Nontorsion Points of Low Height on Elliptic Curves over Quadratic Fields

TL;DR

This paper presents low-height points on elliptic curves over quadratic fields, achieved through a search over elliptic divisibility sequences, improving previous height bounds and comparing with known minimal examples.

Contribution

It provides new examples of low-height points on elliptic curves over quadratic fields, surpassing previous height records and utilizing a novel search method over elliptic divisibility sequences.

Findings

Smallest height example dh(P)=0.0077127

Improves on previous smallest height dh(P)=0.0194426

Comparable with smallest height examples over Q

Abstract

We give examples of points with particularly low height on elliptic curves over quadratic fields, recovered by a search over elliptic divisibility sequences. The smallest example identified satisfies dh(P)=0.0077127...: improving on the previous smallest for curves over quadratic fields of dh(P)=0.0194426... given by Everest and Ward; and comparable with some of the examples of smallest height on curves over Q tabulated by Elkies.