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  • Print publication year: 2011
  • Online publication date: January 2012

Potential modularity – a survey

Summary

Introduction

Our main goal in this article is to talk about recent theorems of Taylor and his co-workers on modularity and potential modularity of Galois representations, particularly those attached to elliptic curves. However, so as to not bog down the exposition unnecessarily with technical definitions right from the off, we will build up to these results by starting our story with Wiles' breakthrough paper [Wil95], and working towards the more recent results. We will however assume some familiarity with the general area – for example we will assume the reader is familiar with the notion of an elliptic curve over a number field, and a Galois representation, and what it means for such things to be modular (when such a notion makes sense). Let us stress now that, because of this chronological approach, some theorems stated in this paper will be superseded by others (for example Theorem 1 gets superseded by Theorem 6 which gets superseded by Theorem 7), and similarly some conjectures (for example Serre's conjecture) will become theorems as the story progresses. The author hopes that this slightly non-standard style nevertheless gives the reader the feeling of seeing how the theory evolved.

We thank Toby Gee for reading through a preliminary draft of this article and making several helpful comments, and we also thank Matthew Emerton and Jan Nekovář for pointing out various other inaccuracies and ambiguities.

Semistable elliptic curves over Q are modular

The story, of course, starts with the following well-known result proved in [Wil95] and [TW95].

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