Serre's Modularity Conjecture

TL;DR

This paper provides an overview of Serre's modularity conjecture, its proof by Khare, Wintenberger, and Kisin, and discusses related recent generalizations in the field of Galois representations and modular forms.

Contribution

It summarizes the proof of Serre's conjecture and highlights recent developments and generalizations in the area.

Findings

Proof of Serre's modularity conjecture explained

Key contributions by Khare, Wintenberger, and Kisin summarized

Discussion of recent generalizations of the conjecture

Abstract

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture.