2017/01/27 by DeMarco, Laura, Mavraki, Niki Myrto · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1701.07947
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 hE(P)\not=0. In this article, we show that the function t ↦ hEt(Pt) on B(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 ∈ B(K) where Pt 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 there is a positive lower bound on the height hAt(Pt), after excluding finitely many points t ∈ B, for any "non-special" section P of a family of abelian varieties A → B that split as a product of elliptic curves.