Collision of Orbits on an Elliptic Surface

Dragos Ghioca; Negin Shadgar

paper

Collision of Orbits on an Elliptic Surface

arXiv:2602.10383

Abstract

Let $C$ be a smooth projective curve defined over $\Qbar$, let $Ï:\mathcal{E}\lra C$ be an elliptic surface and let $I ¨ _{P_{1}}, I ¨ _{P_{2}}, I ¨ _{Q}$ be sections of $I ¨ $ (corresponding to points $P_{1}, P_{2}, Q$ of the generic fiber $E$ of $E$ ). 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 $Å\in C(\Qbar)$ with the property that for some nonzero integers $m_{1, A ˚ }, m_{2, A ˚ }$ , we have that $[m_{i, A ˚ }] (I ¨ _{P_{i}} (A ˚ )) = I ¨ _{Q} (A ˚ )$ (for $i = 1, 2$ ) on the smooth fiber $E_{A ˚} $ of $E$ .