Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

• 成果类型: Article
• 署名作者: Gao, Ziyang; Habegger, Philipp
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2019.189.2.3
• 发表日期: 2019
• 页码: 527-604
• 关键词: rational-points; CURVES; THEOREM

摘要

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