Ordinary pseudorepresentations and modular forms

TL;DR

This paper uses techniques from recent work to offer a new proof of certain modularity lifting results for residually reducible Galois representations, linking deformation rings and Hida Hecke algebras.

Contribution

It introduces a novel approach to modularity lifting by connecting deformation rings of ordinary pseudorepresentations with Eisenstein components of Hida Hecke algebras.

Findings

Deformation ring of ordinary pseudorepresentations equals Eisenstein component of Hida Hecke algebra.

Provides a new proof of residually reducible modularity lifting results.

Shows Vandiver's conjecture implies Sharifi's conjecture.

Abstract

In this short note, we observe that the techniques of our recent work "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.