Bounding the torsion in CM elliptic curves

TL;DR

This paper investigates bounds on torsion points of CM elliptic curves over number fields, providing linear estimates based on complex multiplication and supersingular prime results, and discusses potential universal bounds.

Contribution

It offers a linear bound on torsion points for CM elliptic curves over number fields using complex multiplication and supersingular prime results.

Findings

Linear estimate on the number of torsion points depending on degree

Connection between complex multiplication and torsion bounds

Heuristic for universal torsion bounds across elliptic curves

Abstract

Merel has shown that the order of torsion subgroup of an elliptic curve over a number field can be bounded in terms of only the degree of the number field. The purpose of this note is to investigate what could be the `right bound'. In this paper we use the result of Deuring on the supersingular primes for a CM elliptic curves to give estimate on the number of torsion points on CM elliptic curves over any number field which depends only linearly with the degree of the number field. This result is also a simple corollary of a result of A. Silverberg bounding the order of torsion elements which was proved using the main theorem of Complex multiplication. We also give a heuristic for hoping that the number of torsion points on any elliptic curve be a bounded linearly with the degree of the number field.