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 admits the structure of a GMA representation with residual pseudorepresentation . Note that by definition of GMA representation, we have insisted that any such GMA structure satisfies Equation 3.1.1.

Given a GMA structure on , we will write

for the decomposition of as in Equation 3.1.1 and write for the corresponding coordinates of . Similarly, for any GMA representation deforming , we will write for the induced coordinate decomposition.

3.3. Reducibility

It will be important to understand the notion of a reducible pseudorepresentation and the reducibility ideal in for an -valued pseudodeformation of . We call reducible if for characters such that and . Otherwise, is called irreducible. Equivalently, is reducible if for some GMA representation with scalar ring such that is zero.

Since and generate and respectively as -modules, is reducible exactly when the image of in under multiplication vanishes under . This holds true for any choice of GMA structure. Consequently, we call the image of in the reducibility ideal of , and its image in is the reducibility ideal for .

3.4. Ordinary GMA representations

We will say that a representation of on a -dimensional -adic vector space is ordinary if there exists a -dimensional quotient representation such that is unramified. A representation of is ordinary if is ordinary. This notion of ordinariness is relatively restrictive compared to other uses of the term, but it will suit our purpose of studying Galois representations associated to ordinary modular forms.

Relative to the ordering of factors in Equation 3.1.1, we have the following definition of an ordinary Cayley-Hamilton representation.

Definition 3.4.1.

Let be a Cayley-Hamilton representation with scalar ring and induced pseudorepresentation . We call ordinary provided that it admits a GMA structure such that

(1)

, and

(2)

,

where is the -adic cyclotomic character.

Remark 3.4.2.

The condition that and are locally -distinguished, i.e., , is critically necessary to making this definition sensible. This condition is equivalent to the assumption (b) of §2.1.

Example 3.4.3.

Let be a representation on a free rank 2 -module with induced pseudorepresentation . This amounts to a Cayley-Hamilton representation over valued in . We claim that is ordinary according to Definition 3.4.1 if and only if there is a quotient character of such that and . Indeed, the latter condition is true if and only if there exists a basis such that satisfies Definition 3.4.1. Observe that a choice of basis for induces a GMA structure on .

This definition is slightly more restrictive than the definition of “ordinary representation” given by some authors. Nonetheless, our definition can be useful for studying those more general representations (see §5.2).

Example 3.4.4.

The motivating example of an ordinary GMA representation is the -module given by the cohomology of modular curves. There exist isomorphisms of -modules , where is the dualizing module of (see Reference WWE15, §3.4). Because , any such isomorphism determines a GMA representation ; moreover, there exists a choice of isomorphism such that

satisfies conditions (1) and (2) of Definition 3.4.1 relative to the resulting GMA structure.

Here is a summary of the results of Reference WWE15 on ordinary Cayley-Hamilton representations.

Proposition 3.4.5.
(1)

There is a universal ordinary Cayley-Hamilton algebra , a quotient of , such that a Cayley-Hamilton representation with residual pseudorepresentation is ordinary if and only if its map factors through .

(2)

There is a universal reducible ordinary Cayley-Hamilton algebra , a quotient of , such that a Cayley-Hamilton representation with residual pseudorepresentation is reducible ordinary if and only if its map factors through .

(3)

is an ordinary Cayley-Hamilton representation.

Proof.

Statement (1) comes from Reference WWE15, Proposition 5.9.7, (2) comes from Proposition 7.3.1, and (3) comes from Theorem 7.1.2.

For the rest of the paper, we fix a GMA structure on such that (1) and (2) of Definition 3.4.1 are satisfied. By its universal property (part (1) of Proposition 3.4.5), this induces a GMA structure on all ordinary Cayley-Hamilton representations. We refer to an ordinary Cayley-Hamilton representation with this choice of GMA structure as an ordinary GMA representation.

3.5. Ordinary pseudorepresentations

Having established a notion of ordinary GMA representation, we can now define ordinary pseudorepresentations.

Definition 3.5.1.

Let be a pseudorepresentation deforming . Then we call ordinary if there exists an ordinary GMA representation with scalar ring such that .

