An R=T theorem for certain orthogonal Shimura varieties

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).

💡 Research Summary

The paper establishes an “almost minimal” R = T theorem for self‑dual Galois representations valued in the disconnected reductive groups GO(2m) or GSp(2m). The authors introduce a new residual condition called “rigidity”. Roughly, a residual representation ρ̄ is rigid if (i) its image is absolutely irreducible and “adequate” (in the sense of Taylor–Wiles patching for general G) so that the dual Selmer group has the expected dimension, and (ii) any lift of ρ̄ that is crystalline at the primes above the residual characteristic ℓ and unramified outside a prescribed set automatically satisfies the minimal ramification conditions required in the deformation problem.

The main arithmetic input is a regular algebraic self‑dual cuspidal automorphic representation Π of GL₂ₘ over a totally real field F (the authors call this d‑REASDC). For such Π they assume the existence of a super‑cuspidal local component at some finite place v, and either that the associated ℓ‑adic local system on the Shimura variety is constant or that the Shimura variety has dimension zero. Under these hypotheses, for all but finitely many primes ℓ the authors construct a maximal ideal m_ℓ of the Hecke algebra T_{m,Σ∪Σ(ℓ)} such that the localized Hecke algebra T_{ℓ,m_ℓ} is canonically isomorphic to a universal deformation ring R_{ℓ} classifying self‑dual deformations of the residual representation ρ̄_{Π,ℓ} which are crystalline with the prescribed Fontaine–Laffaille weights at ℓ‑adic places and minimally ramified elsewhere. Moreover, the middle‑degree étale cohomology H^d(Sh(V,K),L_{ξ,ℓ})_{m_ℓ} is a finite free O_ℓ‑module.

The proof follows the modern Taylor–Wiles patching framework for G‑valued Galois representations as developed in