#### 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
##### Abstract

Let $F$ be a CM field and let ${\left({\stackrel{̄}{r}}_{\pi ,\lambda }\right)}_{\lambda }$ be the compatible system of residual ${\mathsc{𝒢}}_{n}$-valued representations of ${Gal}_{F}$ attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation $\pi$ of ${GL}_{n}\left(\mathbb{𝔸}\right)$, as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to ${\stackrel{̄}{r}}_{\pi ,\lambda }$ are unobstructed for all places $\lambda$ 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 ${\stackrel{̄}{r}}_{\pi ,\lambda }$ and a certain unitary fixed-type Hecke algebra.

##### Keywords
Galois deformation, automorphic representation
Primary: 11F80
Secondary: 11F70