On the automorphy of ll-adic Galois representations with small residual image

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into $GL_n$. 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.Comment: 59 pages. To appear in the Journal of the Institute of Mathematics of Jussie

Automorphy of some residually dihedral Galois representations

We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called ‘Taylor–Wiles hypothesis’. We apply this to the problem of the modularity of elliptic curves over totally real fields.During the period this research was conducted, Jack Thorne served as a Clay Research Fellow.This is the author accepted manuscript. The final version is available from Springer http://link.springer.com/article/10.1007%2Fs00208-015-1214-z

A pp-adic approach to the existence of level-raising congruences

We construct level-raising congruences between $p$-ordinary automorphic representations, and apply this to the problem of symmetric power functoriality for Hilbert modular forms. In particular, we prove the existence of the $n^\text{th}$ symmetric power lift of a Hilbert modular eigenform of regular weight for each odd integer $n = 1, 3, \dots, 25$

On subquotients of the etale cohomology of Shimura varieties

We study the conditions imposed by conjectures of Arthur and Kottwitz on the Galois representations occurring in the cohomology of Shimura varieties.Comment: Accepted manuscript. To appear in Shimura Varieties volum

On the arithmetic of simple singularities of type E.

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams E 6 , E 7 , E 8 . These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally

Automorphy lifting for residually reducible ll-adic Galois representations, II

We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.Comment: Accepted versio