Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Ziyang Gao
  • Philipp Habegger

External Research Organisations

  • Princeton University
  • University of Basel
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Details

Original languageEnglish
Pages (from-to)527-604
Number of pages78
JournalAnnals of Mathematics
Volume189
Issue number2
Publication statusPublished - 1 Mar 2019
Externally publishedYes

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.

Keywords

    math.NT, 11G10, 11G50, 14G25, 14K15, Functional constancy, Height inequality, Geometric and Relative Bogomolov Conjecture, Point counting, O-minimality

ASJC Scopus subject areas

Cite this

Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. / Gao, Ziyang; Habegger, Philipp.
In: Annals of Mathematics, Vol. 189, No. 2, 01.03.2019, p. 527-604.

Research output: Contribution to journal › Article › Research › peer review

Gao Z, Habegger P. Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. Annals of Mathematics. 2019 Mar 1;189(2):527-604. doi: 10.48550/arXiv.1801.05762, 10.4007/annals.2019.189.2.3
Gao, Ziyang ; Habegger, Philipp. / Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. In: Annals of Mathematics. 2019 ; Vol. 189, No. 2. pp. 527-604.
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