Bounding tangencies of sections on elliptic surfaces

Abstract

Given an elliptic surface $\mathcal{E}\to \mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

v2: corrections and additional application after referee report. 24 pages