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arXiv · 2310.14886

Deformations of $G$-valued pseudocharacters

Julian Quast
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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.

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Julian Quast. 2023-10-23. Deformations of $G$-valued pseudocharacters. https://arxiv.org/abs/2310.14886

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