Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

TL;DR

This paper constructs and analyzes moduli spaces of Galois representations with p-adic coefficients, focusing on potentially semi-stable loci and their relation to pseudorepresentations and deformation rings.

Contribution

It introduces a new framework for moduli of Galois representations and demonstrates the algebraization of the associated morphism over pseudorepresentations.

Findings

Moduli spaces of Galois representations are constructed over pseudorepresentations.

Potentially semi-stable loci are Zariski-closed and descend to deformation rings.

The morphism from Galois representations to pseudorepresentations is algebraizable.

Abstract

We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal deformation ring of the corresponding pseudorepresentation.