Average rank in families of quadratic twists: a geometric point of view

TL;DR

This paper studies the average rank of quadratic twists of a fixed elliptic curve over , using a geometric approach based on the canonical height of the curve's lowest non-torsion rational point.

Contribution

It introduces a geometric perspective to analyze the average rank in quadratic twist families, focusing on ordering by canonical height.

Findings

Provides new insights into the distribution of ranks in quadratic twist families.

Establishes a connection between geometric properties and rank behavior.

Offers potential tools for further research in elliptic curve ranks.

Abstract

We investigate the average rank in the family of quadratic twists of a given elliptic curve defined over $\mathbb{Q}$, when the curves are ordered using the canonical height of their lowest non-torsion rational point.