Vol. 295, No. 1, 2018

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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
Abstract

Given a smooth projective curve $C$ defined over $\overline{ℚ}$ and given two elliptic surfaces ${\mathsc{ℰ}}_{1}\to C$ and ${\mathsc{ℰ}}_{2}\to C$ along with sections ${\sigma }_{{P}_{i}},{\sigma }_{{Q}_{i}}$ (corresponding to points ${P}_{i},{Q}_{i}$ of the generic fibers) of ${\mathsc{ℰ}}_{i}$ (for $i=1,2$), we prove that if there exist infinitely many $t\in C\left(\overline{ℚ}\right)$ 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 ${\mathsc{ℰ}}_{i}$ (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