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Relative Manin–Mumford for Semi-Abelian Surfaces

Part of: Differential and difference algebra Diophantine approximation, transcendental number theory Abelian varieties and schemes

Published online by Cambridge University Press:  18 January 2016

D. Bertrand ,
D. Masser ,
A. Pillay  and
U. Zannier
D. Bertrand
Affiliation:
Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 75252 Paris Cedex 05, France (daniel.bertrand@imj-prg.fr)
D. Masser
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland (david.masser@unibas.ch)
A. Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, 281 Hurley Hall, Notre Dame, IN 46556, USA (anand.pillay.3@nd.edu; a.pillay@leeds.ac.uk)
U. Zannier
Affiliation:
Scuole Normale Superiore, Cl. Sc. Matematiche, Piazza dei Cavalieri, 7-56126 Pisa, Italy (u.zannier@sns.it)
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Abstract

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We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber–Pink conjecture for curves in a mixed Shimura variety of dimension 4, as well as the study of polynomial Pell equations with non-separable discriminants.

Keywords

semi-abelian varietiesManin–Mumford conjectureAndré–Oort conjectureZilber–Pink conjecturedifferential Galois theorypolynomial Pell equations

MSC classification

Secondary: 14K15: Arithmetic ground fields 12H05: Differential algebra 14K20: Analytic theory; abelian integrals and differentials 11J95: Results involving abelian varieties
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 4 , November 2016 , pp. 837 - 875
Copyright
Copyright © Edinburgh Mathematical Society 2016