Heights in families of abelian varieties and the Geometric Bogomolov Conjecture

TL;DR

This paper establishes an inequality between fiberwise Néron-Tate heights and base heights in abelian schemes over curves, leading to a proof of the Geometric Bogomolov Conjecture for function fields over algebraic closures of rationals.

Contribution

It proves a height inequality for subvarieties of abelian schemes and applies it to confirm the Geometric Bogomolov Conjecture in characteristic zero.

Findings

Proved height inequality for subvarieties of abelian schemes.

Confirmed the Geometric Bogomolov Conjecture for function fields over algebraic closures of rationals.

Outlined extension of results to characteristic zero using Moriwaki's height.

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.