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Distribution of Selmer groups of quadratic twists of a family of elliptic curves
- Source :
-
Advances in Mathematics . Oct2008, Vol. 219 Issue 2, p523-553. 31p. - Publication Year :
- 2008
-
Abstract
- 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 . 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 , as . [Copyright &y& Elsevier]
- Subjects :
- *ALGEBRAIC curves
*ELLIPTIC curves
*MATHEMATICAL analysis
*GAUSSIAN distribution
Subjects
Details
- Language :
- English
- ISSN :
- 00018708
- Volume :
- 219
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Advances in Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 33890535
- Full Text :
- https://doi.org/10.1016/j.aim.2008.05.005