The Taylor-Wiles method for reductive groups

Abstract

We construct a local deformation problem for residual Galois representations ρ valued in an arbitrary reductive group Ĝ which we use to develop a variant of the Taylor–Wiles method. Our generalization allows Taylor–Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of Ĝ-adequate subgroup, our corresponding ‘big image’ condition. When Ĝ → GLn is a faithful irreducible representation, we show that a subgroup is Ĝ-adequate if it is GLn-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case Ĝ = GSp4 and prove a modularity lifting theorem for abelian surfaces over a totally real field F which holds under weaker hypotheses than in the work of Boxer–Calegari–Gee–Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of F. We further apply our ideas to V. Lafforgue’s construction of the global Langlands correspondence in characteristic p for a semisimple group Ĝ, proving an automorphy lifting theorem under the assumption of Ĝ-adequate residual image, a weakening of the Ĝ-abundant condition appearing previously in the literature. We deduce potential automorphy of everywhere unramified Galois representations with Ĝ-adequate residual image. We also give results concerning existence of global cyclic base change and the possible finite images of automorphic Galois representations. The main technical difficulty is in proving an instance of local-global compatibility at certain level structures deeper than parahoric level, which we require for our implementation of the Taylor–Wiles method.