A Variant of a Theorem by Ailon-rudnick for Elliptic Curves
نویسنده
چکیده
Given a smooth projective curve C defined over Q and given two elliptic surfaces E1 −→ C and E2 −→ C along with sections Pi, Qi of Ei (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 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 ψ : E1 −→ E2 and also there exist nontrivial endomorphisms φi of Ei (for i = 1, 2) such that φ2(P2) = ψ(φ1(P1)); 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.
منابع مشابه
On a Variant of the Ailon-rudnick Theorem in Finite Characteristic
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