On local Galois deformation rings: generalised reductive groups

TL;DR

This paper investigates the deformation rings of mod p Galois representations valued in generalised reductive group schemes over p-adic fields, establishing their structural properties and component structure.

Contribution

It extends deformation theory to generalised reductive groups, proving that associated deformation rings are complete intersections with normal, well-understood components.

Findings

Deformation rings are complete intersections of expected dimension.

Irreducible components of deformation rings are determined in many cases.

Deformation rings and their special fibres are normal and complete intersections.

Abstract

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised reductive group schemes, such as $L$-groups and $C$-groups. We show that the corresponding deformation rings are complete intersections of expected dimension. We determine their irreducible components in many cases and show that they and their special fibres are normal and complete intersection.