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
Milestones
Received: 20 November 2015
Revised: 2 September 2017
Accepted: 3 September 2017
Published: 2 December 2017
Authors
Antoine Chambert-Loir
Univ. Paris Diderot
Sorbonne Paris Cité
Institut de Mathématiques de Jussieu-Paris Rive Gauche
UMR 7586
F-75013, Paris
France
François Loeser
Sorbonne Universités
UPMC Univ Paris 06, UMR 7586 CNRS, Institut Mathématique de Jussieu-Paris Rive Gauche
F-75005, Paris
France