Specialization of canonical heights on abelian varieties

TL;DR

This paper establishes a precise relationship between Néron-Tate heights on abelian varieties and Weil heights on the base curve, linking height specialization properties to the geometric property of the associated line bundle.

Contribution

It proves that the canonical height along fibers equals a Weil height from an adelic line bundle, connecting height finiteness conjectures to geometric bigness.

Findings

Canonical heights match Weil heights from adelic line bundles.

Finiteness of small-height specializations relates to the bigness of the line bundle.

Provides a geometric criterion for height finiteness conjectures.

Abstract

Given a family of abelian varieties over a quasiprojective smooth curve $T^0$ over a global field and a point $P$ on the generic fiber, we show that the N\'eron-Tate canonical height $h_{X_t}(P_t)$ of $P_t$ along each fiber is exactly equal to a Weil height $h_{\overline M}(t)$ given by an adelic metrized line bundle $\overline M$ on the unique smooth projective curve $T$ containing $T^0$. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of $P$ is equivalent to $\overline M$ being big.