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Computing the Rank of Elliptic Curves over Number Fields

Published online by Cambridge University Press:  01 February 2010

Denis Simon
Denis Simon
Affiliation:
Université de Caen – France, Campus II – Boulevard Maréchal Juin, BP 5186, 14032 Caen cedex, France, [email protected]

Abstract

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This paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields of degree from 1 to 5.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 5 , 2002 , pp. 7 - 17
Copyright
Copyright © London Mathematical Society 2002

References

1Birch, B. J. and Swinnerton–Dyer, H. P. F., ‘Notes on elliptic curves’, J. Reine Angew. Math. 212 (1963) 7–25.CrossRefGoogle Scholar
2Bruin, N., ‘Algae, a program for computing 2-Selmer groups of elliptic curves over number fields’, http://www.cecm.sfu.ca/~bruin/ell.shar.CrossRefGoogle Scholar
3Bruin, N., ‘Visualizing Sha[2] in abelian surfaces’, preprint, 2002.Google Scholar
4Cassels, J. W. S., Lectures on elliptic curves, London Math. Soc. Stud. Texts 24 (Cambridge University Press, 1991).Google Scholar
5Cohen, H., A course in computational algebraic number theory, Grad. Texts in Math. 138, 3rd corrected printing (Springer, 1996).Google Scholar
6Cohen, H., Advanced topics in computational algebraic number theory, Grad. Texts in Math. 193 (Springer, 2000).CrossRefGoogle Scholar
7Cremona, J. E., Algorithms for modular elliptic curves, 2nd edn (Cambridge University Press, 1997).Google Scholar
8Cremona, J. E., ‘Reduction of binary cubic and quartic forms’, LMS J. Comput. Math. 2 (1999) 62–92; http://www.lms.ac.uk/jcm/2/lms1998-007/.CrossRefGoogle Scholar
9Cremona, J. E., ‘Classical invariants and 2-descent on elliptic curves’, J. Symb. Comput. 31 (2001) 71–87.CrossRefGoogle Scholar
10Cremona, J. E., Rusin, D., ‘Efficient solution of rational conics’, Math. Comp. (2002), to appear.CrossRefGoogle Scholar
11Cremona, J. E., Serf, P., ‘Computing the rank of elliptic curves over real quadratic fields of class number 1‘, Math.Comp. 68 (1999) 1187–1200.CrossRefGoogle Scholar
12Cremona, J. E. and Whitley, E., ‘Periods of cusp forms and elliptic curves over imaginary quadratic fields’, Math.Comp. 62 (1994) 407–429.CrossRefGoogle Scholar
13Djabri, Z. and Smart, N. P., ‘A comparison of direct and indirect methods for computing Selmer groups of an elliptic curve’, ANTS-III, Lecture Notes in Comput. Sci. 1423 (ed. Buhler, J., Springer, 1998) 502–513.Google Scholar
14Djabri, Z., Schaefer, E. F. and Smart, N. P., ‘Computing the P-Selmer group of an elliptic curve’, Trans. Amer. Math. Soc. 352 (2000) 5583–5597.CrossRefGoogle Scholar
15Lang, S., Algebraic number theory, Grad. Texts in Math. 110, 2nd edn (Springer,1994).CrossRefGoogle Scholar
16Merriman, J. R., Siksek, S. and Smart, N. P., ‘Explicit 4-descent on an elliptic curve’, Acta Arith. 11 (1996) 385–404.CrossRefGoogle Scholar
17Poonen, B. and Schaefer, E. F., ‘Explicit descent for Jacobians of cyclic covers of the projective line’, J.Reine Angew. Math. 488 (1997) 141–188.Google Scholar
18Schaefer, E. F., ‘2-descent on the Jacobians of hyperelliptic curves’, J.Number Theory 51 (1995) 219–232.CrossRefGoogle Scholar
19Schaefer, E. F., ‘Class groups and Selmer groups‘, J.Number Theory 56 (1996) 79–114.CrossRefGoogle Scholar
20Schaefer, E. F., and Stoll, M., ‘HOW to do a p-descent on an elliptic curve’, preprint, 2001.Google Scholar
21Simon, D., ‘Equations dans les corps de nombres et discriminants minimaux’, Doctoral Thesis, Université Bordeaux I, France (1998).Google Scholar
22Simon, D., ‘Solving norm equations using S-units’, Math.Comp. (2002), to appear.CrossRefGoogle Scholar
23Simon, D., ‘The gp-program’, http://www.math.unicaen.fr/~simon/ell.gp.Google Scholar
24Smart, N. P., The algorithmic resolution of diophantine equations, London Math. Soc. Stud. Texts 41 (Cambridge University Press, 1998).Google Scholar
25Stoll, M., ‘Implementing 2-descent for Jacobians of elliptic curves’, Acta Arith. 98 (2001) 245–277.Google Scholar