The distribution of 2-Selmer ranks of quadratic twists of elliptic curves with partial two-torsion

TL;DR

This paper investigates the distribution of 2-Selmer ranks in quadratic twists of elliptic curves with partial two-torsion over arbitrary number fields, revealing that at least half have arbitrarily large 2-Selmer ranks, differing from curves with no or full rational two-torsion.

Contribution

It establishes a new distribution result for 2-Selmer ranks in quadratic twists of elliptic curves with partial two-torsion, especially when no cyclic 4-isogeny exists over the two-division field.

Findings

At least half of quadratic twists have arbitrarily large 2-Selmer rank.

Distribution differs from curves with no or full rational two-torsion.

Results hold over arbitrary number fields.

Abstract

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.