Characterizing perfectoid covers of abelian varieties

TL;DR

This paper characterizes all perfectoid covers of abelian varieties using the Hodge-Tate filtration, computes the geometric Sen morphism for p-adic Lie torsors, and proves a related conjecture, extending to semi-abeloid varieties and other cases.

Contribution

It provides a simple characterization of perfectoid covers of abelian varieties and proves a conjecture on their perfectoidness, with new methods involving the Hodge-Tate filtration and the geometric Sen morphism.

Findings

Characterization of perfectoid covers via Hodge-Tate filtration

Proof of Rodrf3guez Camargo's conjecture on perfectoidness

Extension of results to semi-abeloid varieties and p-divisible groups

Abstract

We give a simple characterization of all perfectoid profinite \'{e}tale 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 Rodr\'{i}guez 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.