Selmer groups and Tate–Shafarevich groups for the congruent number problem

Abstract

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.