Unobstructedness of Galois deformation rings associated to RACSDC automorphic representations

TL;DR

This paper proves that certain Galois deformation rings associated with automorphic representations over CM fields are unobstructed for almost all primes, advancing understanding of deformation theory and automorphic forms.

Contribution

It establishes the unobstructedness of universal deformation rings for residual Galois representations linked to RACSDC automorphic representations, and develops a general framework and an $R=T$-theorem.

Findings

Universal deformation rings are unobstructed for a density 1 set of primes.

Develops a new framework for proving unobstructedness in Galois deformation theory.

Establishes an $R=T$-theorem relating deformation rings and Hecke algebras.

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 $\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $\pi$ of $\operatorname{GL}_n(\mathbb{A})$, 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.