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

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.

arXiv admin note: text overlap with arXiv:1203.1030

References in corpus (2)

• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
• A Markov model for Selmer ranks in families of twists

Cited by in corpus (4)

• On 2-Selmer groups of twists after quadratic extension
• Elements of given order in Tate-Shafarevich groups of abelian varieties in quadratic twist families
• The $4$-rank of class groups of $K(\sqrt{n})$
• The $3$-isogeny Selmer groups of the elliptic curves $y^2=x^3+n^2$