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 - 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 -