  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 a non-repeating 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. We also
prove a general equidistribution theorem for adelically metrized
$\mathbb{R}$-divisors on curves (over a number field), extending the
equidistribution theorems of Chambert-Loir, Thuillier, and Yuan, and using
results of Moriwaki.
