2-Selmer Groups and the Birch-Swinnerton-Dyer Conjecture for the Congruent Number Curve

Details 2-Selmer Groups and the Birch-Swinnerton-Dyer Conjecture for the Congruent Number Curve 2-Selmer Groups and the Birch-Swinnerton-Dyer Conjecture for the Congruent Number Curve

2007-06-29

arxiv.org/abs/0706.4344v1

Mathematics

Number Theory

14 pages

Scientific paper

We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence'' of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.

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