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arXiv · 1801.05762

Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

Ziyang Gao·Philipp Habegger
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Abstract

On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise N\'eon-Tate height. If the base curve is inside a projective space, we also have a height on its $\overline{\mathbb Q}$-points that serves as a measure of each fiber, an abelian variety. Silverman proved an asymptotic equality between these two heights on a curve in the abelian scheme. In this paper we prove an inequality between these heights on a subvariety of any dimension of the abelian scheme. As an application we prove the Geometric Bogomolov Conjecture for the function field of a curve defined over $\overline{\mathbb Q}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.

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Ziyang Gao, Philipp Habegger. 2018-01-17. Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. https://arxiv.org/abs/1801.05762

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