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Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined overa number field, for smooth projective curve $B$. Let $k$ denote the functionfield $\overline{\mathbb{Q}}(B)$ 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 $X \inE(k)\otimes \mathbb{R}$. We prove non-degeneracy of the Arakelov-Zhangintersection numbers $\overline{D}_X\cdot \overline{D}_Y$, as a biquadraticform on $E(k)\otimes \mathbb{R}$. As a consequence, we have the followingBogomolov-type statement for the N\'eron-Tate height functions on the fibers$E_t(\overline{\mathbb{Q}})$ of $\mathcal{E}$ over $t \inB(\overline{\mathbb{Q}})$: given points $P_1, \ldots, P_m \in E(k)$ with $m\geq2$, there exist an infinite sequence $t_n\in B(\overline{\mathbb{Q}})$ andsmall-height perturbations $P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}})$ ofspecializations $P_{i,t_n}$ 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 andCapuano and extends our earlier results. In the Appendix, we prove anequidistribution theorem for adelically metrized $\mathbb{R}$-divisors onprojective varieties (over a number field) using results of Moriwaki, extendingthe equidistribution theorem of Yuan.

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