A nonarchimedean Ax–Lindemann theorem

Abstract

Motivated by the André–Oort conjecture, Pila has proved an analogue of the Ax–Lindemann theorem for the uniformization of classical modular curves. In this paper, we establish a similar theorem in nonarchimedean geometry. Precisely, we give a geometric description of subvarieties of a product of hyperbolic Mumford curves such that the irreducible components of their inverse image by the Schottky uniformization are algebraic, in some sense. Our proof uses a $p$-adic analogue of the Pila–Wilkie theorem due to Cluckers, Comte and Loeser, and requires that the relevant Schottky groups have algebraic entries.