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arXiv · 1709.04275

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

Jorge António
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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 $π_1^\textrm{ét}(X)$. 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].

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Jorge António. 2020-05-01. Moduli of $p$-adic representations of a profinite group. https://arxiv.org/abs/1709.04275

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