The Taylor-Wiles Method and the Modularity of Elliptic Curves

Abstract

In this article, we discuss a proof of the modularity theorem for semistable elliptic curves with a focus on the patching arguments, and on highlighting the overall structure of the original arguments of Wiles and Taylor-Wiles. We start by introducing necessary background concerning the deformation theory of Galois representations. In particular, we discuss explicit constructions of universal deformation rings following an argument of Faltings, and explain the relationship between tangent spaces of certain universal deformation rings and Selmer groups. From there, we review the construction of Galois representations arising from cuspidal Hecke eigenforms, as well as several structural results on Hecke Algebras. The last chapter is devoted to the proof of the modularity theorem for semistable elliptic curves; here we discuss residual modularity via the Langlands-Tunnell theorem, a variant of the Taylor-Wiles patching argument due to Diamond, and a numerical freeness criterion also due to Diamond. The variants of Taylor-Wiles patching and Wiles' numerical coincidence due to Diamond simplify the proof, in the sense that the deep "strong multiplicity 1 theorems" are no longer needed as input into the patching argument, but rather are obtained as a byproduct of the patching construction itself. Finally, we complete the proof with an explanation of the 3-5 switch.