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.

N1 - Funding Information:
We thank Myrto Mavraki and Joe Silverman for several useful conversations. We are grateful to the anonymous referee for numerous comments and suggestions which improved our paper. The research of Ghioca was partially supported by an NSERC Discovery grant. Hsia was supported by MOST grant 104-2115-M-003-004-MY2. Tucker was partially supported by NSF Grant DMS-0101636.
Publisher Copyright:
© 2018 Mathematical Sciences Publishers.

PY - 2018

Y1 - 2018

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

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U2 - 10.2140/pjm.2018.295.1

DO - 10.2140/pjm.2018.295.1

M3 - Article

AN - SCOPUS:85044200457

SN - 0030-8730

VL - 295

SP - 1

EP - 15

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 1

ER -