On the independence of Heegner points associated to distinct quadratic imaginary fields on CM elliptic curves

TL;DR

This paper extends the criteria for the independence of Heegner points from non-CM elliptic curves to CM elliptic curves, including a generalization for points associated with orders of fixed conductor in quadratic imaginary fields.

Contribution

It demonstrates that the independence criteria for Heegner points apply to CM elliptic curves and generalizes these criteria for points linked to specific orders in quadratic imaginary fields.

Findings

Criteria for independence of Heegner points on CM elliptic curves.

Generalization of independence criteria for points from orders of fixed conductor.

Extension of Rosen-Silverman's results to CM elliptic curves.

Abstract

This paper is complementary to the work Rosen-Silverman, which derives a criteria on the number fields for the independence of Heegner points associated to them on non-CM elliptic curves. We show that the same criteria holds for CM elliptic curves. A generalisation of the independence criteria for the Heegner points associated to an order of fixed conductor of quadratic imaginary fields can also be found in this paper.