Variation of canonical height and equidistribution

Let π: E → B be an elliptic surface defined over a number field K, where B is a smooth projective curve, and let P: B → E be a section defined over K with canonical height ĥ_E (P) ≠ 0. In this article, we show that the function t ↦→ ĥ_Et (P_t) on B(K) is the height induced from an adelically metrized line bundle with continuous semipositive metrics on B. The proof builds on work of Silverman and results from complex dynamical systems. Applying arithmetic equidistribution theorems (of Chambert-Loir, Thuillier, and Yuan), we obtain the equidistribution of points t ∈ B(K) where P_t is torsion, and we give an explicit description of the limiting distribution on B(C). Finally, combined with results of Masser and Zannier, we show that—for any non-special section P of a family of abelian varieties A → B that split as a product of elliptic curves—there is a positive lower bound on the height ĥ_At (P_t), after excluding finitely many points t ∈ B, thus addressing a conjecture of Zhang from 1998.