Moduli of $p$-adic representations of a profinite group

Abstract

Let $X$ be a smooth and proper scheme over an algebraically closed field. The purpose of the current text is twofold. First, we construct the moduli stack parametrizing rank $n$ continuous $p$-adic representations of the étale fundamental group ${\pi }_1^{\text{eˊt}}(X)$. Our construction realizes such object as a ${\mathbb{Q}}_p$-analytic stack, denoted ${\mathrm{LocSys}}_{p,n}(X)$. Secondly, we prove that ${\mathrm{LocSys}}_{p,n}(X)$ admits a canonical derived structure. This derived structure allow us to intrinsically recover the deformation theory of continuous $p$-adic representations, studied in [GV18]. Our proof of geometricity of ${\mathrm{LocSys}}_{p,n}(X)$ uses in an essential way the ${\mathbb{Q}}_p$-analytic analogue of Lurie-Artin representability, proved in [PY17].

Last edit 28/05/2018

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• Higher and derived stacks: a global overview
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• Derived non-archimedean analytic spaces
• The derived Riemann-Hilbert correspondence
• Structures symplectiques et de Poisson sur les champs en catégories
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Cited by in corpus (6)

• Coherent sheaves on the stack of Langlands parameters
• Structures symplectiques et de Poisson sur les champs en catégories
• Presentations of non-commutative deformation rings via $A_{\infty }$-algebras and applications to deformations of Galois representations and pseudorepresentations
• Derived ${\mathcal{O}}_k$-adic geometry and derived Raynaud localization theorem
• Moduli of $\ell$-adic pro-étale local systems for smooth non-proper schemes
• Spreading out the Hodge filtration in non-archimedean geometry