How to facet a gemstone: from potential modularity to the proof of Serre's modularity conjecture

Abstract

In this survey paper we present recent results obtained by Khare, Wintenberger and the author that have led to a proof of Serre's conjecture, such as existence of compatible families, modular upper bounds for universal deformation rings and existence of minimal lifts, prime switching and modularity propagation, weight reduction (via existence of conjugates) and (iterated) killing ramification. The main tools used in the proof of these results are modularity lifting theorems a la Wiles and a result of potential modularity due to R. Taylor.

Survey based on two talks I gave last summer, at the Segundas Jornadas de Teoria de Numeros (Madrid) and at the Summer School on Serre's Modularity Conjecture (Luminy)