An R=T theorem for certain orthogonal Shimura varieties

TL;DR

This paper establishes an R=T theorem for certain self-dual Galois representations, utilizing Taylor--Wiles patching methods for Galois groups related to orthogonal and symplectic groups, advancing understanding in number theory.

Contribution

It proves an almost minimal R=T theorem for self-dual Galois representations with a new rigidity property, and demonstrates this property for a broad class of residual representations.

Findings

Proved R=T theorem for self-dual Galois representations.

Established rigidity property for many residual Galois representations.

Applied Taylor--Wiles patching to orthogonal and symplectic groups.

Abstract

We prove an almost minimal R=T theorem for self-dual Galois representations with coefficients in a finite field satisfying a property called rigid. We also prove the rigidity property for a large family of residual Galois representations attached to regular algebraic self-dual representations. Our theorem is based on a Taylor--Wiles patching argument for G-valued Galois representation, where G equals GO(2m) or GSp(2m).