Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves
Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves
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椭圆曲线对二次扭曲的 Mordell-Weil 等级
DOI：
10.1006/jnth.2002.2788
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发表时间：
2002
影响因子：
0.7
通讯作者：
Jorge Jiménez
中科院分区：
文献类型：
--
作者：
Gwynneth H. Coogan;Jorge Jiménez
受Mazur猜想的启发，Kuwata和Wang证明了对于j-不变量不同时为0或1728的椭圆曲线E1和E2，存在无穷多个无平方数的整数d，使得E1和E2的d-二次扭的Mordell-Weil群的秩满足：rk（Ed 1，Q）>0和rk（Ed 2，Q）>0。在这里，我们提出的相关问题的结果：是否有无穷多个无平方的整数d，其中：rk（Ed 1，Q）=0和rk（Ed 2，Q）=0？并且，是否存在无限多个无平方数的整数d，使得：rk（Ed 1，Q）=0且rk（Ed 2，Q）>0？
Abstract Motivated by a conjecture of Mazur, Kuwata and Wang proved that for elliptic curves E1 and E2 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 E1 and E2 satisfy: rk(Ed1, Q )>0 and rk(Ed2, Q )>0. Here we present results for the related questions: Are there infinitely many square-free integers d for which: rk(Ed1, Q )=0 and rk(Ed2, Q )=0? And, are there infinitely many square-free integers d for which: rk(Ed1, Q )=0 and rk(Ed2, Q )>0?