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.
The key technical innovation behind our previous work Reference WWE15 was our definition of an ordinary pseudorepresentation of -dimensional 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.
1.1. Ordinary pseudorepresentations
A pseudorepresentation of -dimensional 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 if and only if -module) is Gorenstein, which is not always true. Nonetheless, always induces an ordinary pseudorepresentation deforming the residual pseudorepresentation -valued This pseudorepresentation extends to . resulting in a surjection ,.
This map is naturally a morphism of augmented where -algebras, is an Iwasawa algebra. The augmentation ideals
correspond to the Eisenstein family of modular forms and the reducible locus of Galois representations, respectively. We can show that certain Iwasawa class groups surject onto -adic 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 Hecke algebras and modular forms. -adic
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:
if then , and ,
if then ,.
A subscript or on a module refers to the eigenspace for an action of A superscript . will denote the for complex conjugation. Let -eigenspace 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 cyclotomic character. -adic
Fix a system of primitive roots of unity such that -th 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 as defined in -modulesReference Wak15a, §2.1.3. Namely, as but -modules, acts on (resp. as ) (resp. acts on ) We sometimes, especially when using duality, are forced to consider . with characters other than -modules 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 character -valued 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 version of Poitou-Tate duality. It is a generalization of -adicReference 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.
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.
The following is well-known to experts.
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 , residual pseudorepresentation induced by the Galois action on -valued Write . for the pseudodeformation ring for Reference WWE15, §5.4 with universal object In this section, . will denote a Noetherian local with residue field -algebra 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 isomorphism -algebra
which we call a GMA structure. That is, we have as for some -modules -modules and and there is an , map -linear 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 and such that in matrix coordinates, -GMA, 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 , pseudorepresentation, denoted -valued given by , and cf. ;Reference WE15, Prop. 2.23.
A Cayley-Hamilton representation with scalar ring and residual pseudorepresentation is the data of a pair where , is an associative such that -algebra 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 and -module, is continuous for the natural adic topology from on .
admits various structures making -GMA a GMA representation over .
In particular, any Cayley-Hamilton representation with residual pseudorepresentation