Potential Automorphy for $G{L}_n$

Abstract

We prove potential automorphy results for a single Galois representation $G_F\to G{L}_n({\overline{\mathbb{Q}}}_l)$ 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}.