# Variation of canonical height and equidistribution

## TL;DR

This paper studies the variation of canonical heights in elliptic surfaces over number fields, showing they induce a non-negative curvature height function, leading to equidistribution results and bounds on heights in families of abelian varieties.

## Contribution

It establishes that the canonical height variation is an adelically metrized line bundle with non-negative curvature, enabling equidistribution theorems and explicit height bounds in families of abelian varieties.

## Key findings

- Canonical height function is induced from an adelically metrized line bundle with non-negative curvature.
- Points where the section is torsion are equidistributed with an explicit limiting distribution.
- There is a positive lower bound on heights in non-special families, excluding finitely many points.

## Abstract

Let $\pi : E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P: B \to E$ be a section defined over $K$ with canonical height $\hat{h}_E(P)\not=0$. In this article, we show that the function $t \mapsto \hat{h}_{E_t}(P_t)$ on $B(\overline{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$. Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points $t \in B(\overline{K})$ where $P_t$ is torsion, and we give an explicit description of the limiting distribution on $B(\mathbb{C})$. Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height $\hat{h}_{A_t}(P_t)$, after excluding finitely many points $t \in B$, for any "non-special" section $P$ of a family of abelian varieties $A \to B$ that split as a product of elliptic curves.