### Abstract

Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε_{1} → C and ε_{2} → C along with sections σ_{Pi}, σ_{Qi} (corresponding to points P_{i}, Q_{i} of the generic fibers) of ε_{i} (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m_{1,t},m_{2,t}, we have [m_{i,t}](σP_{i}(t)) = σ_{Qi} (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε_{1}→ε_{2} and ψ:ε_{2}→ε_{2} such that ϕ(P1)=(P2) 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.

Original language | English |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Pacific Journal of Mathematics |

Volume | 295 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2018 Jan 1 |

### Keywords

- Elliptic surfaces
- Heights
- Unlikely intersections in arithmetic dynamics

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Ghioca, D., Hsia, L. C., & Tucker, T. J. (2018). A variant of a theorem by Ailon-Rudnick for elliptic curves.

*Pacific Journal of Mathematics*,*295*(1), 1-15. https://doi.org/10.2140/pjm.2018.295.1