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

TL;DR

This paper investigates the distribution of 2-Selmer groups for a family of elliptic curves related to the congruent number problem and explores their connection to the Birch-Swinnerton-Dyer conjecture, using number-theoretic independence assumptions.

Contribution

It provides an asymptotic analysis of the size of 2-Selmer groups for the congruent number curves and links this to the BSD conjecture, extending prior work by Heath-Brown.

Findings

Asymptotic formulas for the count of n with specific 2-Selmer group sizes

Connection established between Selmer group sizes and BSD conjecture verification

Dependence on Legendre symbol independence for asymptotic results

Abstract

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.