We define a deformation space of Lafforgue’s G-valued pseudocharacters of a profinite group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> for a possibly disconnected reductive group G. We show that this definition generalizes Chenevier’s construction. We show that the universal pseudodeformation ring is noetherian and that the functor of continuous G-pseudocharacters on affinoid <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {Q}}_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Q</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-algebras is represented by a quasi-Stein rigid analytic space, whenever <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> satisfies Mazur’s condition <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi _p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Φ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and G satisfies a certain invariant-theoretic condition. For <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$G = \operatorname {Sp}_{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <msub> <mo>Sp</mo> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation>, 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.

Deformations of G-Valued Pseudocharacters

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2026年1月4日

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Julian Quast

摘要

We define a deformation space of Lafforgue’s G-valued pseudocharacters of a profinite group $\Gamma $ $Γ$ 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 ${\mathbb {Q}}_p$ $Q_{p}$ -algebras is represented by a quasi-Stein rigid analytic space, whenever $\Gamma $ $Γ$ is topologically finitely generated. We also show that the pseudodeformation ring is noetherian when $\Gamma $ $Γ$ satisfies Mazur’s condition $\Phi _p$ $Φ_{p}$ and G satisfies a certain invariant-theoretic condition. For $G = \operatorname {Sp}_{2n}$ $G = {Sp}_{2 n}$ , 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.