The Taylor-Wiles method for reductive groups

TL;DR

This paper extends the Taylor-Wiles method to arbitrary reductive groups, allowing for broader applications in modularity lifting and Galois representation deformation, especially for groups like GSp4.

Contribution

It develops a generalized deformation theory for residual Galois representations valued in reductive groups and introduces the notion of G-adequate subgroups, broadening the scope of the Taylor-Wiles method.

Findings

Constructed local deformation problems for residual Galois representations in reductive groups.

Proved modularity lifting theorems for GSp4 and abelian surfaces.

Established weaker conditions for modularity results over totally real fields.

Abstract

We construct a local deformation problem for residual Galois representations $\bar{\rho}$ valued in an arbitrary reductive group $\hat{G}$ 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 $\hat{G}$-adequate subgroup, our corresponding 'big image' condition. When $\hat{G}$ is a simply connected simple group of type $\mathrm{C}$ or of exceptional type and $\hat{G} \to \mathrm{GL}_n$ is a faithful irreducible representation of minimal dimension, we show that a subgroup is $\hat{G}$-adequate if it is $\mathrm{GL}_n$-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case $\hat{G} = \mathrm{GSp}_4$ 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 $F$.