Characterizing perfectoid covers of abelian varieties

TL;DR

The paper resolves when profinite étale covers of abelian varieties admit perfectoid structures by linking perfectoidness to the dual Hodge–Tate map on $p$-divisible and semi-abeloid groups. It provides an explicit criterion via the inverse image $f^{-1}(H_p\otimes C(-1))$ of the dual Hodge–Tate map, and it computes the geometric Sen morphism $\kappa_{X_\infty}$ for relevant torsors, enabling a proof of Rodríguez Camargo's conjecture in this setting. The approach combines pro-étale/diamond techniques with Raynaud uniformization and Scholze’s canonical subgroup framework, and extends to semi-abeloid varieties and $p$-divisible rigid analytic groups, as well as to varieties with globally generated 1-forms through Albanese considerations. The results have implications for the structure of infinite-level towers, canonical subgroups, and the interplay between Hodge–Taylor filtrations and perfectoid geometry in non-archimedean analytic contexts.

Abstract

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 $\mathbb{Q}_p$. 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.

Theorems & Definitions (81)

• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• proof
• Remark 1.8
• Remark 1.9
• Remark 1.10
• Corollary 1.11: See \ref{['corollary.variety with gg diff']}