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

Dragos Ghioca, Liang Chung Hsia, Thomas J. Tucker

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalPacific Journal of Mathematics
Volume295
Issue number1
DOIs
Publication statusPublished - 2018 Jan 1

    Fingerprint

Keywords

  • Elliptic surfaces
  • Heights
  • Unlikely intersections in arithmetic dynamics

ASJC Scopus subject areas

  • Mathematics(all)

Cite this