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 E1/C and E2/C along with sections Pi and Qi of Ei (for i = 1,2), we prove that if there exist infinitely many algebraic points t on C such that for some integers m1,t and m2,t, we have that [mi,t](Pi)t = (Qi)t on Ei (for i = 1,2), then at least one of the following conclusions must hold: either (i) there exists an isogeny f between E1 and E2 and also there exists a nontrivial endomorphism g of E2 such that f(P1) = g(P2); or (ii) Qi is a multiple of Pi for some i = 1,2. A special case of our result answers a conjecture made by Silverman.
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