Pro-étale uniformisation of abelian varieties

2021/05/26 by Ben Heuer, Heuer, Ben
Computer Science · Mathematics · #11G07 #11G20 #14G45 #14K15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2105.12604

openalex publication_date 2021/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For an abelian variety A over an algebraically closed non-archimedean field K of residue characteristic p, we show that the isomorphism class of the pro-étale perfectoid cover \widetilde A=\varprojlim_[p]A is locally constant as A varies p-adically in the moduli space. This gives rise to a pro-étale uniformisation of abelian varieties as diamonds A^\diamond=\widetilde A/T_pA that works uniformly for all A without any assumptions on the reduction of A. More generally, we determine all morphisms between pro-finite-étale covers of abeloid varieties. For example, over \mathbb C_p, all abeloids can be uniformised in terms of universal covers that only depend on the isogeny class of the semi-stable reduction over \mathbb F_p.

Citations

• Relative p-adic Hodge theory, II: Imperfect period rings
• Perfectoid covers of abelian varieties
• The pro-étale topology for schemes
• Line bundles on perfectoid covers: case of good reduction

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• Characterizing perfectoid covers of abelian varieties
• Perfectoid covers of abelian varieties
• The universal vector extension of an abeloid variety
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