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

Abstract

Let $F$ be a CM field and let ${\left({\bar{r}}_{\pi ,\lambda }\right)}_{\lambda }$ be the compatible system of residual ${\mathcal{𝒢}}_n$-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 ${\bar{r}}_{\pi ,\lambda }$ 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 ${\bar{r}}_{\pi ,\lambda }$ and a certain unitary fixed-type Hecke algebra.