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 log log X + O(1), as X → ∞.
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Maosheng Xiong, Alexandru Zaharescu, Selmer groups and Tate–Shafarevich groups for the congruent number problem. Comment. Math. Helv. 84 (2009), no. 1, pp. 21–56DOI 10.4171/CMH/151