Mordell-Weil ranks of quadratic twists of pairs of elliptic curves

[摘要] Motivated by a conjecture of Mazur, Kuwata and Wang proved that for elliptic curves E-1 and E-2 whose j-invariants are not simultaneously 0 or 1728, there exist infinitely many square-free integers d for which the rank of the Mordell-Weil group of the d-quadratic twists of E-1 and E-2 satisfy: rk(E-d(1), Q) > 0 and rk(E-d(2), Q) > 0. Here we present results for the related questions: Are there infinitely many square-free integers d for which: rk(E-d(1), Q) = 0 and rk(E-d(2), Q) = 0? And, are there infinitely many square-free integers d for which: rk(E-d(1), Q) = 0 and rk (E-d(2), Q) > 0? (C) 2002 Elsevier Science (USA).

[发布日期] 2002-10-01