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