Potential Automorphy for $GL_n$

TL;DR

This paper establishes potential automorphy results for Galois representations over CM fields by modifying motives and using log geometry, advancing the understanding of automorphic forms and Galois representations.

Contribution

It introduces a novel approach using the $p,q$ switch trick and motive modifications to prove potential automorphy for $GL_n$ representations over CM fields.

Findings

Potential automorphy results for Galois representations over CM fields.

Use of the $p,q$ switch trick to break self-duality in motives.

Proof of ordinarity of certain $p$-adic representations using log geometry techniques.

Abstract

We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in \cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain $p$-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in \cite{tap}.