The Taylor-Wiles method for reductive groups

2022/05/10 by Whitmore, Dmitri
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.05062

Abstract

We construct a local deformation problem for residual Galois representations ρ valued in an arbitrary reductive group G 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 G-adequate subgroup, our corresponding 'big image' condition. When G is a simply connected simple group of type C or of exceptional type and G → GL_n is a faithful irreducible representation of minimal dimension, we show that a subgroup is G-adequate if it is GL_n-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case G = 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.

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