科研之友
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摘要
Given a smooth projective curve C defined over (Q)over-bar and given two elliptic surfaces epsilon(1) -> C and epsilon(2) -> C along with sections sigma(Pi), sigma(Q)(i) (corresponding to points P-i, Q(i) of the generic fibers) of epsilon(i) (for i = 1, 2), we prove that if there exist infinitely many t is an element of C(Q)over-bar such that for some integers m(1,t), m(2,t), we have [m(i,t)] (sigma(Pi) (t) = sigma(Q)(i) (t) on epsilon(i) (for i = 1, 2), then at least one of the following conclusions must hold:
i. There exist isogenies phi : E-l -> E-2 and psi : E-2 -> E-2 such that phi(P-1) = psi(P-2).
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.
- 出版日期2018-7
