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

Maosheng Xiong, Alexandra Zaharescu

Research output: Contribution to journalArticlepeer-review


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 : 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 1/2 log log X + O(l), as X → ∞.

Original languageEnglish (US)
Pages (from-to)21-56
Number of pages36
JournalCommentarii Mathematici Helvetici
Issue number1
StatePublished - 2009


  • Congruent number problem
  • Elliptic curves
  • Erdös-kac theorem
  • Selmer group
  • Tate-shafarevich group

ASJC Scopus subject areas

  • Mathematics(all)


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