Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Collison of Orbits on an Elliptic Surface

Dragos Ghioca, Negin Shadgar

TL;DR

This work analyzes collisions of orbits on fibers of an elliptic surface, characterizing when there are infinitely many base points λ for which a target point Q lies in both generated orbits of P_1 and P_2 under End(E). It gives a precise, CM-sensitive criterion: either a global relation [k](P_i)=Q, a shared End(E)-relation [k_1](P_1)=[k_2](P_2), or, in the CM case, a triple α_1,α_2,β in the endomorphism ring with α_1/α_2 not rational such that [α_1](P_1)=[α_2](P_2)=[β](Q). The argument combines unlikely-intersection philosophy with CM-arithmetic tools, using Betti-coordinates and endomorphism dynamics to move from fiberwise statements to global relations, and then proves the converse by case analysis, including CM refinements. The results extend the Masser–Zannier/DeMarco–Mavraki paradigm to a collision-of-orbits setting for elliptic curves, providing a complete description of when infinite base points yield a shared orbit membership across fibers. This advances understanding of arithmetic dynamics in families of elliptic curves and highlights the distinct role of complex multiplication in orbit-collision phenomena.

Abstract

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 $[m_{i,ł}](σ_{P_{i}}(ł))=σ_{Q}(ł)$ (for $i=1,2$) on the smooth fiber $E_ł$ of $\mathcal{E}$.

Collison of Orbits on an Elliptic Surface

TL;DR

This work analyzes collisions of orbits on fibers of an elliptic surface, characterizing when there are infinitely many base points λ for which a target point Q lies in both generated orbits of P_1 and P_2 under End(E). It gives a precise, CM-sensitive criterion: either a global relation [k](P_i)=Q, a shared End(E)-relation [k_1](P_1)=[k_2](P_2), or, in the CM case, a triple α_1,α_2,β in the endomorphism ring with α_1/α_2 not rational such that [α_1](P_1)=[α_2](P_2)=[β](Q). The argument combines unlikely-intersection philosophy with CM-arithmetic tools, using Betti-coordinates and endomorphism dynamics to move from fiberwise statements to global relations, and then proves the converse by case analysis, including CM refinements. The results extend the Masser–Zannier/DeMarco–Mavraki paradigm to a collision-of-orbits setting for elliptic curves, providing a complete description of when infinite base points yield a shared orbit membership across fibers. This advances understanding of arithmetic dynamics in families of elliptic curves and highlights the distinct role of complex multiplication in orbit-collision phenomena.

Abstract

Let be a smooth projective curve defined over , let be an elliptic surface and let be sections of (corresponding to points of the generic fiber of ). We obtain a precise characterization, expressed solely in terms of the dynamical relations between the points with respect to the endomorphism ring of , so that there exist infinitely many with the property that for some nonzero integers , we have that (for ) on the smooth fiber of .
Paper Structure (12 sections, 15 theorems, 42 equations)

This paper contains 12 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Notation and setup for our problem
  3. Our main result
  4. Unlikely intersections in arithmetic dynamics
  5. Previous results and further connections to our work
  6. Plan for our paper
  7. Arithmetic properties for an elliptic curve with complex multiplication
  8. Proof of the direct implication in Theorem \ref{['thm:main']}
  9. Proof of the converse implication in Theorem \ref{['thm:main']}
  10. A useful result
  11. Proof of Proposition \ref{['prop:converse']} assuming conditions (A) or (B) hold
  12. Proof of Proposition \ref{['prop:converse']} assuming condition (C) holds

Key Result

Theorem 1.1

Let $C$ be a smooth projective curve defined over ${\overline{\mathbb Q}}$, let $\mathcal{E}\longrightarrow C$ be an elliptic surface and let $P_1,P_2, Q\in E({\overline{\mathbb Q}}(C))$ be non-torsion points on the generic fiber $E$ of $\pi$. Furthermore, if $\mathcal{E}$ is isotrivial, then not a

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • ...and 19 more