Pro-étale uniformisation of abelian varieties.

Abstract: Any complex g-dimensional abelian variety admits a uniformisation in terms of its topological universal cover (which is C^{g}) and a lattice. From the work of Raynaud, Bosch, Lütkebohmert we know that for non-archimedean case there exist a 'uniformization' of abelian varieties, in rigid analytic category in terms of a semi-abelian rigid space and and a discrete lattice. While in the complex uniformization, the universal cover was isomorphic for all g-dimensional abelian varieties, the rigid analytic uniformization is not even locally constant (i.e. in the moduli space of abelian varieties). In the category of diamonds introduced by P.Scholze, there exists a new kind of "pro-étale uniformization" in terms of the perfectoid tilde limit and the p-adic Tate module of the abelian variety, which remains locally constant.
The talk is based on https://arxiv.org/pdf/2105.12604.pdf by Ben Heuer, where the above result is proved. I shall try to explain the main idea and techniques of the proof.