A nonarchimedean Ax–Lindemann theorem

Chambert-Loir, Antoine; Loeser, François

doi:10.2140/ant.2017.11.1967

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A nonarchimedean Ax–Lindemann theorem

Antoine Chambert-Loir and François Loeser

Vol. 11 (2017), No. 9, 1967–1999

DOI: 10.2140/ant.2017.11.1967

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

Vol. 11, No. 9, 2017