A Family of Elliptic Curves with a Lower Bound on 2-Selmer Ranks of Quadratic Twists

Abstract

For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim{\mathbb{F}}_{2}Se{l}_2(E^F/K)\ge dim{\mathbb{F}}_2{E}^F(K)[2]+r_2$ for every quadratic twist E^F of every curve E in this family, where r_2 is the number of complex places of K. This provides a counterexample to a conjecture appearing in work of Mazur and Rubin.

arXiv commentLatex, 9 pages. To appear in Math Research Letters

Cited by in corpus (1)

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