Torsion Points on CM Elliptic Curves Over Real Number Fields
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.
Keywords
Cite
@article{arxiv.1411.2742,
title = {Torsion Points on CM Elliptic Curves Over Real Number Fields},
author = {Abbey Bourdon and Pete L. Clark and James Stankewicz},
journal= {arXiv preprint arXiv:1411.2742},
year = {2015}
}
Comments
Updated to include many more new results, including on number fields whose degree is the square of an odd prime