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Heights in families of abelian varieties and the geometric Bogomolov Conjecture
JournalArticle (Originalarbeit in einer wissenschaftlichen Zeitschrift) |
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ID |
4496810 |
Author(s) |
Gao, Ziyang; Habegger, Philipp |
Author(s) at UniBasel |
Habegger, Philipp
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Year |
2018 |
Title |
Heights in families of abelian varieties and the geometric Bogomolov Conjecture |
Journal |
Annals of mathematics |
Pages / Article-Number |
78 pages |
Abstract |
On an abelian scheme over a smooth curve over $\overline{\mathbb{Q}}$ a symmetric relatively ample line bundle defines a fiberwise N\'{e}ron--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 $0$. |
ISSN/ISBN |
0003-486X |
Full Text on edoc |
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Additional Information |
Accepted for publication in 2018 |
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