix multiplication. In this case, is called the scalar subring of and is called an -GMA.

A GMA representation with coefficients in and residual pseudorepresentation is a homomorphism , such that is an -GMA, and such that in matrix coordinates, is given by

with , , and . We emphasize the fact that we fix the order of the diagonal characters.

Given such a , there is an induced -valued pseudorepresentation, denoted , given by and ; cf. Reference WE15, Prop. 2.23.

3.2. Universality

A Cayley-Hamilton representation with scalar ring and residual pseudorepresentation is the data of a pair , where is an associative -algebra such that is a pseudorepresentation deforming . These data must satisfy an additional Cayley-Hamilton condition that, for all , must satisfy the characteristic polynomial associated to by . If is a GMA representation, then is a Cayley-Hamilton representation.

For our purposes, the important properties of Cayley-Hamilton representations are the following (see Reference WE15, Prop. 3.6).

There is a universal Cayley-Hamilton representation with residual pseudorepresentation , and the induced pseudorepresentation of is equal to the universal deformation of .

is finitely generated as an -module, and is continuous for the natural adic topology from on .

admits various -GMA structures making a GMA representation over .

In particular, any Cayley-Hamilton representation with residual pseudorepresentation