Disparity in Selmer ranks of quadratic twists of elliptic curves

2013/04/04 by Zev Klagsbrun, Barry Mazur, Karl Rubin · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Analytic Number Theory Research

paper · pdf · doi:10.4007/annals.2013.178.1.5

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 Fp-representations of the absolute Galois group of K by characters of order p.

Citations

• Finding large Selmer rank via an arithmetic theory of local constants

Cited by

• A Markov model for Selmer ranks in families of twists
• Strong Selmer companion elliptic curves
• p-twisted Selmer near-companion curves

Related

• On 2-Selmer ranks of quadratic twists of elliptic curves
• Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion
• On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
• A Markov model for Selmer ranks in families of twists
• Increasing the p-Selmer rank by twisting