Automorphy of some residually S₅ Galois representations

We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic 5, the residual representation has projective image PGL_2(F_5), and the fixed field of its kernel contains a primitive 5th root of unity. In this case the Taylor-Wiles patching method does not suffice and we prove our theorem by combining patching with the technique of automorphy by p-adic approximation.