Vector bundles with numerically flat reduction on rigid analytic varieties and p-adic local systems

2019/10/09 by Matti Würthen, Würthen, Matti
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Alkaloids: synthesis and pharmacology #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1910.03727

openalex publication_date 2019/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show how to functorially attach continuous p-adic representations of the profinite fundamental group to vector bundles with numerically flat reduction on a proper rigid analytic variety over ℂ_p. This generalizes results by Deninger and Werner for vector bundles on smooth algebraic varieties. Our approach uses fundamental results on the pro-étale site of a rigid analytic variety introduced by Scholze. This enables us to get rid of the smoothness condition and to work in the analytic category. Moreover, under some mild conditions, the functor we construct gives a full embedding of the category of vector bundles with numerically flat reduction into the category of continuous ℂ_p-representations. This provides new insights into the p-adic Simpson correspondence in the case of a vanishing Higgs-field.

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