Unobstructedness of Galois deformation rings associated to regular algebraic conjugate self-dual cuspidal automorphic representations

Abstract

Let F be a CM field and let (rπ,λ)λ be the compatible system of residual 𝒢n-valued representations of GalF attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation π of GLn(𝔸), as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to rπ,λ are unobstructed for all places λ in a subset of Dirichlet density 1, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framework for proving unobstructedness (with future applications in mind) and an R=T-theorem, relating the universal crystalline deformation ring of rπ,λ and a certain unitary fixed-type Hecke algebra.