Disparity in Selmer ranks of quadratic twists of elliptic curves

Zev Klagsbrun,Barry Mazur,Karl Rubin

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

Highlights

The type of question that we consider in this paper has its roots in a conjecture of Goldfeld [6, Conj
B] on the distribution of Mordell-Weil ranks in the family of quadratic twists of an arbitrary elliptic curve over Q and a result of Heath-Brown [7, Th. 2] on the distribution of 2-Selmer ranks in the family of quadratic twists over Q of the elliptic curve y2 = x3 − x
We study here the distribution of the parities of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K

Summary

Introduction

The type of question that we consider in this paper has its roots in a conjecture of Goldfeld [6, Conj. (This is consistent with the behavior of the global root numbers of twists of E.) In particular, the map that sends a character χ ∈ Hom(GK, {±1}) to the parity of dimF2 Sel2(Eχ/K) factors through the finite quotient v|2∆E∞ Hom(GKv , {±1}) Using this fact we are able to deduce Theorem A. We make essential use of a recent observation of Poonen and Rains [17] that the local conditions that define the 2-Selmer groups we are studying are maximal isotropic subspaces for a natural quadratic form on the local cohomology groups H1(Kv, E[2]) We use this in a crucial way in the proof of Theorem 3.9, which extends a result from [12] to include the case p = 2.

Notation

Metabolic spaces

Metabolic structures and Selmer structures

Twisted Selmer groups

Example: twists of elliptic curves

Local and global characters