We write for the scalar ring of , which admits a pseudorepresentation defined as the composition of with . We have shown in Reference WWE15, Thm. 5.10.4 that the ring represents the functor of ordinary pseudodeformations of , with universal object . We write for the reducibility ideal of .

Remark 3.5.2.

The reason for introducing GMA representations is to make Definition 3.5.1: not every pseudodeformation of comes from a representation, but every pseudodeformation comes from a GMA representation.

By definition, we see that the modular pseudorepresentation arising from (as defined in Example 3.4.4) is ordinary. It can be extended to an -valued pseudorepresentation with the following properties.

Proposition 3.5.3.

There is a pseudorepresentation that is ordinary, deforms , and satisfies . The corresponding map is:

(1)

a map of augmented -algebras, where the augmentation ideals are the reducibility ideals of and of , and

(2)

surjective.

Proof.

The pseudorepresentation is constructed by gluing together with the Eisenstein pseudorepresentation, and it follows that and that the reducibility ideal is Reference WWE15, Cor. 7.1.3.

Then (2) follows from Reference WWE15, Lem. 7.1.4, and follows from the fact that Reference WWE15, Prop. 7.3.1.

Consequently, the functor of reducible ordinary pseudorepresentations is represented by , and is the scalar ring of .

4. New results

With the overview of Reference WWE15 complete, we prove the main theorems.

4.1. Reducible representations and class groups

Let us write and in GMA form as

Our goal is to control using Galois cohomology. For this, we use the surjection discussed in §3.3. We also use the fact that the map , which exists because the target receives a reducible ordinary GMA representation, is surjective Reference WWE15, Prop. 7.3.1(4). (Actually, it is an isomorphism, but that will not be used here.) Composing these surjections, we have

(cf. Reference WWE15, Prop. 5.7.2). The main result is the following.

Proposition 4.1.2.

is determined as follows.

(1)

There exists a natural isomorphism

(2)

Assume that . Then there exists a natural isomorphism

Moreover, is free of rank over .

First, some lemmas. We will need the following notation, which is particular to §4.1. We will abbreviate and to and , respectively. For , we will let denote , where runs over prime divisors of . Likewise, for a -module (with trivial -action), we will write for and write for .

Lemma 4.1.3.

For any finitely generated -module and any we have . In particular, .

Proof.

Let , where is the maximal ideal of . By Nakayama’s lemma, it suffices to show that . By the Euler characteristic formula and local Tate duality, we have

The Galois action on and is via the characters and , respectively. Both these characters are non-trivial at because is trival at and we assume that is primitive. This implies that the groups appearing in Equation 4.1.4 are , so .

Lemma 4.1.5.

Functorially in finitely generated -modules , we have isomorphisms

and

Proof.

For Equation 4.1.7, Reference BC09, Thm. 1.5.5 tells us that there is a natural -linear injective map when is a cyclic module. But nothing about the proof depends upon being cyclic, so we have injectivity in general.

An element of results in a short exact sequence of -modules . Choose an element mapping to and write for the map . Then we have a GMA representation

By the universal property of , there exists a unique map induced by this representation. This construction is an inverse to the construction of in Reference BC09, Thm. 1.5.5, so we have proved that Equation 4.1.7 is an isomorphism.

Recall the definition of as the cohomology of a mapping cone. Note that for any . This follows from the fact that the residual character is non-trivial on decomposition groups at all primes dividing . Consequently, is naturally isomorphic to the subset of whose restriction to is zero. By Lemma 4.1.3, .

We have an injection

like above, because any extension of by realized by a GMA map

induces a trivial extension of -representations. The same argument as above shows that is surjective.

Now, we prove Proposition 4.1.2.

Proof.

We will use the Yoneda lemma for finitely generated -modules to determine and . Indeed, and are finitely generated by the fact that is a finitely generated -module (see §3.2) and the construction of .

Let’s begin with . Let be an arbitrary finitely generated -module. We will use the spectral sequence of Proposition 2.2.1:

From Reference WWE15, Cor. 6.3.1, we have

The spectral sequence degenerates to yield a functorial isomorphism

From this and Equation 4.1.6, we see that and represent the same functor . Then the Yoneda lemma implies that .

