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

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 $\bar{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In this article, we construct adelically metrized $\mathbb{R}$-divisors ${\bar{D}}_X$ 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 ${\bar{D}}_X\cdot {\bar{D}}_Y$, 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 $E_t(\bar{\mathbb{Q}})$ of $\mathcal{E}$ over $t \in B(\overline{$\mathbb{Q}$})$:givenpoints$P_1, \ldots, P_m \in E(k)$with$m\geq 2$,thereexistaninfinitesequence$t_n\in B(\overline{$\mathbb{Q}$})$ and small-height perturbations $P_{i,t_n}^{\mathrm{\prime }}\in E_{t_n}(\bar{\mathbb{Q}})$ of specializations $P_{i,t_n}$ so that the set ${P_{1, t_n}‘, \ldots, P_{m,t_n}’}$satisfiesatleasttwoindependentlinearrelationsforall$n$, 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.