Deformation rings and images of Galois representations

TL;DR

This paper investigates the structure of deformation rings and images of Galois representations associated with reductive algebraic groups over Witt rings, establishing classification and universality results under certain conditions.

Contribution

It provides a classification of closed subgroups and surjective homomorphisms of Galois representation groups, generalizing prior results and introducing an axiomatic framework.

Findings

Closed subgroups with full residual image are conjugate to groups over local subrings.

Surjective homomorphisms are induced by ring homomorphisms, up to conjugation.

The identity map on the deformation ring is the universal deformation of the residual representation.

Abstract

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ . (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity map on $\mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $\mathcal{G}(R)$ given by the reduction map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $\mathcal{G}$, and we study in the case at hand in great detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$ are necessary for the above results to hold.