Deformations of $G$-valued pseudocharacters

Abstract

We define a deformation space of V. Lafforgue's $G$-valued pseudocharacters of a profinite group $\hat{I}“$ 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 $\hat{I}“$ is topologically finitely generated. We also show that the pseudodeformation ring is noetherian, when $\hat{I}“$ satisfies Mazur's condition $\hat{I}{¦}_p$ 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.

52 pages, minor changes, numbering agrees with published version