Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-19T05:31:49.920Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative Manin–Mumford for Semi-Abelian Surfaces

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

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)
Get access Rights & Permissions [Opens in a new window]

Abstract

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)