#### Vol. 11, No. 9, 2017

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A nonarchimedean Ax–Lindemann theorem

### Antoine Chambert-Loir and François Loeser

Vol. 11 (2017), No. 9, 1967–1999
##### 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.

##### Keywords
Schottky group, Ax–Lindemann theorem, Pila–Wilkie theorem, nonarchimedean analytic geometry
##### Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 03C98, 11D88, 11J91, 14G22, 14G35