Heights in families of abelian varieties and the geometric Bogomolov conjecture

Gao, Ziyang and Habegger, Philipp. (2019) Heights in families of abelian varieties and the geometric Bogomolov conjecture. Preprints Fachbereich Mathematik, 2019 (02).

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Official URL: https://edoc.unibas.ch/69809/

On an abelian scheme over a smooth curve over $\bar{\mathbb{Q}}$ a symmetric relatively ample line bundle defines a fiberwise Néron–Tate height. If the base curve is inside a projective space, we also have a height on its $\bar{\mathbb{Q}}$-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 $\bar{\mathbb{Q}}$. Using Moriwaki's height we sketch how to extend our result when the base field of the curve has characteristic 0.