An R=T theorem for certain orthogonal Shimura varieties
Hao Peng, Dmitri Whitmore
TL;DR
This work establishes an almost minimal $R=\mathbb{T}$ theorem for self-dual Galois representations valued in a finite field under a rigidity criterion, and shows rigidity for residual representations arising from symmetric powers of elliptic curves and from regular algebraic self-dual automorphic representations with a supercuspidal component at some place when the residual characteristic is large. The authors develop a Taylor–Wiles patching framework for $G$-valued Galois representations with $G\in\{\mathrm{GO}(2m),\ \mathrm{GSp}(2m)\}$ and extend the unitary-method toolkit to orthogonal Shimura varieties, including Fontaine–Laffaille and minimally ramified local conditions, level-raising, and adequate subgroups. They prove finiteness and patching results for deformation rings, and deduce an isomorphism $\mathsf R_{\mathscr S}\cong \mathbb T_{\mathfrak m}$ acting on the cohomology of orthogonal Shimura varieties, thereby extending classical $R=\mathbb T$ results to a broader self-dual orthogonal setting. These results provide new inputs toward Beilinson–Bloch–Kato-type conjectures for large families of automorphic Galois representations and illuminate the arithmetic of orthogonal Shimura varieties under rigidity constraints.
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).