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 $E_n:y^2=x^3-n^{2}x$. 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 $n\le X$ is $\frac{1}{2}\log \log X+O(1)$, as $X\to \infty$.