Potential modularity---a survey

TL;DR

This survey reviews the recent proof that all elliptic curves over totally real fields are potentially modular, discussing the proof strategy and historical context.

Contribution

It provides a comprehensive overview of the proof strategy for potential modularity of elliptic curves over totally real fields, including historical background.

Findings

Theorem that all elliptic curves over totally real fields are potentially modular.

Summary of the proof strategy and key ideas.

Historical development leading to the recent theorem.

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.