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

Abstract

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into $G{L}_n$. 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.

59 pages. To appear in the Journal of the Institute of Mathematics of Jussieu