On the automorphy of $l$-adic Galois representations with small residual image

TL;DR

This paper develops new automorphy lifting theorems for Galois representations with smaller residual images, extending previous results by weakening the big image requirement through an enhanced Taylor-Wiles approach.

Contribution

It introduces a strengthened Taylor-Wiles method enabling automorphy lifting for Galois representations with small residual images, broadening applicability.

Findings

Proved automorphy lifting theorems under weaker residual image conditions

Extended the Taylor-Wiles method for broader Galois representations

Achieved new automorphy results for conjugate self-dual Galois representations

Abstract

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into $GL_n$. Existing theorems require that the residual representation have 'big' image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor-Wiles method which allows one to weaken this hypothesis.