Vol. 14, No. 6, 2020

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

David-Alexandre Guiraud

Vol. 14 (2020), No. 6, 1331–1380

Let F be a CM field and let (r̄π,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 r̄π,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 r̄π,λ and a certain unitary fixed-type Hecke algebra.

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Galois deformation, automorphic representation
Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F70
Received: 30 June 2017
Revised: 13 August 2019
Accepted: 10 February 2020
Published: 30 July 2020
David-Alexandre Guiraud