On local Galois deformation rings: generalised tori

TL;DR

This paper investigates the deformation rings of mod p Galois representations valued in generalised tori over p-adic fields, revealing their smoothness, dimension, and irreducible components.

Contribution

It provides a detailed analysis of the structure and properties of deformation rings for Galois representations in the context of generalised tori, extending existing deformation theory.

Findings

Deformation rings are formally smooth over a group algebra of a finite abelian p-group.

Computed the dimension of these deformation rings.

Identified the set of irreducible components of the deformation rings.

Abstract

We study deformation theory of mod $p$ Galois representations of $p$-adic fields with values in generalised tori, such as $L$-groups of (possibly non-split) tori. We show that the corresponding deformation rings are formally smooth over a group algebra of a finite abelian $p$-group. We compute their dimension and the set of irreducible components.