Skip to main content
SpringerLink
Log in

A comparison of direct and indirect methods for computing Selmer groups of an elliptic curve

  • Conference paper
  • First Online:
Algorithmic Number Theory (ANTS 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1423))

Included in the following conference series:

Abstract

In this paper we examine differences between the two standard methods for computing the 2-Selmer group of an elliptic curve. In particular we focus on practical differences in the timings of the two methods. In addition we discuss how to proceed if one fails to determine the rank of the curve from the 2-Selmer group. Finally we mention briefly ongoing research into generalizing such methods to the case of computing the 3-Selmer group.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Batut, D. Bernardi, H. Cohen, and M. Olivier. GP/PARI version 1.39.03. Université Bordeaux I, 1994.

    Google Scholar 

  2. B.J. Birch and H.P.P. Swinnerton-Dyer. Notes on elliptic curves. I. J. Reine Angew. Math., 212:7–25, 1963.

    MATH  MathSciNet  Google Scholar 

  3. B.J. Birch and H.P.F. Swinnerton-Dyer. Notes on elliptic curves. II. J. Reine Angew. Math., 218:79–108, 1965.

    MATH  MathSciNet  Google Scholar 

  4. J.W.S. Cassels. Second descents for elliptic curves. J. Reine Angew. Math., 494:101–127, 1998.

    MATH  MathSciNet  Google Scholar 

  5. J.W.S. Cassels. Diophantine equations with special reference to elliptic curves. J. of LMS, 41:193–291, 1966.

    MathSciNet  Google Scholar 

  6. J.W.S. Cassels. Lectures on Elliptic Curves. LMS Student Texts, Cambridge University Press, 1991.

    Google Scholar 

  7. J.E. Cremona. Classical invariants and 2-descent on elliptic curves. Preprint 1997.

    Google Scholar 

  8. J.E. Cremona. Reduction of cubic and quartic polynomials. Preprint 1998.

    Google Scholar 

  9. J.E. Cremona. Algorithms for Modular Elliptic Curves. Cambridge University Press, 1992.

    Google Scholar 

  10. H. Davenport. On the minimum of a ternary cubic form. J. London Math. Soc., 19:13–18, 1944.

    MATH  MathSciNet  Google Scholar 

  11. E.B. Elliott. An Introduction to the Algebra of Quantics. Oxford University Press, 1895.

    Google Scholar 

  12. J. Gebel, A. Pethó, and H.G. Zimmer. Computing integral points on elliptic curves. Acta. Arith., 68:171–192, 1994.

    MATH  MathSciNet  Google Scholar 

  13. D. Hilbert. Theory of Algebraic Invariants. Cambridge University Press, 1993.

    Google Scholar 

  14. G. Julia. étude sur les formes binaires non quadratiques. Mem. Acad. Sci. l'Inst. France, 55:1–293, 1917.

    Google Scholar 

  15. LiDIA Group. LiDIA v1.3 — a library for computational number theory. TH Darmstadt, 1997.

    Google Scholar 

  16. J.R. Merriman, S. Siksek, and N.P. Smart. Explicit 4-descents on an elliptic curve. Acta. Arith., 77:385–404, 1996.

    MATH  MathSciNet  Google Scholar 

  17. G. Salmon. Higher Plane Curves. Hodges, Foster and Figgis, 1879.

    Google Scholar 

  18. E.F. Schaefer. Computing a Selmer group of a Jacobian using functions on the curve. Preprint.

    Google Scholar 

  19. P. Serf. The rank of elliptic curves over real quadratic number fields of class number 1, Phd Thesis, UniversitÄt des Saarlandes, 1995.

    Google Scholar 

  20. S. Siksek. Infinite descent on elliptic curves. Rocky Mountain Journal of Maths, 25:1501–1538, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Siksek and N.P. Smart. On the complexity of computing the 2-Selmer group of an elliptic curve. Glasgow Math. Journal., 39:251–258, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  22. N.P. Smart. S-integral points on elliptic curves. Proc. Camb. Phil. Soc., 116:391–399, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  23. R.J. Stroeker and N. Tzanakis. Solving elliptic diophantine equations by estimating linear forms in elliptic logarithms. Acta. Arith., 67:177–196, 1994.

    MATH  MathSciNet  Google Scholar 

  24. P. Swinnerton-Dyer. Private communication. 1996.

    Google Scholar 

  25. J. Top. Descent by 3-isogeny and the 3-rank of quadratic fields. In F.Q. Gouvea and N. Yui, editors, Advances in Number Theory, pages 303–317. Clarendon Press, Oxford, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Maths and Statistics, University of Kent at Canterbury, CT2 7NF, Canterbury, Kent, UK

    Z. Djabri

  2. Hewlett-Packard Laboratories, Filton Road, BS12 6QZ, Stoke Gifford, Bristol, UK

    N. P. Smart

Authors
  1. Z. Djabri
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. P. Smart
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Joe P. Buhler

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Djabri, Z., Smart, N.P. (1998). A comparison of direct and indirect methods for computing Selmer groups of an elliptic curve. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054888

Download citation

  • DOI: https://doi.org/10.1007/BFb0054888

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64657-0

  • Online ISBN: 978-3-540-69113-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation