Pro-\'etale uniformisation of abelian varieties

TL;DR

This paper establishes a uniform pro-étale uniformisation for abelian varieties over non-archimedean fields, showing local constancy of their perfectoid covers and describing morphisms between such covers.

Contribution

It introduces a pro-étale uniformisation of abelian varieties as diamonds, valid uniformly across all varieties without reduction assumptions, and classifies morphisms between pro-finite-étale covers.

Findings

The isomorphism class of the perfectoid cover is locally constant in the moduli space.

A uniform pro-étale uniformisation of abelian varieties as diamonds is constructed.

All morphisms between pro-finite-étale covers of abeloid varieties are determined.

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-\'etale 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-\'etale 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-\'etale 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 $\bar{\mathbb F}_p$.