Distribution of Selmer groups of quadratic twists of a family of elliptic curves

Maosheng Xiong, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

Abstract

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q. We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate-Shafarevich group and a large Tate-Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1-2) (2005) 139-154 (2006)], this shows that the mean value of the 2-rank of the large Tate-Shafarevich group for square-free positive integers n less than X is frac(1, 2) log log X + O (1), as X → ∞.

Original languageEnglish (US)
Pages (from-to)523-553
Number of pages31
JournalAdvances in Mathematics
Volume219
Issue number2
DOIs
StatePublished - Oct 1 2008

Keywords

  • Elliptic curves
  • Erdo{combining double acute accent}s-Kac Theorem
  • Selmer group
  • Tate-Shafarevich group

ASJC Scopus subject areas

  • General Mathematics

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