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?)

  • 1. M. Bertolini and G. Canuto, Good reduction of elliptic curves defined over $\mathbb{Q}(\sqrt[3]{2})$, Arch. Math. 50 (1988), 42-50. MR 89d:10046
  • 2. Salvador Comalada, Elliptic curves with trivial conductor over quadratic fields, Pacific J. Math. 144 (1990), no. 2, 237–258. MR 1061318
  • 3. J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, 1997. CMP 98:14
  • 4. M. Kida and T. Kagawa, Nonexistence of elliptic curves with good reduction everywhere over real quadratic fields, J. Number Theory 66 (1997) 201-210. CMP 98:02
  • 5. S. Kwon, Degree of isogenies of elliptic curves with complex multiplication, Preprint.
  • 6. J. Masley, On the class number of cyclotomic fields, Ph.D. Thesis, Princeton Univ., 1972.
  • 7. Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 0387283

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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
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