Collision of Orbits on an Elliptic Surface

TL;DR

This paper characterizes when infinitely many fibers of an elliptic surface contain points where two sections simultaneously collide with a third, based on the dynamical relations of their associated points in the generic fiber.

Contribution

It provides a precise criterion, in terms of endomorphism ring relations, for the existence of infinitely many such collision points on fibers of an elliptic surface.

Findings

Characterization of infinite collision points in terms of dynamical relations.

Connection between endomorphism ring and fiber intersections.

Conditions for simultaneous point collisions on elliptic surface fibers.

Abstract

Let $C$ be a smooth projective curve defined over $\Qbar$, let $\pi:\mathcal{E}\lra C$ be an elliptic surface and let $\sigma_{P_1},\sigma_{P_2},\sigma_{Q}$ be sections of $\pi$ (corresponding to points $P_1,P_2, Q$ of the generic fiber $E$ of $\mathcal{E}$). We obtain a precise characterization, expressed solely in terms of the dynamical relations between the points $P_1,P_2,Q$ with respect to the endomorphism ring of $E$, so that there exist infinitely many $\l\in C(\Qbar)$ with the property that for some nonzero integers $m_{1,\l},m_{2,\l}$, we have that $[m_{i,\l}](\sigma_{{P_{i}}}(\l))=\sigma_{Q}(\l)$ (for $i=1,2$) on the smooth fiber $E_\l$ of $\mathcal{E}$.