Hecke algebras associated to Λ-adic modular forms

Abstract

We show that if an Eisenstein component of the p-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds.

Ordinary pseudorepresentations and modular forms

“…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 H is Gorenstein when X + = 0, an implication that was known previously only after assuming Sharifi's conjecture [Wak15b]. Previous partial results in this direction by Skinner-Wiles [SW97] and Ohta [Oht05] require much stronger conditions on class groups.…”

Congruences with Eisenstein series and -invariants

“…We also note that Vandiver's conjecture implies that both (Goren) and (Cusp Goren) hold, (see Propostion 3.7) and so we can offer no examples where these hypotheses fail when N = 1. However, when N > 1 one does not expect these hypotheses to hold generally (see [34,Corollary 1.4]).…”

Eisenstein Hecke algebras and conjectures in Iwasawa theory

“…Indeed, if X χ = 0, then H = Λ, the domain and codomain of Υ are 0, and ξ χ is a unit (cf. [W,Remark 1.3]).…”

“…Since it is now known that H is not always Gorenstein ( [W,Corollary 1.4]), one may wonder if Sharifi's conjecture should be weakened to the statement "Υ is pseudo-isomorphism" (cf. Conjecture 4.2 below).…”

“…First, one finds an imaginary quadratic field with non-cyclic p-class group; this provides a character χ of order 2 such that X θ = 0 and such that H is not Gorenstein (cf. [W,Corollary 1.4]). However, h will be weakly Gorenstein by Lemma 5.8 below (or else we have found a counterexample to a famous conjecture!).…”