We now similarly calculate . Let be any finitely generated -module, and consider the spectral sequence

As in the proof of Proposition 2.2.6, we have . Similar arguments show that and, using the weak Leopoldt conjecture (see Reference NSW08, Thm. 10.3.22), that . Then the spectral sequence degenerates to yield . As above, using Equation 4.1.7 and the Yoneda lemma, we obtain .

4.2. A version of Wiles’s numerical criterion

Having controlled in terms of Iwasawa class groups, we will now make use of a version of Wiles’s numerical criterion Reference Wil95, Appendix to prove our theorem. We thank Eric Urban for suggesting that the numerical criterion might be used to improve an earlier version. We follow the exposition of Reference dSRS97.

Consider the diagram

where arises from Proposition 3.5.3. In this situation, de Smit, Rubin, and Schoof prove the following theorem on the way to giving a proof of Wiles’s criterion.

Theorem 4.2.1 (Reference dSRS97, Thm. of §3 (p. 9)).

The map is an isomorphism of complete intersections if and only if .

We will use this theorem to show that is an isomorphism under the assumption that . The proof is inspired by the proof of Criteria I in Reference dSRS97. We first prove some preliminary results. The notation that appears in the theorem refers to Fitting ideals, which are reviewed in Reference dSRS97, §1. We will make frequent use of the following well-known properties of Fitting ideals (see Reference dSRS97, Prop. 1.1).

Lemma 4.2.2.

Let be a ring, let be a finitely presented -module, and let be an -algebra. Then:

(1)

.

(2)

.

Before stating the next lemma, we summarize some known results about the Eisenstein ideals and (see Reference WWE15, Prop. 3.2.5 and Lem. 3.2.9): the natural map is an isomorphism, , and . In particular, is a faithful -module.

Lemma 4.2.3.

We have as ideals of .

Proof.

By Lemma 4.2.2(1), we have

Since , we have .

Since is a faithful -module, Lemma 4.2.2(2) implies that . Applying Lemma 4.2.2(1), we have . Recalling that , another application of the same lemma gives .

Proposition 4.2.4.

Assume that . Then the -modules , and are all isomorphic and they all have Fitting ideal over equal to .

Remark 4.2.5.

Sharifi has studied a map from class groups to similar to the one that appears in the following proof; see the map of Reference Sha07, Thm. 5.2.

Proof.

From Equation 4.1.1, we have the surjections

By Proposition 4.1.2, we have and , and so by Proposition 2.2.6, we have . Hence we have a surjection

But by the previous lemma, . This implies that the is finite (see Reference Wak15a, Lem. A.7 for example), and hence , since has no finite submodule by Ferrero-Washington Reference FW79. Thus is an isomorphism, and therefore so are all of the maps in Equation 4.2.6.

Since the modules are all isomorphic, it suffices to compute , which is well-known to be . Indeed, by Ferrero-Washington and Lemma 2.2.5, has projective dimension 1. Therefore the ideal is principal. This implies (see Reference Wak15a, Lem. A.6, for example) which is by definition.

Lemma 4.2.7.

We have and, in particular, . The restriction of to induces an isomorphism

Proof.

That and the fact that is an isomorphism follows from Reference WWE15, Prop. 3.2.5. To see that , note that is a surjection of free -modules of distinct rank, and so is a non-zero -free direct summand of .

Theorem 4.2.8.

Assume that . Then is an isomorphism of complete intersections.

Proof.

By Theorem 4.2.1 it suffices to show that

By Lemma 4.2.2(1) and Proposition 4.2.4, we have

On the other hand, Lemma 4.2.2(2) implies that , and, since is surjective, we have . This implies that . Now we have

But we know that , and , so we have

Since is an isomorphism by Lemma 4.2.7, we conclude that . It then follows from Lemma 4.2.7 that .

Remark 4.2.9.

