Deformations of $G$-valued pseudocharacters

TL;DR

This paper develops a deformation theory for G-valued pseudocharacters of profinite groups, generalizing previous constructions, and analyzes the structure and properties of the associated deformation rings and spaces.

Contribution

It introduces a generalized deformation space for G-valued pseudocharacters, proves the noetherian property of the universal pseudodeformation ring, and describes obstructed loci for symplectic groups.

Findings

Universal pseudodeformation ring is noetherian.

The functor of continuous G-pseudocharacters is represented by a quasi-Stein rigid analytic space.

Describes obstructed loci and bounds their dimension for G=Sp_{2n}.

Abstract

We define a deformation space of V. Lafforgue's $G$-valued pseudocharacters of a profinite group $\Gamma$ for a possibly disconnected reductive group $G$. We show, that this definition generalizes Chenevier's construction. We show that the universal pseudodeformation ring is noetherian and that the functor of continuous $G$-pseudocharacters on affinoid $\mathbb{Q}_p$-algebras is represented by a quasi-Stein rigid analytic space, whenever $\Gamma$ is topologically finitely generated. We also show that the pseudodeformation ring is noetherian, when $\Gamma$ satisfies Mazur's condition $\Phi_p$ and $G$ satisfies a certain invariant-theoretic condition. For $G = \mathrm{Sp}_{2n}$ we describe three types of obstructed loci in the special fiber of the universal pseudodeformation space of an arbitrary residual pseudocharacter and give upper bounds for their dimension.