Unobstructedness of Galois deformation rings associated to RACSDC automorphic representations

Abstract

Let $F$ be a CM field and let $({\overline{r}}_{\pi ,\lambda }{)}_{\lambda }$ be the compatible system of residual ${\mathcal{G}}_n$-valued representations of ${\mathrm{Gal}}_F$ attached to a RACSDC automorphic representation $\pi$ of ${\mathrm{GL}}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylor and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to ${\overline{r}}_{\pi ,\lambda }$ 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 (which could be useful for other applications in future) and an $R=T$-theorem, relating the universal crystalline deformation ring of ${\overline{r}}_{\pi ,\lambda }$ and a certain unitary fixed-type Hecke algebra.