This answers in the affirmative a question of Sharifi Reference Sha09, §5 whether is Gorenstein when . As noted in Reference Sha09, it follows from work of Ohta that is Gorenstein when , and so our result improves on Ohta’s. It was proven in Reference Wak15b, Thm. 1.2 that is Gorenstein when under the additional assumption of Sharifi’s conjecture.

5. Applications

5.1. Toward Sharifi’s conjecture

We have the following immediate corollary.

Corollary 5.1.1.

Assume that and that is cyclic. Then the ideals , , and are all principal, and both and are complete intersections.

Proof.

By Proposition 4.2.4 we have , and so if is cyclic, then each ideal is principal by Nakayama’s lemma. Since is faithful as an -module, it must be generated by a non-zero divisor. Then since is a complete intersection and is generated by a regular sequence, is a complete intersection (see Reference BH93, Thm. 2.3.4(a), p. 75, for example). Theorem 4.2.8 implies that is a complete intersection.

These results have applications to Sharifi’s conjecture. Recall that Sharifi’s conjecture states that two maps and are isomorphisms Reference Sha11. In Reference FK12Reference FKS14 this conjecture was refined to state that and are mutually inverse. (See also Reference WWE15, §8.1 for a review of Sharifi’s conjecture using the same notation as this paper.)

Corollary 5.1.2.

Consider the maps

defined by Sharifi. If , then is an isomorphism. If, in addition, is cyclic, then is an isomorphism as well. Finally, if, in addition, has no multiple root, then they are mutual inverses.

Proof.

By Theorem 4.2.8, if , then is Gorenstein. It is known that if is Gorenstein, then is an isomorphism (see Reference Sha11, Prop. 4.10). If is cyclic, then the previous corollary implies that and are complete intersections, and hence Gorenstein. The result now follows by work of Fukaya-Kato.

Indeed, since is Gorenstein and is a dualizing module over (see Reference WWE15, Cor. 3.4.2), we see that , which has no -torsion. Moreover, the fact that is Gorenstein implies the condition of Fukaya-Kato by Reference FK12, §7.2.10. Then the final two claims follow from Theorems 7.2.8 and 7.2.7 of Reference FK12, respectively.

5.2. Residually reducible modularity

In this subsection, we deduce from the main Theorem 4.2.8 a new proof of a known modularity result for ordinary Galois representations. This may be viewed as a verification that our notion of ordinary GMA representation in §3.4 can be usefully applied to ordinary Galois representations as they are usually known. We emphasize that from now on we use the word “ordinary” exclusively to refer to Definition 3.4.1.

Remark 5.2.1.

Unlike the rest of this paper, we are not assuming the running assumptions about , , and from §2.1 in this subsection. Rather, we will verify that the running assumptions follow from the assumptions of Theorem 5.2.4.

One of the main results of Skinner and Wiles’s work Reference SW99 is

Theorem 5.2.2 (Skinner-Wiles).

Suppose that is continuous, irreducible, and ramified at finitely many primes, where is a -dimensional -vector space, is a finite extension, and is odd. Suppose that and that

(1)

,

(2)

is conjugate to ,

(3)

is odd, where is a finite order character and .

Then comes from a modular form.

Remark 5.2.3.

Our conventions for modular forms and their Galois representations follow those of Reference FK12 and differ slightly from those of Reference SW99. For us, the phrase comes from a modular form means that is isomorphic to , where the map is induced by a cuspidal eigenform with coefficients in . By Reference FK12, 1.2.9, 1.5.8, we see that if is modular, then the weight of must be the integer of (3).

The representation comes from a modular form in our sense if and only if comes from a modular form in the sense of Reference SW99. With this in mind, the theorem is a restatement of the theorem stated in Reference SW99, p. 6.

We can give a new proof of this result in certain cases, following directly from Theorem 4.2.8. Among the additional restrictions, the serious ones are:

(iii)

that is a lift of “minimal level”,

(iv)

that a Vandiver conjecture type condition holds for the relevant isotypic parts of the class group.

Now fix and as in Theorem 5.2.2. Let be the prime-to- part of the conductor of , and write . We will treat these both as characters of and as Dirichlet characters of modulus .

