Average rank of quadratic twists with a rational point of almost minimal height

TL;DR

This paper studies the average rank of quadratic twists of a fixed elliptic curve over A9, focusing on those with a rational point of nearly minimal height, and finds that the average analytic rank exceeds one.

Contribution

It provides new insights into the average rank of quadratic twists with specific rational points, revealing that the average analytic rank is greater than one.

Findings

Average analytic rank of these twists is greater than one.

Focus on twists with a rational point of almost minimal height.

Enhances understanding of rank distribution in elliptic curve families.

Abstract

Given a family of quadratic twists of a fixed elliptic curve defined over $\mathbb{Q}$, we investigate the average rank in the subfamily of twists having a nontorsion rational point of almost minimal height. We show in particular that the average analytic rank is greater than one.