Quadratic Twists of Elliptic Curves with Small Selmer Rank

TL;DR

This paper proves the existence of quadratic twists of elliptic curves over rationals with small 2-Selmer rank, providing new insights into the distribution of ranks and Selmer groups of elliptic curves.

Contribution

It establishes the existence of quadratic twists with 2-Selmer rank ≤ 1 for elliptic curves without rational 2-torsion points, advancing understanding of Selmer groups.

Findings

Existence of quadratic twists with 2-Selmer rank ≤ 1

Lower bounds on the number of such twists

Discussion on the difficulty of trivial Selmer groups

Abstract

Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower bound on the number of D's for which the twists E(D) have 2-Selmer rank <= 1. We include in the introduction our (brief) opinion about why it is supposed to be hard to push our technique to make the Selmer group trivial.