Deformations of G-Valued Pseudocharacters

Abstract

We define a deformation space of Lafforgue’s G-valued pseudocharacters of a profinite group

\mathrm{\Gamma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}

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

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {Q}}_p$$\end{document}

-algebras is represented by a quasi-Stein rigid analytic space, whenever

\mathrm{\Gamma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}

is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when

\mathrm{\Gamma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}

satisfies Mazur’s condition

{\mathrm{\Phi }}_p

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Phi _p$$\end{document}

and G satisfies a certain invariant-theoretic condition. For

G={Sp}_{2n}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G = \operatorname {Sp}_{2n}$$\end{document}

, 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.

Keywords

Pseudorepresentation / Pseudocharacter / Determinant law / Deformation theory / Galois representation / Primary 11F80 / 14J10 / Secondary 20G25 / 11S25 / 14M35

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Funding

Deutsche Forschungsgemeinschaft(444845124)

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