Commentarii Mathematici Helvetici

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Volume 84, Issue 1, 2009, pp. 21–56
DOI: 10.4171/CMH/151

Published online: 2009-03-31

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

Maosheng Xiong[1] and Alexandru Zaharescu[2]

(1) Pennsylvania State University, University Park, United States
(2) University of Illinois at Urbana-Champaign, United States

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 = x3n2x. 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 nX is ½ log log X + O(1), as X → ∞.

Keywords: Elliptic curves, congruent number problem, Selmer group, Tate–Shafarevich group, Erdös–Kac Theorem

Xiong Maosheng, Zaharescu Alexandru: Selmer groups and Tate–Shafarevich groups for the congruent number problem. Comment. Math. Helv. 84 (2009), 21-56. doi: 10.4171/CMH/151