Pro-étale uniformisation of abelian varieties

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 $\tilde{A}={\underleftarrow{\lim }}_{[p]}A$ 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 ${\bar{\mathbb{F}}}_p$.

References in corpus (1)

• Perfectoid covers of abelian varieties and the weight-monodromy conjecture