Theorem 5.2.4.

In addition to the conditions of Theorem 5.2.2, assume that

(i)

and ;

(ii)

is ramified at when ;

(iii)

is ramified only at primes dividing ;

(iv)

, and if is unramified at , assume also that .

Then comes from a modular form.

The proof of Theorem 5.2.4 relies on applying Theorem 4.2.8. In order to apply that theorem, we will need to relate condition (2) of Theorem 5.2.2 – which is the “ordinary” condition in Reference SW99 – to our ordinary condition on GMA representations and on pseudorepresentations, established in Reference WWE15 and reviewed in §3. Note that condition (1) implies condition (b) of §2.1, so our definition of ordinary applies in this situation. We will see that either is ordinary in our sense or is unramified at and is ordinary in our sense.

The following lemma is standard, following from Clifford theory. We provide a proof for lack of a reference.

Lemma 5.2.5.

Given any satisfying the conditions of Theorem 5.2.2, is reducible. Moreover, there is a unique quotient character of such that .

Proof.

Let and . By conditions (2)-(3) of Theorem 5.2.2, we have , and is a quotient of . Since and has finite order, we see that and are distinct. Moreover, since they are restrictions of distinct characters of , they are not Frobenius conjugate.

The result now follows from Clifford theory, which states that the restriction of an irreducible representation to a normal subgroup is semi-simple and consists of conjugate representations. Indeed, when is indecomposable, must be reducible and indecomposable, and we let be the quotient character of . We see that . Otherwise, is semi-simple, and since and are not conjugate, must be reducible and semi-simple. We let be the summand of satisfying .

We note that our definition of ordinary in Definition 3.4.1 depends on the residual pseudorepresentation. Below, we will sometimes write is ordinary” to mean is ordinary as a deformation of ”. Let and let , where . We are now ready to prove the following.

Lemma 5.2.6.

Let be as in Theorem 5.2.2 and also satisfy condition (iii) of Theorem 5.2.4. If is ramified at , then is an ordinary deformation of . If is unramified at , then either is an ordinary deformation of or is an ordinary deformation of .

Proof.

We will use the criterion for a representation to be ordinary given in Example 3.4.3. By Lemma 5.2.5, we see that is reducible. We will label the Jordan-Hölder factors in two different ways. First we write with . By the same lemma, we see that is a quotient of .

We also write with and . (The labeling is meant to reflect the coordinate labels of a -matrix and the ordering convention of Equation 3.1.1.) We see that if , then is a quotient character such that and , so is ordinary by Example 3.4.3. Moreover, if is ramified at , we see that , and so, since , we must have .

Now assume that (and so is unramified at ), and let . Since is unramified at , one can verify that satisfies the conditions of Theorem 5.2.2. We apply Lemma 5.2.5 to , and we get with a quotient and . Since is unramified at , we must have . We also write with and . We see that . Hence we have

so is ordinary by Example 3.4.3.

The following lemma prepares us to apply Theorem 4.2.8 to prove Theorem 5.2.4.

Lemma 5.2.7.

Assume that , , and satisfy the conditions imposed by Theorems 5.2.2 and 5.2.4. Then , , and satisfy the running assumptions of §2.1. Also, if is unramified at , then satisfies these running assumptions.

Proof.

The conditions and imposed in §2.1 are implied by condition (i) of Theorem 5.2.4 and the definition of as the tame level of .

As we have already noted, and satisfy condition (1) of Theorem 5.2.2, which is equivalent to condition (b) of §2.1. Thus it remains to verify conditions (a) and (c).

Suppose that . We claim that the existence of implies that cannot be or . Indeed, in these cases, , and Stickelberger’s theorem implies that (see e.g. Reference Was82, Prop. 6.16 and Thm. 6.17, p. 102, and note that ). However, Reference Rib76, Prop. 2.1 implies that the irreducible representation leaves stable a lattice such that the resulting residual representation is not diagonalizable and has as a subrepresentation. Moreover, because satisfies the conditions of Theorem 5.2.2, and since is ramified at , the argument of Lemma 5.2.6 implies that has as a quotient representation. This implies that is a non-trivial extension of by that is split upon restriction to . This extension gives rise to a non-zero element of , so , a contradiction. Hence, if , then and is primitive (so (a) is true), and (so (c) is true).

