Ordinary pseudorepresentations and modular forms

Abstract

Abstract. In this note, we observe that the techniques of our paper "Pseudomodularity 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.

Pseudorepresentations of weight one are unramified

“…The idea behind this definition, as we shall see shortly below, is to capture the notion that the product (h − ψ(h))(φ − α) is identically zero, rather than just of the form 0 * 0 0 . There is presumably a close relationship between this definition and the definition of ordinary pseudo-characters in Wake, Wang-Erickson (see [WWE17] and §7.3 of [WE18]), although in our context it is important that we can work in non-p distinguished situations by choosing an eigenvalue of Frobenius, which amounts to a partial resolution of the corresponding deformation rings (presumably such modifications could also be adapted to [WE18]). On the other hand, we do exploit the crucial idea due to Wang-Erickson that the notion of ordinarity for pseudo-representations should be a global rather than local condition.…”

Deformation Conditions for Pseudorepresentations

“…The following lemma gives the general procedure for doing this. A special case of this lemma was employed in [WWE18a,WWE17a].…”

“…In our previous work [WWE18a,WWE17a], we considered the ordinary condition on two-dimensional global Galois representations (see also [CS19] and [WE18a,Section 7]). However, the ordinary condition is of a rather different flavor, as it does not apply readily to a finite-cardinality Z p [G]-module without extra structure.…”

Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

“…Combine Lemma 7.12 and Theorem 7.9. See [WWE15a] for a deformation-theoretic definition of ordinary pseudorepresentation.…”

“…This is well-illustrated through the explicit example of a 2-dimensional global ordinary pseudodeformation ring, which we discuss in §7.3. These ordinary pseudodeformation rings are compared to Hecke algebras in [WWE15b,WWE15a].…”