Elliptic surfaces and intersections of adelic $\mathbb{R}$-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 Néron-Tate height functions on the fibers $E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t\in B(\overline{\mathbb{Q}})$: given points $P_1,\dots ,P_m\in E(k)$ with $m\ge 2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ and small-height perturbations $P_{i,t_n}^{\prime }\in E_{t_n}(\overline{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set $\{P_{1,t_n}^{\prime },\dots ,P_{m,t_n}^{\prime }\}$ satisfies at least two independent linear relations for all $n$, if and only if the points $P_1,\dots ,P_m$ 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.

Added an appendix to prove an equidistribution theorem for $\mathbb{R}$-divisors in all dimensions

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