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On an abelian scheme over a smooth curve over $\overline {\mathbb {Q}}$ 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 $\overline {\mathbb {Q}}$-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 $0$.
@article{10_4007_annals_2019_189_2_3,
author = {Ziyang Gao and Philipp Habegger},
title = {Heights in families of abelian varieties and the {Geometric} {Bogomolov} {Conjecture}},
journal = {Annals of mathematics},
pages = {527--604},
publisher = {mathdoc},
volume = {189},
number = {2},
year = {2019},
doi = {10.4007/annals.2019.189.2.3},
mrnumber = {3922127},
zbl = {07041750},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.3/}
}
TY - JOUR AU - Ziyang Gao AU - Philipp Habegger TI - Heights in families of abelian varieties and the Geometric Bogomolov Conjecture JO - Annals of mathematics PY - 2019 SP - 527 EP - 604 VL - 189 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.3/ DO - 10.4007/annals.2019.189.2.3 LA - en ID - 10_4007_annals_2019_189_2_3 ER -
%0 Journal Article %A Ziyang Gao %A Philipp Habegger %T Heights in families of abelian varieties and the Geometric Bogomolov Conjecture %J Annals of mathematics %D 2019 %P 527-604 %V 189 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2019.189.2.3/ %R 10.4007/annals.2019.189.2.3 %G en %F 10_4007_annals_2019_189_2_3
Ziyang Gao; Philipp Habegger. Heights in families of abelian varieties and the Geometric Bogomolov Conjecture. Annals of mathematics, Tome 189 (2019) no. 2, pp. 527-604. doi: 10.4007/annals.2019.189.2.3
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