Deformation rings and images of Galois representations

Arias-de-Reyna, Sara; Böckle, Gebhard

doi:10.2140/ant.2026.20.697

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Deformation rings and images of Galois representations

Sara Arias-de-Reyna and Gebhard Böckle

Vol. 20 (2026), No. 4, 697–745

DOI: 10.2140/ant.2026.20.697

Abstract

Let $𝒢$ be a connected reductive almost simple group over the Witt ring $W (𝔽)$ for $𝔽$ a finite field of characteristic $p$ . Let $R$ and $R^{'}$ be complete noetherian local $W (𝔽)$ -algebras with residue field $𝔽$ . Under a mild condition on $p$ in relation to structural constants of $𝒢$ , we show the following results: (1) Every closed subgroup $H$ of $𝒢 (R)$ with full residual image $𝒢 (𝔽)$ is a conjugate of a group $𝒢 (A)$ for $A \subset R$ a closed subring that is local and has residue field $𝔽$ . (2) Every surjective homomorphism $𝒢 (R) \to 𝒢 (R^{'})$ is, up to conjugation, induced from a ring homomorphism $R \to R^{'}$ . (3) The identity map on $𝒢 (R)$ represents the universal deformation of the representation of the profinite group $𝒢 (R)$ given by the reduction map $𝒢 (R) \to 𝒢 (𝔽)$ . This generalizes results of Dorobisz, Eardley and Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $𝒢 (R)$ with residually full image.

We provide an axiomatic framework to study this type of question, also for slightly more general $𝒢$ , and we study in the case at hand in great detail what conditions on $𝔽$ or on $p$ in relation to $𝒢$ are necessary for the above results to hold.

Keywords

deformations of group representations, universal deformation rings, inverse deformation problem, reductive groups

Mathematical Subject Classification

Primary: 11F80, 14D15, 20G40

Milestones

Received: 4 October 2020

Revised: 6 May 2024

Accepted: 2 April 2025

Published: 30 April 2026

Authors

Sara Arias-de-Reyna
Facultad de Matemáticas Universidad de Sevilla Sevilla Spain

Gebhard Böckle
Interdisciplinary Center for Scientific Computing Universität Heidelberg Heidelberg Germany

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Vol. 20, No. 4, 2026