Serre’s Modularity Conjecture

These notes are based on lectures given by the author 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 [36] [37] [39], as well as of the results of other mathematicians that played an important role in the proof. Along the way we will remark on some recent work concerning generalizations of the conjecture. We have tried as much as possible to concentrate on giving a broad picture of the structure of the arguments and have ignored technical details in places. Some results are given incomplete statements, where we have chosen not to list technical hypotheses; we request the reader’s forbearance. Let F be a totally real number field. We will denote by GF the absolute Galois group Gal(Q/F ). It was shown in Prof. Böckle’s lectures in this volume how, under some hypotheses, a Hilbert modular eigenform f over F gives rise to a compatible system {ρf,v} of p-adic Galois representations; see [34] for the most general theorem. These representations are extracted from the cohomology of a suitable algebraic variety, and this construction is more or less the only method we have for obtaining p-adic Galois representations. Therefore the following question is of acute interest: given a Galois representation ρ : GF → GL2(Qp), when is ρ modular , i.e. when does there exist a Hilbert modular eigenform f and a place v|p of F such that ρ ' ρf,v? This question is a very difficult one. We will split it into two questions, which are still very difficult, by introducing the notion of the reduction of a Galois representation. The following result is classical; the proof given here is attributed to N. Katz and appears in print, for instance, at the beginning of section 2 of [55].