Pro-\'etale uniformisation of abelian varieties

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 $\bar{\mathbb F}_p$.