2025/01/07 by Rebecca Bellovin, Bellovin, Rebecca, Hanlin Cai +5 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2501.03974
openalex publication_date 2025/01/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give a simple characterization of all perfectoid profinite étale covers of abelian varieties in terms of the Hodge-Tate filtration on the p-adic Tate module. We also compute the geometric Sen morphism for all profinite p-adic Lie torsors over an abelian variety, and combine this with our characterization to prove a conjecture of Rodríguez Camargo on perfectoidness of p-adic Lie torsors in this case. We obtain complementary results for covers of semi-abeloid varieties, p-divisible rigid analytic groups, and varieties with globally generated 1-forms. Our proof of perfectoidness for covers of abelian varieties is based on results of Scholze on the canonical subgroup and holds for an arbitrary abelian variety over an algebraically closed non-archimedean extension of ℚp. In an appendix authored by Tongmu He, an alternate proof is presented in the case of abelian varieties that can be defined over a discretely valued subfield by combining our computation of the geometric Sen morphism with previous pointwise perfectoidness and purity of perfectoidness results of He.