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

Abstract: Given a smooth projective curve C defined over a number field and given two elliptic surfaces E_1/C and E_2/C along with sections P_i and Q_i of E_i (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m_{1,t} and m_{2,t}, we have that m_{i,t}_t = (Q_i)_t on E_i (for i = 1,2), then at least one of the following conclusions must hold: either (i) there exists an isogeny f between E_1 and E_2 and also there exists a nontrivial endomorphism g of E_2 such that f(P_1) = g(P_2); or (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.