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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Abstract
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.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Zahlentheorie (Habegger) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Habegger, Philipp 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
edoc DOI:  
Last Modified:  01 Apr 2019 15:13 
Deposited On:  01 Apr 2019 15:13 
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