Bounding the torsion in CM 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.

Cited by in corpus (6)

• On the minimal degree of definition of p-primary torsion subgroups of elliptic curves
• Torsion points on elliptic curves with complex multiplication
• Right triangles with algebraic sides and elliptic curves over number fields
• Bounds for torsion on abelian varieties with integral moduli
• Anatomy of torsion in the CM case
• Torsion bounds for elliptic curves and Drinfeld modules