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Computing the rank of elliptic curves over real quadratic number fields of class number 1
Author(s):
J.
E.
Cremona;
P.
Serf.
Journal:
Math. Comp.
68
(1999),
1187-1200.
MSC (1991):
Primary 11G05, 11Y16, 11Y50, 14G25, 14H52, 14Q05
Posted:
February 15, 1999
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Abstract:
In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over  . Several examples are included.
References:
- 1.
- B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on Elliptic Curves I, J. Reine Angew. Math. 212 (1963), 7-25. MR 26:3669
- 2.
- J. E. Cremona, Algorithms for Modular Elliptic Curves (Second Edition), Cambridge University Press, 1997.
- 3.
- J. E. Cremona, Classical invariants and elliptic curves, preprint, 1996.
- 4.
- P. B. Garrett, Holomorphic Hilbert modular forms, Wadsworth, 1990. MR 90k:11058
- 5.
- H. Graf, Konstruktion elliptischer Kurven hohen Ranges über quadratischen Zahlkörpern der Klassenzahl eins, Diploma thesis, Universität des Saarlandes, Saarbrücken, 1995.
- 6.
- D. Husemöller, Elliptic curves, Springer-Verlag, 1987. MR 88h:11039
- 7.
- K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevitch groups, Proc. Amer. Math. Soc. 89 (1983), No. 3, 379-386. MR 85d:14059
- 8.
- P. Serf, The rank of elliptic curves over real quadratic number fields of class number
 , PhD thesis, Universität des Saarlandes, Saarbrücken, 1995. - 9.
- S. Siksek, Infinite Descent on Elliptic Curves, Rocky Mountain J. Math. 25, (1995) No. 4, 1501-1538. MR 97g:11053
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- 11.
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Additional Information:
J.
E.
Cremona
Affiliation:
Department of Mathematics, University of Exeter, Laver Building, North Park Road, Exeter EX4 4QE, U.K.
Email:
cremona@maths.exeter.ac.uk
P.
Serf
Affiliation:
Fachbereich 9 Mathematik, Universität des Saarlandes, Postfach 151150, D-66041 Saarbrücken, Germany
Email:
pascale@math.uni-sb.de
DOI:
10.1090/S0025-5718-99-01055-8
PII:
S 0025-5718(99)01055-8
Keywords:
Elliptic curves,
Mordell-Weil,
real quadratic fields
Received by editor(s):
June 7, 1996
Received by editor(s) in revised form:
January 22, 1998
Posted:
February 15, 1999
Additional Notes:
The second author was supported in part by DFG grant 5130097383.
Copyright of article:
Copyright
1999,
American Mathematical Society
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