### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- Elliptic surfaces
- Heights
- Unlikely intersections in arithmetic dynamics

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

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

**A variant of a theorem by Ailon-Rudnick for elliptic curves.** / Ghioca, Dragos; Hsia, Liang Chung; Tucker, Thomas J.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 295, no. 1, pp. 1-15. https://doi.org/10.2140/pjm.2018.295.1

}

TY - JOUR

T1 - A variant of a theorem by Ailon-Rudnick for elliptic curves

AU - Ghioca, Dragos

AU - Hsia, Liang Chung

AU - Tucker, Thomas J.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(P2) ii. Qi is a multiple of Pi for some i = 1,2. A special case of our result answers a conjecture made by Silverman.

AB - Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε1 → C and ε2 → C along with sections σPi, σQi (corresponding to points Pi, Qi of the generic fibers) of εi (for i = 1, 2), we prove that if there exist infinitely many t ∈ C(ℚ) such that for some integers m1,t,m2,t, we have [mi,t](σPi(t)) = σQi (t) on εi (for i = 1, 2), then at least one of the following conclusions must hold: i. There exist isogenies ϕ:ε1→ε2 and ψ:ε2→ε2 such that ϕ(P1)=(P2) ii. Qi is a multiple of Pi for some i = 1,2. A special case of our result answers a conjecture made by Silverman.

KW - Elliptic surfaces

KW - Heights

KW - Unlikely intersections in arithmetic dynamics

UR - http://www.scopus.com/inward/record.url?scp=85044200457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044200457&partnerID=8YFLogxK

U2 - 10.2140/pjm.2018.295.1

DO - 10.2140/pjm.2018.295.1

M3 - Article

AN - SCOPUS:85044200457

VL - 295

SP - 1

EP - 15

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -