Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Ziyang Gao
  • Philipp Habegger

Externe Organisationen

  • Princeton University
  • Universität Basel
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Details

OriginalspracheEnglisch
Seiten (von - bis)527-604
Seitenumfang78
FachzeitschriftAnnals of Mathematics
Jahrgang189
Ausgabenummer2
PublikationsstatusVeröffentlicht - 1 März 2019
Extern publiziertJa

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.

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Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. / Gao, Ziyang; Habegger, Philipp.
in: Annals of Mathematics, Jahrgang 189, Nr. 2, 01.03.2019, S. 527-604.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Gao Z, Habegger P. Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. Annals of Mathematics. 2019 Mär 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 ; Jahrgang 189, Nr. 2. S. 527-604.
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