On the automorphy of l -adic Galois representations with small residual image With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Jack Thorne

Source abstract

Abstract 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.