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

Given a smooth projective curve C defined over ℚ and given two elliptic surfaces ε₁ → C and ε₂ → 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:ε₁→ε₂ and ψ:ε₂→ε₂ 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.