WorkshopsOn the distribution of Selmer groups for a family of elliptic curves
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In the arithmetic of elliptic curves, to find the rank is of fundamental importance. Ranks can be bounded by Selmer groups, which are easier to handle and also contain information on the mysterious Tate- Shafarevich groups. The purpose of this talk is to survey various results related with the title of this talk. We will focus on the recent work of the author and his collaborators on the distribution of Selmer groups arising from a 2-isogeny for quadratic twists of an elliptic curve which possesses a non-trivial 2-torsion point over the rationals. It turns out that whether or not the elliptic curve has full 2-torsions over the rationals influences the distribution. Combining with other people’s results on Selmer groups, this implies in many families that the corresponding Tate-Shafarevich group is quite large in average.
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Mao-Sheng Xiong
2012-01-16
17:10:00 - 18:00:00
101 , Mathematics Research Center Building (ori. New Math. Bldg.)
In the arithmetic of elliptic curves, to find the rank is of fundamental importance. Ranks can be bounded by Selmer groups, which are easier to handle and also contain information on the mysterious Tate- Shafarevich groups. The purpose of this talk is to survey various results related with the title of this talk. We will focus on the recent work of the author and his collaborators on the distribution of Selmer groups arising from a 2-isogeny for quadratic twists of an elliptic curve which possesses a non-trivial 2-torsion point over the rationals. It turns out that whether or not the elliptic curve has full 2-torsions over the rationals influences the distribution. Combining with other people’s results on Selmer groups, this implies in many families that the corresponding Tate-Shafarevich group is quite large in average.
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