Potential modularity---a survey

Abstract

A Spitalfields Day at the Newton Institute was organised on the subject of the recent theorem that any elliptic curve over any totally real field is potentially modular. This article is a survey of the strategy of the proof, together with some history.

Written at the request of the editors of the volume "Nonabelian Fundamental Groups and Iwasawa Theory", a book arising from the six month programme "Non-abelian fundamental groups in arithmetic geometry" at the Isaac Newton Institute in 2009. This (final) version of the paper fixes a few typos and a historical inaccuracy