On local Galois deformation rings

TL;DR

This paper proves that certain local Galois deformation rings are complete intersections with well-understood geometric properties and uses these results to analyze the density of specific p-adic Hodge theoretic loci.

Contribution

It establishes that framed deformation rings of mod p Galois representations are complete intersections and describes their geometric structure and irreducible components.

Findings

Deformation rings are complete intersections of expected dimension.

They are normal and have well-characterized irreducible components.

Density results for loci with specific p-adic Hodge properties are proved.

Abstract

We show that framed deformation rings of mod $p$ representations of the absolute Galois group of a $p$-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application we prove density results of loci with prescribed $p$-adic Hodge theoretic properties.