Automorphy of some residually S_5 Galois representations

TL;DR

This paper establishes automorphy lifting theorems for certain 2-dimensional Galois representations over totally real fields, specifically addressing an exceptional case where traditional methods are insufficient.

Contribution

It introduces a novel approach combining patching with p-adic approximation to handle the exceptional residual representations of projective image PGL_2(F_5).

Findings

Proves automorphy lifting in the exceptional case for residual PGL_2(F_5) representations.

Develops a new method combining patching with automorphy by p-adic approximation.

Extends automorphy results to cases previously inaccessible by standard techniques.

Abstract

We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic 5, the residual representation has projective image PGL_2(F_5), and the fixed field of its kernel contains a primitive 5th root of unity. In this case the Taylor-Wiles patching method does not suffice and we prove our theorem by combining patching with the technique of automorphy by p-adic approximation.