Lower bounds of the canonical height on quadratic twists of elliptic curves

TL;DR

This paper establishes a lower bound for the canonical height on quadratic twists of elliptic curves and presents a method to construct explicit rational points, demonstrating their primitivity in the Mordell-Weil group.

Contribution

It introduces a new lower bound for canonical heights and a simple construction method for rational points on quadratic twists of elliptic curves.

Findings

Lower bounds for canonical heights are derived.

Explicit rational points can be constructed and shown to be primitive.

The method applies to quadratic twists from cubic polynomials.

Abstract

We compute a lower bound of the canonical height on quadratic twists of certain elliptic curves. Also we show a simple method for constructing families of quadratic twists with an explicit rational point. % from cubic polynomials. Using the above lower bound, we show that the explicit rational point is primitive as an element of the Mordell--Weil group.