Torsion points on CM elliptic curves over real number fields

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. However, if we fix an odd integer d and consider number fields of degree $dp$ as p ranges over all prime numbers, all but finitely many torsion subgroups that appear for CM elliptic curves actually occur in a degree dividing d. This implies an absolute bound on the size of torsion subgroups of CM elliptic curves defined over number fields of degree $dp$. In the case where $d=1$, there are six “Olson groups” which arise as torsion subgroups of CM elliptic curves over Q, and there are precisely $17$ “non-Olson” CM elliptic curves defined over a number field of (variable) prime degree.