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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On fields of definition of torsion points of elliptic curves with complex multiplication


Authors: Luis Dieulefait, Enrique González-Jiménez and Jorge Jiménez Urroz
Journal: Proc. Amer. Math. Soc. 139 (2011), 1961-1969
MSC (2010): Primary 11G05; Secondary 11F80
Published electronically: November 9, 2010
MathSciNet review: 2775372
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Abstract: For any elliptic curve $ E$ defined over the rationals with complex multiplication (CM) and for every prime $ p$, we describe the image of the mod $ p$ Galois representation attached to $ E$. We deduce information about the field of definition of torsion points of these curves; in particular, we classify all cases where there are torsion points over Galois number fields not containing the field of definition of the CM.


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Additional Information

Luis Dieulefait
Affiliation: Departament d’Algebra i Geometria, Universitat de Barcelona, G. V. de les Corts Catalanes 585, 08007 Barcelona, Spain
Email: ldieulefait@ub.edu

Enrique González-Jiménez
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), 28049 Madrid, Spain
Email: enrique.gonzalez.jimenez@uam.es

Jorge Jiménez Urroz
Affiliation: Departament de Matemàtica Aplicada IV, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain
Email: jjimenez@ma4.upc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10621-4
PII: S 0002-9939(2010)10621-4
Keywords: Elliptic curves, torsion, Galois representation
Received by editor(s): March 16, 2010
Received by editor(s) in revised form: June 2, 2010
Published electronically: November 9, 2010
Additional Notes: The authors were partially supported by the grants MTM2006-04895 (Ministerio de Educación y Ciencia of Spain), MTM2009-07291 (Ministerio de Ciencia e Innovación of Spain), CCG08-UAM/ESP-3906 (Universidad Autónoma de Madrid and Comunidad de Madrid), and MTM2009-11068 (Ministerio de Ciencia e Innovación of Spain) respectively.
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.