Automorphy of some residually dihedral Galois representations

TL;DR

This paper proves the automorphy of certain 2-dimensional Galois representations over totally real fields without relying on the Taylor--Wiles hypothesis, advancing the understanding of elliptic curve modularity.

Contribution

It extends automorphy results to residually dihedral Galois representations, bypassing the Taylor--Wiles hypothesis, and applies this to elliptic curves over totally real fields.

Findings

Automorphy established for specific residually dihedral Galois representations.

Results apply to modularity of elliptic curves over totally real fields.

Advances in automorphy lifting techniques without Taylor--Wiles hypothesis.

Abstract

We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called `Taylor--Wiles hypothesis'. We apply this to the problem of the modularity of elliptic curves over totally real fields