TY - JOUR

T1 - Distribution of Selmer groups of quadratic twists of a family of elliptic curves

AU - Xiong, Maosheng

AU - Zaharescu, Alexandru

N1 - Funding Information:
* Corresponding author. E-mail addresses: xiong@math.psu.edu (M. Xiong), zaharesc@math.uiuc.edu (A. Zaharescu). 1 Supported by NSF grant number DMS-0456615, and by CNCSIS grant GR106/2007, code 1116, of the Romanian Ministry of Education and Research.

PY - 2008/10/1

Y1 - 2008/10/1

N2 - 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 → ∞.

AB - 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 → ∞.

KW - Elliptic curves

KW - Erdo{combining double acute accent}s-Kac Theorem

KW - Selmer group

KW - Tate-Shafarevich group

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U2 - 10.1016/j.aim.2008.05.005

DO - 10.1016/j.aim.2008.05.005

M3 - Article

AN - SCOPUS:48949097878

VL - 219

SP - 523

EP - 553

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 2

ER -