Mathematics of Computation

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Reduction of elliptic curves over certain real quadratic number fields

Author(s): Masanari Kida.
Journal: Math. Comp. 68 (1999), 1679-1685.
MSC (1991): Primary 11G05
Posted: May 21, 1999
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Abstract: The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a -rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at 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:

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.: S. Comalada, Elliptic curves with trivial conductor over quadratic fields, Pacific J. Math. 144 (1990) 237-258. MR 91e:11058
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.: J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972) 259-331.MR 52:8126

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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: 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
Posted: 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.
Copyright of article: Copyright 1999, American Mathematical Society


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