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.
- Elliptic surfaces
- Unlikely intersections in arithmetic dynamics
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