Variation of canonical height and equidistribution

Let $\pi :E\to B$ be an elliptic surface defined over a number field $K$, where $B$ is a smooth projective curve, and let $P:B\to E$ be a section defined over $K$ with canonical height ${\widehat{h}}_E(P)≠0$. In this article, we show that the function $t\mapsto {\widehat{h}}_{E_t}(P_t)$ on $B(\bar{K})$ is the height induced from an adelically metrized line bundle with non-negative curvature on $B$. Applying theorems of Thuillier and Yuan, we obtain the equidistribution of points $t\in B(\bar{K})$ where $P_t$ is torsion, and we give an explicit description of the limiting distribution on $B(\mathbb{C})$. Finally, combined with results of Masser and Zannier, we show there is a positive lower bound on the height ${\widehat{h}}_{A_t}(P_t)$, after excluding finitely many points $t\in B$, for any “non-special” section $P$ of a family of abelian varieties $A\to B$ that split as a product of elliptic curves.