Remarks on Automorphy of Residually Dihedral Representations

TL;DR

This paper establishes automorphy lifting results for certain geometric Galois representations over totally real fields, especially those induced from quadratic subfields, overcoming limitations of traditional patching methods.

Contribution

It proves automorphy lifting for residual dihedral representations that do not satisfy Taylor-Wiles hypotheses, expanding the class of automorphic Galois representations.

Findings

Automorphy lifting for residually dihedral representations.

Application to elliptic curves with specific 7-isogeny properties.

Overcoming patching technique limitations.

Abstract

We prove automorphy lifting results for geometric representations $\rho:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\bar{\rho}$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(\zeta_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $GL_2(\mathbb{F}_7)$.