On 2-Selmer ranks of quadratic twists of elliptic curves

TL;DR

This paper investigates the distribution of 2-Selmer ranks in quadratic twists of elliptic curves over number fields, establishing conditions for their structure and providing bounds related to rational 2-torsion points.

Contribution

It proves that the set of 2-Selmer ranks in quadratic twists forms a half-infinite arithmetic progression starting from a certain point, and identifies conditions for this set to be all integers above a threshold.

Findings

The set of 2-Selmer ranks is an interval of integers with the same parity starting from a minimal value.

Conditions are provided under which the set of 2-Selmer ranks equals all integers above a certain threshold.

An upper bound for the minimal 2-Selmer rank is given when all 2-torsion points are rational.

Abstract

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all integers larger than $c$ and having the same parity as $c$. We also find sufficient conditions on $A_E$ such that $A_E$ is equal to $\Z_{\ge t_E}$ for some number $t_E$. When all points in $E[2]$ are rational, we give an upper bound for $t_E$.