A variant of a theorem by Ailon–Rudnick for elliptic curves

Dragos Ghioca, Liang-Chung Hsia and Thomas J. Tucker

Vol. 295 (2018), No. 1, 1–15

DOI: 10.2140/pjm.2018.295.1

Abstract

Given a smooth projective curve $C$ defined over $\overline{\mathbb{Q}}$ and given two elliptic surfaces ${\mathcal{ℰ}}_1\to C$ and ${\mathcal{ℰ}}_2\to C$ along with sections ${\sigma }_{P_i},{\sigma }_{Q_i}$ (corresponding to points $P_i,Q_i$ of the generic fibers) of ${\mathcal{ℰ}}_i$ (for $i=1,2$), we prove that if there exist infinitely many $t\in C\left(\overline{\mathbb{Q}}\right)$ such that for some integers $m_{1,t},m_{2,t}$, we have $\left[m_{i,t}\right]\left({\sigma }_{P_i}\left(t\right)\right)={\sigma }_{Q_i}\left(t\right)$ on ${\mathcal{ℰ}}_i$ (for $i=1,2$), then at least one of the following conclusions must hold:

i. There exist isogenies $\phi :E_1\to E_2$ and $\psi :E_2\to E_2$ such that $\phi \left(P_1\right)=\psi \left(P_2\right)$. ii. $Q_i$ is a multiple of $P_i$ for some $i=1,2$.

A special case of our result answers a conjecture made by Silverman.

Keywords

heights, elliptic surfaces, unlikely intersections in arithmetic dynamics

Mathematical Subject Classification 2010

Primary: 11G50

Secondary: 11G35, 14G25

Milestones

Received: 8 March 2017

Revised: 10 November 2017

Accepted: 3 January 2018

Published: 13 March 2018