Collision of Orbits on an Elliptic Surface

Let $C$ be a smooth projective curve defined over $\Qbar$, let $π:\mathcal{E}\lra C$ be an elliptic surface and let $σ_{P_1},σ_{P_2},σ_{Q}$ be sections of $π$ (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 $ł\in C(\Qbar)$ with the property that for some nonzero integers $m_{1,ł},m_{2,ł}$, we have that $m_{i,ł}=σ_{Q}(ł)$ (for $i=1,2$) on the smooth fiber $E_ł$ of $\mathcal{E}$.

💡 Research Summary

The paper studies a precise “collision of orbits” problem on an elliptic surface π : 𝔈 → C defined over ℚ̅. Given three sections σ_{P₁}, σ_{P₂}, σ_Q of π (corresponding to points P₁, P₂, Q in the generic fiber E), the authors ask when there exist infinitely many points λ∈C(ℚ̅) such that, on the smooth fiber E_λ, the specialization Q_λ lies simultaneously in the cyclic subgroups generated by P₁,λ and by P₂,λ. In other words, for each such λ there are non‑zero integers m_{1,λ}, m_{2,λ} with