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On the complexity of computing the 2-Selmer group of an elliptic curve

  • S. Siksek (a1) and N. P. Smart (a2)

Abstract

In this paper we give an algorithm for computing the 2-Selmer group of an elliptic curve

which has complexity O(LD(0·5),c1)), where D is the absolute discriminant of the curve. Our algorithm is unconditional but the complexity estimate assumes the GRH and a standard conjecture on the distribution of smooth reduced ideals. This improves on the corresponding algorithm of Birch and Swinnerton-Dyer, which has complexity of O(√D).

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References

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1.Birch, B. J. and Swinnerton-Dyer, H. P. F., Notes on elliptic curves I, J. Reine Angew. Math. 212 (1963), 725.
2.Brumer, A. and Kramer, K., The rank of elliptic curves, Duke Math. J., 44 (1977), 715743.
3.Buchmann, J., A subexponential algorithm for the determination of class groups and regulators of algebraic number fields, in Séminaire de théorie des nombres, Paris (19881989), 2841.
4.Buchmann, J. and Lenstra, H. W. Jnr, Approximating rings of integers in number fields Journal de Theorie des Nombres de Bordeaux, 6 (1994), 221260.
5.Cassels, J. W. S., Lectures on elliptic curves, LMS Student text 24 (1991).
6.Cohen, H., A course in computational algebraic number theory (Springer-Verlag, 1993).
7.Cremona, J. E., Algorithms for modular elliptic curves (Cambridge University Press, 1992).
8.Cremona, J. E., Classical invariants and 2-descent on elliptic curves, J. Symbolic Computation, 1997, to appear.
9.Merriman, J. R., Siksek, S. and Smart, N. P., Explicit 4-descents on an elliptic curve, Ada. Arith., 77 (1996), 385404.
10.Schaefer, E. F., 2-descent on the Jacobians of hyperelliptic curves, J. Number Theory, 51 (1995), 219232.
11.Silverman, J. H., The arithmetic of elliptic curves (Springer-Verlag, 1986).
12.Thiel, C., Under the assumption of the Generalized Riemann Hypothesis verifying the class number belongs to , in ANTS-1: Algorithmic Number Theory, Eds Adelman, L. M. and Huang, M-D., Lecture Notes In Computer Science No. 877, (Springer-Verlag, 1994), 234247.

On the complexity of computing the 2-Selmer group of an elliptic curve

  • S. Siksek (a1) and N. P. Smart (a2)

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