Some Improvements to 4-Descent on an Elliptic Curve

  • Tom Fisher
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5011)


The theory of 4-descent on elliptic curves has been developed in the PhD theses of Siksek [18], Womack [21] and Stamminger [20]. Prompted by our use of 4-descent in the search for generators of large height on elliptic curves of rank at least 2, we explain how to cut down the number of class group and unit group calculations required, by using the group law on the 4-Selmer group.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    An, S.Y., Kim, S.Y., Marshall, D.C., Marshall, S.H., McCallum, W.G., Perlis, A.R.: Jacobians of genus one curves. J. Number Theory 90(2), 304–315 (2001)Google Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24, 235–265 (1997), Scholar
  3. 3.
    Cassels, J.W.S.: Lectures on Elliptic Curves. LMS Student Texts, vol. 24. CUP Cambridge (1991)Google Scholar
  4. 4.
    Cassels, J.W.S.: Second descents for elliptic curves. J. reine angew. Math. 494, 101–127 (1998)Google Scholar
  5. 5.
    Cremona, J.E.: Classical invariants and 2-descent on elliptic curves. J. Symbolic Comput. 31, 71–87 (2001)Google Scholar
  6. 6.
    Cremona, J.E., Fisher, T.A.: On the equivalence of binary quartics (submitted)Google Scholar
  7. 7.
    Cremona, J.E., Fisher, T.A., O’Neil, C., Simon, D., Stoll, M.: Explicit n-descent on elliptic curves, I Algebra. J. reine angew. Math. 615, 121–155 (2008), II Geometry, to appear in J. reine angew. Math.; III Algorithms (in preparation)Google Scholar
  8. 8.
    Donnelly, S.: Computing the Cassels-Tate pairing (in preparation)Google Scholar
  9. 9.
    Fisher, T.A.: Testing equivalence of ternary cubics. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 333–345. Springer, Heidelberg (2006)Google Scholar
  10. 10.
    Fisher, T.A.: The Hessian of a genus one curve (preprint)Google Scholar
  11. 11.
    Fisher, T.A.: Finding rational points on elliptic curves using 6-descent and 12-descent (submitted)Google Scholar
  12. 12.
    Fisher, T.A., Schaefer, E.F., Stoll, M.: The yoga of the Cassels-Tate pairing (submitted)Google Scholar
  13. 13.
    Hilbert, D.: Theory of Algebraic Invariants. CUP, Cambridge (1993)Google Scholar
  14. 14.
    Merriman, J.R., Siksek, S., Smart, N.P.: Explicit 4-descents on an elliptic curve. Acta Arith. 77(4), 385–404 (1996)Google Scholar
  15. 15.
    O’Neil, C.: The period-index obstruction for elliptic curves. J. Number Theory 95(2), 329–339 (2002)Google Scholar
  16. 16.
    Pílniková, J.: Trivializing a central simple algebra of degree 4 over the rational numbers. J. Symbolic Comput. 42(6), 579–586 (2007)Google Scholar
  17. 17.
    Poonen, B., Schaefer, E.F.: Explicit descent for Jacobians of cyclic covers of the projective line. J. reine angew. Math. 488, 141–188 (1997)Google Scholar
  18. 18.
    Siksek, S.: Descent on Curves of Genus 1. PhD thesis, University of Exeter (1995)Google Scholar
  19. 19.
    Simon, D.: Computing the rank of elliptic curves over number fields. LMS J. Comput. Math. 5, 7–17 (2002)Google Scholar
  20. 20.
    Stamminger, S.: Explicit 8-Descent on Elliptic Curves. PhD thesis, International University Bremen (2005)Google Scholar
  21. 21.
    Womack, T.: Explicit Descent on Elliptic Curves. PhD thesis, University of Nottingham (2003)Google Scholar
  22. 22.
    Zarhin, Y.G.: Noncommutative cohomology and Mumford groups. Math. Notes 15, 241–244 (1974)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Tom Fisher
    • 1
  1. 1.DPMMS, Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK

Personalised recommendations