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Torsion points on CM elliptic curves over real number fields


Authors: Abbey Bourdon, Pete L. Clark and James Stankewicz
Journal: Trans. Amer. Math. Soc. 369 (2017), 8457-8496
MSC (2010): Primary 11G05; Secondary 11G15
DOI: https://doi.org/10.1090/tran/6905
Published electronically: May 1, 2017
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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. 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 $ \mathbb{Q}$, and there are precisely $ 17$ ``non-Olson'' CM elliptic curves defined over a number field of (variable) prime degree.


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Additional Information

Abbey Bourdon
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605
Email: abourdon@uga.edu

Pete L. Clark
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30605
Email: plclark@gmail.com

James Stankewicz
Affiliation: Heilbronn Institute for Mathematical Research, University of Bristol, Bristol BS8 1TW, United Kingdom
Email: j.stankewicz@bristol.ac.uk

DOI: https://doi.org/10.1090/tran/6905
Received by editor(s): March 24, 2015
Received by editor(s) in revised form: October 4, 2015, and January 21, 2016
Published electronically: May 1, 2017
Additional Notes: The first author was supported in part by NSF grant DMS-1344994 (RTG in Algebra, Algebraic Geometry, and Number Theory, at the University of Georgia). The third author was supported by the Villum Fonden through the network for Experimental Mathematics in Number Theory, Operator Algebras, and Topology.
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