A non-archimedean Ax-Lindemann theorem

TL;DR

This paper establishes a non-archimedean Ax-Lindemann theorem for products of Mumford curves, characterizing bi-algebraic subvarieties via their fundamental groups within p-adic uniformization.

Contribution

It extends the Ax-Lindemann theorem to a non-archimedean setting for Mumford curves with specific fundamental groups, providing new insights into their algebraic and uniformization structures.

Findings

Characterization of bi-algebraic subvarieties in the non-archimedean setting.

Extension of Ax-Lindemann theorem to Mumford curves with non-abelian Schottky groups.

Identification of fundamental groups within PGL(2, Q_p) related to uniformization.

Abstract

We prove a statement of Ax-Lindemann type for the uniformization of products of Mumford curves whose associated fundamental groups are non-abelian Schottky subgroups of $\mathop{\rm PGL}(2,\bar{\mathbf Q_p})$ contained in $\mathop{\rm PGL}(2,\bar{\mathbf Q})$. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization.