On the Tate–Shafarevich group of semistable elliptic curves with a rational 3-torsion

Aoki, Noboru

doi:10.4064/aa112-3-1

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2009

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“…Even the finiteness of the group is not known in general. Various families of elliptic curves with large Tate-Shafarevich groups were identified by a number of authors (see [2], [3], [6], [7], [24], [26], [28], [30]). Moments ( [10]), heuristic results ( [9]), and upper bounds ( [17], [18]) on the order of Tate-Shafarevich groups were also considered.…”

Section: Introductionmentioning

confidence: 99%

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

Xiong¹,

Zaharescu²

2009

Comment. Math. Helv.

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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 W y 2 D 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 Ä X is 1 2 log log X C O.1/, as X ! 1.

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