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Relative Manin–Mumford for Semi-Abelian Surfaces

Published online by Cambridge University Press:  18 January 2016

D. Bertrand
Affiliation:
Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 75252 Paris Cedex 05, France ([email protected])
D. Masser
Affiliation:
Mathematisches Institut, Universität Basel, Rheinsprung 21, 4051 Basel, Switzerland ([email protected])
A. Pillay
Affiliation:
Department of Mathematics, University of Notre Dame, 281 Hurley Hall, Notre Dame, IN 46556, USA ([email protected]; [email protected])
U. Zannier
Affiliation:
Scuole Normale Superiore, Cl. Sc. Matematiche, Piazza dei Cavalieri, 7-56126 Pisa, Italy ([email protected])

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.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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