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Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)$
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Umberto Zannier

Print publication date: 2012

Print ISBN-13: 9780691153704

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153704.001.0001

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Unlikely Intersections in Elliptic Surfaces and Problems of Masser

Unlikely Intersections in Elliptic Surfaces and Problems of Masser

Chapter:
(p.62) Chapter 3 Unlikely Intersections in Elliptic Surfaces and Problems of Masser
Source:
Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)
Author(s):

Umberto Zannier

David Masser

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691153704.003.0004

This chapter turns from the multiplicative-group context to the context of abelian varieties. There are here entirely similar results and conjectures: we have already recalled the Manin–Mumford conjecture, and pointed out that the Zilber conjecture also admits an abelian exact analogue. Actually, abelian varieties have moduli, which introduce new issues with respect to the toric case. The chapter focuses mainly on some new problems, raised by Masser, which represent a relative case of Manin–Mumford–Raynaud, where the relevant abelian variety is no longer fixed but moves in a family. The unlikely intersections of Masser's questions occur in the special case of elliptic surfaces (i.e., families of elliptic curves), and can be dealt with by a method that has been recently introduced.

Keywords:   Manin–Mumford conjecture, abelian analogue, Manin–Mumford–Raynaud, elliptic surfaces

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