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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Families of elliptic curves over quartic number fields with prescribed torsion subgroups


Authors: Daeyeol Jeon, Chang Heon Kim and Yoonjin Lee
Journal: Math. Comp. 80 (2011), 2395-2410
MSC (2010): Primary 11G05; Secondary 11G18
Published electronically: April 29, 2011
MathSciNet review: 2813367
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Abstract: We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields.


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Additional Information

Daeyeol Jeon
Affiliation: Department of Mathematics Education, Kongju National University, Kongju, Chungnam, South Korea
Email: dyjeon@kongju.ac.kr

Chang Heon Kim
Affiliation: Department of mathematics and Research institute for natural sciences, Hanyang University, Seoul, South Korea
Email: chhkim@hanyang.ac.kr

Yoonjin Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, South Korea
Email: yoonjinl@ewha.ac.kr

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02493-2
PII: S 0025-5718(2011)02493-2
Keywords: Elliptic curve, torsion, quadratic number field, quartic number field, modular curve
Received by editor(s): June 29, 2010
Received by editor(s) in revised form: October 18, 2010
Published electronically: April 29, 2011
Additional Notes: The first author was supported by the research grant of the Kongju National University in 2009
The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0001654)
The third author is the corresponding author and was supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827)
Article copyright: © Copyright 2011 American Mathematical Society