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On Selmer groups and Tate-Shafarevich groups for elliptic curves \(y^{2}=x^{3}-n^3\). (English) Zbl 1281.11053
Summary: We study the distribution of the size of Selmer groups and Tate-Shafarevich groups arising from a 2-isogeny and its dual 2-isogeny for elliptic curves \(E_n: y^2=x^3-n^3\). We show that the 2-ranks of these groups all follow the same distribution. The result also implies that the mean value of the 2-rank of the corresponding Tate-Shafarevich groups for square-free positive integers \(n\leq X\) is \(X\sqrt{\frac{1}{2} \log \log X}\) as \(X \rightarrow \infty \). This is quite different from quadratic twists of elliptic curves with full 2-torsion points over \(\mathbb Q\) [M. Xiong and A. Zaharescu, Adv. Math. 219, No. 2, 523–553 (2008; Zbl 1194.11066)], where one Tate-Shafarevich group is almost always trivial while the other is much larger.

MSC:
11G05 Elliptic curves over global fields
11L40 Estimates on character sums
11N45 Asymptotic results on counting functions for algebraic and topological structures
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