Selmer groups and Tate–Shafarevich groups for the congruent number problem
Selmer groups and Tate–Shafarevich groups for the congruent number problem
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DOI：
10.4171/cmh/151
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发表时间：
2009-03
影响因子：
0.9
通讯作者：
Maosheng Xiong;A. Zaharescu
中科院分区：
文献类型：
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作者：
Maosheng Xiong;A. Zaharescu
我们研究了来自椭圆曲线\(E_n:y^2 = x^3 - n^2x\)的三个\(2\)-同构及其对偶\(2\)-同构所产生的塞尔默群的规模分布。我们表明其中三个几乎总是平凡的，而另外三个的\(2\)-秩遵循高斯分布。这意味着三个几乎总是平凡的泰特 - 沙法列维奇群和三个大的泰特 - 沙法列维奇群。当与希思 - 布朗得到的一个结果相结合时，我们表明对于无平方因子的正奇数\(n = X\)，大泰特 - 沙法列维奇群的\(2\)-秩的平均值为\(\frac{1}{2}\log\log X + O(1)\)，当\(X\rightarrow\infty\)。
最后“as \(X = 8\)”这里应该是\(X\rightarrow\infty\)（趋于无穷）的笔误，否则逻辑不通。
We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve En: y2 = x3 - n2x. We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate�Shafarevich groups and three large Tate�Shafarevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate�Shafarevich groups for square-free positive odd integers n = X is ½ loglog X + O(1), as X = 8.