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Heights on a subvariety of an abelian variety

✍ Scribed by Takashi Ichikawa

Publisher: Elsevier Science
Year: 2004
Tongue: English
Weight: 202 KB
Volume: 104
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2003.07.002

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✦ Synopsis

Extending Ullmo-Zhang's result on the Bogomolov conjecture, we give conditions that a closed subvariety of an abelian variety A defined over a number field is isomorphic to an abelian variety in terms of the value distribution of a Neron-Tate height function on the subvariety. As a corollary of the result, we prove the Bogomolov conjecture which claims that if an irreducible curve X in A is not isomorphic to an elliptic curve, then for the pseudodistance defined by the Neron-Tate height, the distribution of algebraic points on X is uniformly discrete. These results can be extended in the case where base fields are finitely generated over Q via Moriwaki's height theory.

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## Abstract In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main consequence we obtain an involution on the set of