Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion

TL;DR

This paper studies the distribution of 2-Selmer ranks among quadratic twists of elliptic curves over number fields, revealing conditions under which all non-negative integers appear infinitely often as ranks.

Contribution

It establishes new results on the frequency of 2-Selmer ranks in quadratic twist families, especially concerning the presence of cyclic 4-isogenies and the influence of rational 2-torsion.

Findings

All non-negative integers appear infinitely often as 2-Selmer ranks under certain conditions.

Results depend on the existence of cyclic 4-isogenies over the base field.

Provides bounds on the number of twists with rank 0 and 1.

Abstract

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over K(E[2]), then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of E. If E has a cyclic 4-isogeny defined over K(E[2]) but not over K, then we prove the same result for 2-Selmer ranks greater than or equal to r_2, the number of complex places of K. We also obtain results about the minimum number of twists of E with rank 0, and subject to standard conjectures, the number of twists with rank 1, provided E does not have a cyclic 4-isogeny defined over K.