Now suppose , making (c) satisfied. Assumption (i) implies that our running assumption that is satisfied. We have that is primitive of modulus either or by definition of . Assumption (ii) rules out the case that is primitive of modulus , so (a) holds.

When is unramified at , we wish to show that satisfies the running assumptions. Since is odd, the assumption that is unramified at implies that . As , the conductor of is either or . Since is unramified at , this implies that is ramified at , making its conductor and satisfying assumption (a).

Proof of Theorem 5.2.4.

In Lemma 5.2.7 we checked that the running assumptions of the paper about , , , and are satisfied. By Lemma 5.2.6, we can break the proof into two cases.

Case 1 ( is ordinary).

We consider the case where is ordinary. In this case is an ordinary pseudorepresentation, because is an ordinary (GMA) representation inducing it. Therefore there is a unique map corresponding to . Assumption (iv) allows us to apply Theorem 4.2.8 so that . Then the ordinary -adic modular eigenform determined by satisfies , which implies since is irreducible. Condition (3) implies that this modular form has weight ; consequently, is classical by Reference Hid86, Thm. I, and is modular.

Case 2 ( is unramified at and is ordinary).

We want to apply Theorem 4.2.8 to now. Assumption (iv) allows us to apply Theorem 4.2.8, so that there is an isomorphism , where is for . Then we have a unique map corresponding to , and the rest of the argument is the same as above.

Remark 5.2.8.

We see in the proof that for each , only one of the two groups and must be zero. Moreover, if is ramified at , we can be certain that it is that must be .

Before Reference SW99, Skinner and Wiles gave a different proof of the modularity of under different hypotheses Reference SW97. Among their assumptions is that . This is equivalent to by the reflection principle, and so it is a much stronger assumption than our assumption that . In this way, Theorem 5.2.4 may be seen as an improvement of the method of Reference SW97.

Acknowledgments

The authors would like to thank Romyar Sharifi and Eric Urban for helpful conversations. They also thank the referee for many helpful comments and corrections. Both authors would like to recognize the Simons Foundation for support in the form of AMS-Simons travel grants. The first author was supported by the National Science Foundation under the Mathematical Sciences Postdoctoral Research Fellowship No. 1606255.

Mathematical Fragments

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 .

Equation (2.2.4)
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.

Proposition 2.2.6.

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

Equation (3.1.1)
Definition 3.4.1.

Let be a Cayley-Hamilton representation with scalar ring and induced pseudorepresentation . We call ordinary provided that it admits a GMA structure such that

(1)

, and

(2)

,

where is the -adic cyclotomic character.

Example 3.4.3.

Let be a representation on a free rank 2 -module with induced pseudorepresentation . This amounts to a Cayley-Hamilton representation over valued in . We claim that is ordinary according to Definition 3.4.1 if and only if there is a quotient character of such that and . Indeed, the latter condition is true if and only if there exists a basis such that satisfies Definition 3.4.1. Observe that a choice of basis for induces a GMA structure on .

This definition is slightly more restrictive than the definition of “ordinary representation” given by some authors. Nonetheless, our definition can be useful for studying those more general representations (see §5.2).

Example 3.4.4.

The motivating example of an ordinary GMA representation is the -module given by the cohomology of modular curves. There exist isomorphisms of -modules , where is the dualizing module of (see Reference WWE15, §3.4). Because , any such isomorphism determines a GMA representation ; moreover, there exists a choice of isomorphism such that

satisfies conditions (1) and (2) of Definition 3.4.1 relative to the resulting GMA structure.

Proposition 3.4.5.
(1)

There is a universal ordinary Cayley-Hamilton algebra , a quotient of , such that a Cayley-Hamilton representation with residual pseudorepresentation is ordinary if and only if its map factors through .

