Disparity in Selmer ranks of quadratic twists of elliptic curves

TL;DR

This paper investigates the distribution of parity in 2-Selmer ranks among quadratic twists of elliptic curves over number fields, providing explicit formulas and demonstrating density of certain fractions.

Contribution

It establishes the existence and explicit computation of the stable limit fraction of twists with even 2-Selmer rank and extends results to p-Selmer ranks of Galois representations.

Findings

Fraction of twists with even 2-Selmer rank exists as a stable limit.

Explicit product formula for the fraction based on local factors.

Examples show the fractions are dense in [0, 1] over varying fields.

Abstract

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0, 1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.