Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

TL;DR

This paper constructs adelic divisors on elliptic surfaces over number fields, proves their intersection form is non-degenerate, and derives a Bogomolov-type result linking point dependencies to specializations, extending prior work.

Contribution

It introduces a new adelic divisor framework on elliptic surfaces, proves intersection non-degeneracy, and establishes a Bogomolov-type theorem for fiber points, extending previous results.

Findings

Non-degeneracy of Arakelov-Zhang intersection numbers

Existence of infinite sequences with linear relations among specialized points

Extension of equidistribution theorems to adelic $ ext{R}$-divisors

Abstract

Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors $\overline{D}_X$ on the base curve $B$ over a number field, for each $X \in E(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhang intersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadratic form on $E(k)\otimes \mathbb{R}$. As a consequence, we have the following Bogomolov-type statement for the N\'eron-Tate height functions on the fibers $E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t \in B(\overline{\mathbb{Q}})$: given points $P_1, \ldots, P_m \in E(k)$ with $m\geq 2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\{P_{1, t_n}', \ldots, P_{m,t_n}'\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1, \ldots, P_m$ are linearly dependent in $E(k)$. This gives a new proof of results of Masser and Zannier and of Barroero and Capuano and extends our earlier results. In the Appendix, we prove an equidistribution theorem for adelically metrized $\mathbb{R}$-divisors on projective varieties (over a number field) using results of Moriwaki, extending the equidistribution theorem of Yuan.