Torsion Points on CM Elliptic Curves Over Real Number Fields

TL;DR

This paper classifies torsion subgroups of CM elliptic curves over real number fields, revealing infinite possibilities in odd degrees and finite classifications in prime degrees, including Olson and non-Olson groups.

Contribution

It provides a complete classification of torsion subgroups of CM elliptic curves over real fields, distinguishing between infinite and finite cases based on degree and identifying specific Olson and non-Olson groups.

Findings

Infinite torsion groups over odd degree fields

Finitely many torsion groups over prime degree fields

Identification of 6 Olson groups and 17 non-Olson groups

Abstract

We study torsion subgroups of elliptic curves with complex multiplication (CM) defined over number fields which admit a real embedding. We give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a CM elliptic curve defined over a number field of odd degree: there are infinitely many. Restricting to the case of prime degree, we show that there are only finitely many isomorphism classes. More precisely, there are six "Olson groups" which arise as torsion subgroups of CM elliptic curves over number fields of every degree, and there are precisely 17 "non-Olson" CM elliptic curves defined over a prime degree number field.