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

 

Reduction of elliptic curves
over certain real quadratic number fields


Author: Masanari Kida
Journal: Math. Comp. 68 (1999), 1679-1685
MSC (1991): Primary 11G05
Published electronically: May 21, 1999
MathSciNet review: 1654021
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Abstract | References | Similar Articles | Additional Information

Abstract: The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a $2$-rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at $2$ is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.


References [Enhancements On Off] (What's this?)

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Additional Information

Masanari Kida
Affiliation: Department of Mathematics, The University of Electro-Communications, Chofu, Tokyo 182-8585, Japan
Email: kida@matha.e-one.uec.ac.jp

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01129-1
PII: S 0025-5718(99)01129-1
Received by editor(s): January 31, 1997
Received by editor(s) in revised form: January 2, 1998
Published electronically: May 21, 1999
Additional Notes: This research was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan.
Article copyright: © Copyright 1999 American Mathematical Society