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

Guiraud, David-Alexandre

doi:10.2140/ant.2020.14.1331

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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

DOI: 10.2140/ant.2020.14.1331

Abstract

Let $F$ be a CM field and let ${({\bar{r}}_{π, λ})}_{λ}$ be the compatible system of residual $𝒢_{n}$ -valued representations of ${Gal}_{F}$ attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation $π$ of ${GL}_{n} (𝔸)$ , as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to ${\bar{r}}_{π, λ}$ 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 ${\bar{r}}_{π, λ}$ and a certain unitary fixed-type Hecke algebra.

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Keywords

Galois deformation, automorphic representation

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11F70

Milestones

Received: 30 June 2017

Revised: 13 August 2019

Accepted: 10 February 2020

Published: 30 July 2020

Authors

David-Alexandre Guiraud
Baden-Baden Germany

Vol. 14, No. 6, 2020