Lecture 18: Overview of the Taylor-Wiles method

Taylor’s lecture on proving modularity without Ihara’s lemma

Taylor will give three lectures, to be held respectively on April 5 at 1:00, April 7 at 3:00, and on April 14 at 3:00. The title is “Proving modularity without Ihara’s lemma”, which is the subject of the preprint [3]. Furthermore, Harris and Taylor will give lectures on April 21, 2:00 to 4:00, entitled “A family of Calabi-Yau varieties and the Sato-Tate conjecture”. The aim of Taylor’s lectures...

Modularity Lifting beyond the Taylor–wiles Method. Ii

18.783 Elliptic Curves Spring 2013 Lecture #24 05/09/2013

Andrew V. Sutherland In this lecture we give a brief overview of modular forms, focusing on their relationship to elliptic curves. This connection is crucial to Wiles’ proof of Fermat’s Last Theorem [7]; the crux of his proof is that every semistable elliptic curve over Q is modular.1 In order to explain what this means, we need to delve briefly into the theory of modular forms. Our goal in doi...

Deformations of Hilbert Modular Galois Representations and Adjoint Selmer Groups

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to a Hilbert modular form under mild assumptions on the residual image. In particular, we do not assume the residual image satisfies the Taylor–Wiles hypothesis. Using this, we deduce that the localization and completion of the universal deformation ring for the r...

ON THE AUTOMORPHY OF l-ADIC GALOIS REPRESENTATIONS WITH SMALL RESIDUAL IMAGE

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor-Wiles method which allows one to weaken this hypothesis.