(2)

There is a universal reducible ordinary Cayley-Hamilton algebra , a quotient of , such that a Cayley-Hamilton representation with residual pseudorepresentation is reducible ordinary if and only if its map factors through .

(3)

is an ordinary Cayley-Hamilton representation.

Definition 3.5.1.

Let be a pseudorepresentation deforming . Then we call ordinary if there exists an ordinary GMA representation with scalar ring such that .

Proposition 3.5.3.

There is a pseudorepresentation that is ordinary, deforms , and satisfies . The corresponding map is:

(1)

a map of augmented -algebras, where the augmentation ideals are the reducibility ideals of and of , and

(2)

surjective.

Equation (4.1.1)
Proposition 4.1.2.

is determined as follows.

(1)

There exists a natural isomorphism

(2)

Assume that . Then there exists a natural isomorphism

Moreover, is free of rank over .

Lemma 4.1.3.

For any finitely generated -module and any we have . In particular, .

Equation (4.1.4)
Lemma 4.1.5.

Functorially in finitely generated -modules , we have isomorphisms

and

Theorem 4.2.1 (Reference dSRS97, Thm. of §3 (p. 9)).

The map is an isomorphism of complete intersections if and only if .

Lemma 4.2.2.

Let be a ring, let be a finitely presented -module, and let be an -algebra. Then:

(1)

.

(2)

.

Proposition 4.2.4.

Assume that . Then the -modules , and are all isomorphic and they all have Fitting ideal over equal to .

Equation (4.2.6)
Lemma 4.2.7.

We have and, in particular, . The restriction of to induces an isomorphism

Theorem 4.2.8.

Assume that . Then is an isomorphism of complete intersections.

Corollary 5.1.1.

Assume that and that is cyclic. Then the ideals , , and are all principal, and both and are complete intersections.

Corollary 5.1.2.

Consider the maps

defined by Sharifi. If , then is an isomorphism. If, in addition, is cyclic, then is an isomorphism as well. Finally, if, in addition, has no multiple root, then they are mutual inverses.

Theorem 5.2.2 (Skinner-Wiles).

Suppose that is continuous, irreducible, and ramified at finitely many primes, where is a -dimensional -vector space, is a finite extension, and is odd. Suppose that and that

(1)

,

(2)

is conjugate to ,

(3)

is odd, where is a finite order character and .

Then comes from a modular form.

Theorem 5.2.4.

In addition to the conditions of Theorem 5.2.2, assume that

(i)

and ;

(ii)

is ramified at when ;

(iii)

is ramified only at primes dividing ;

(iv)

, and if is unramified at , assume also that .

Then comes from a modular form.

Lemma 5.2.5.

Given any satisfying the conditions of Theorem 5.2.2, is reducible. Moreover, there is a unique quotient character of such that .

Lemma 5.2.6.

Let be as in Theorem 5.2.2 and also satisfy condition (iii) of Theorem 5.2.4. If is ramified at , then is an ordinary deformation of . If is unramified at , then either is an ordinary deformation of or is an ordinary deformation of .

Lemma 5.2.7.

Assume that , , and satisfy the conditions imposed by Theorems 5.2.2 and 5.2.4. Then , , and satisfy the running assumptions of §2.1. Also, if is unramified at , then satisfies these running assumptions.

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Article Information

MSC 2010
Primary: 11F33 (Congruences for modular and -adic modular forms), 11F80 (Galois representations), 11R23 (Iwasawa theory)
Author Information
Preston Wake
Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California 90095-1555
wake@math.ucla.edu
MathSciNet
Carl Wang-Erickson
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
c.wang-erickson@imperial.ac.uk
ORCID
MathSciNet
Communicated by
Romyar T.~Sharifi
Journal Information
Proceedings of the American Mathematical Society, Series B, Volume 4, Issue 6, ISSN 2330-1511, published by the American Mathematical Society, Providence, Rhode Island.
Publication History
This article was received on , revised on , , , , , and published on .
Copyright Information
Copyright 2017 by the authors under Creative Commons Attribution 3.0 License (CC BY 3.0)
Article References
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