The universal vector extension of an abeloid variety

TL;DR

This paper describes the universal cover of the universal vector extension of an abelian variety over a non-Archimedean field, linking it to the reduction behavior of the variety.

Contribution

It provides a detailed description of the universal cover of the universal vector extension of an abelian variety in the non-Archimedean setting.

Findings

Universal cover reflects reduction behavior of the abelian variety.

Framework will be used to prove rigidity of analytic functions on the extension.

Connects Berkovich space topology with algebraic properties.

Abstract

Let $A$ be an abelian variety over a complete non-Archimedean field $K$. The universal cover of the Berkovich space attached to $A$ reflects the reduction behaviour of $A$. In this paper the universal cover of the universal vector extension $E(A)$ of $A$ is described. In a forthcoming paper ( arXiv:2007.04659), this will be one of the crucial tools to show that rigid analytic functions on $E(A)$ are all constant.