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

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

The Arithmetic of Simple Singularities

We investigate some arithmetic orbit problems in representations of linear algebraic groups arising from Vinberg theory. We aim to give a description of the orbits in these representations using methods with an emphasis on representation theory rather than algebraic geometry, in contrast to previous works of other authors. It turns out that for the representations we consider, the orbits are related to the arithmetic of the Jacobians of certain algebraic curves, which appear as the smooth nearby fibers of deformations of simple singularities. We calculate these families of algebraic curves, and show that the 2-torsion in their Jacobians is canonically identified with the stabilizers of certain orbits in the corresponding representations.Mathematic

A 2-adic automorphy lifting theorem for unitary groups over CM fields

We prove a ‘minimal’ type automorphy lifting theorem for 2-adic Galois representations of unitary type, over imaginary CM fields. We use this to improve an automorphy lifting theorem of Kisin for GL2 .Clay Mathematics Institut

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

E 8 and the average size of the 3‐Selmer group of the Jacobian of a pointed genus‐2 curve

Abstract: We prove that the average size of the 3‐Selmer group of a genus‐2 curve with a marked Weierstrass point is 4. We accomplish this by studying rational and integral orbits in the representation associated to a stably Z / 3 Z ‐graded simple Lie algebra of type E 8 . We give new techniques to construct integral orbits, inspired by the proof of the fundamental lemma and by the twisted vertex operator realisation of affine Kac–Moody algebras