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How to do a $p$-descent on an elliptic curve


Authors: Edward F. Schaefer and Michael Stoll
Journal: Trans. Amer. Math. Soc. 356 (2004), 1209-1231
MSC (2000): Primary 11G05; Secondary 14H25, 14H52, 14Q05
DOI: https://doi.org/10.1090/S0002-9947-03-03366-X
Published electronically: October 27, 2003
MathSciNet review: 2021618
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we describe an algorithm that reduces the computation of the (full) $p$-Selmer group of an elliptic curve $E$ over a number field to standard number field computations such as determining the ($p$-torsion of) the $S$-class group and a basis of the $S$-units modulo $p$th powers for a suitable set $S$ of primes. In particular, we give a result reducing this set $S$ of `bad primes' to a very small set, which in many cases only contains the primes above $p$. As of today, this provides a feasible algorithm for performing a full $3$-descent on an elliptic curve over $\mathbb Q$, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of $E[p]$ is favorable, simplifications are possible and $p$-descents for larger $p$ are accessible even today. To demonstrate how the method works, several worked examples are included.

References [Enhancements On Off] (What's this?)

  • 1. M.F. Atiyah and C.T.C. Wall, Cohomology of groups, in: Algebraic Number Theory, Ed. J.W.S. Cassels and A. Fröhlich, Academic Press, London, 1967, pp. 94-115. MR 36:2593
  • 2. K.S. Brown, Cohomology of groups, Springer, GTM vol. 87, 1982. MR 83k:20002
  • 3. N. Bruin, Chabauty methods and covering techniques applied to generalised Fermat equations, Ph.D. dissertation, Leiden, 1999. MR 2003i:11042
  • 4. J.W.S. Cassels, Second descents for elliptic curves, J. reine angew. Math. 494 (1998), 101-127. MR 99d:11058
  • 5. J.W.S. Cassels, Arithmetic on curves of genus 1. I. On a conjecture of Selmer, J. reine angew. Math. 202 (1959), 52-99. MR 22:24
  • 6. J.W.S. Cassels, Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer, J. reine angew. Math. 217 (1965), 180-199. MR 31 #3420
  • 7. Y-M.J. Chen, The Selmer groups and the ambiguous ideal class groups of cubic fields, Bull. Austral. Math. Soc. 54 (1996), 267-274. MR 98a:11072
  • 8. Y-M.J. Chen, The Selmer groups of elliptic curves and the ideal class groups of quadratic fields, Comm. Algebra 25 (1997), 2157-2167. MR 98d:11058
  • 9. J.E. Cremona, Algorithms for modular elliptic curves, $2^{\text{nd}}$ ed., Cambridge University Press, 1997. MR 99e:11068
  • 10. J.E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), 13-28. MR 2001g:11083
  • 11. M. DeLong, A formula for the Selmer group of a rational three-isogeny, Acta Arith. 105 (2002), 119-131. MR 2003i:11069
  • 12. Z. Djabri, E.F. Schaefer and N.P. Smart, Computing the $p$-Selmer group of an elliptic curve, Trans. Amer. Math. Soc. 352 (2000), 5583-5597. MR 2001b:11047
  • 13. T. Dokchitser, Deformations on $p$-divisible groups and $p$-descent on elliptic curves, Ph.D. dissertation, Universiteit Utrecht, 2000.
  • 14. T. Fisher, On 5 and 7 descents for elliptic curves, Ph.D. thesis, Cambridge, UK, 2000.
  • 15. E.V. Flynn and J.L. Wetherell, Finding rational points on bielliptic genus 2 curves, Manuscr. Math. 100 (1999), 519-533. MR 2001g:11098
  • 16. G. Frey, Die Klassengruppen quadratischer und kubischer Zahlkvrper und die Selmergruppen gewisser elliptischer Kurven, Manuscripta Math. 16 (1975), 333-362. MR 52:409
  • 17. KANT/KASH is described in M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner and K. Wildanger, KANT V4, J. Symbolic Comp. 24 (1997), 267-283. MR 99g:11150
  • 18. MAGMA is described in W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997), 235-265. (Also see the Magma home page at http://www.maths.usyd.edu.au:8000/u/magma/.)
  • 19. J.R. Merriman, S. Siksek and N.P. Smart, Explicit 4-descents on an elliptic curve, Acta Arith. 77 (1996), 385-404. MR 97j:11027
  • 20. J.S. Milne, Arithmetic duality theorems, Academic Press, Boston, 1986. MR 88e:14028
  • 21. L.J. Mordell, On the rational solutions of the indeterminate equations of the 3rd and 4th degrees, Proc. Camb. Phil. Soc. 21 (1922), 179-192.
  • 22. J. Nekovár, Class numbers of quadratic fields and Shimura's correspondence, Math. Ann. 287 (1990), 577-594. MR 91k:11051
  • 23. PARI homepage: http://www.parigp-home.de/
  • 24. B. Poonen and E.F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. reine angew. Math. 488 (1997), 141-188. MR 98k:11087
  • 25. K. Rubin, The one-variable main conjecture for elliptic curves with complex multiplication, in: L-functions and arithmetic, Ed. J. Coates and M.J. Taylor, LMS Lecture Notes Series, vol. 153, Cambridge University Press, Cambridge, 1991, pp. 353-371. MR 92j:11055
  • 26. K. Rubin, Descents on elliptic curves with complex multiplication, in: Théorie des nombres, Séminaire Paris 1985/86, Ed. C. Goldstein, Progress in Mathematics, vol. 71, Birkhäuser, 1987, pp. 165-173. MR 90g:11073
  • 27. P. Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), 294-317. MR 87i:11070
  • 28. E.F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), 447-471. MR 99h:11063
  • 29. E.F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), 79-114. MR 97e:11068
  • 30. J.H. Silverman, The arithmetic of elliptic curves, Springer GTM 106, 1986. MR 87g:11070
  • 31. D. Simon, Équations dans les corps de nombres et discriminants minimaux, Thèse, Bordeaux, 1998.
  • 32. D. Simon, Computing the rank of elliptic curves over number fields, to appear in LMS J. Comput. Math. 5 (2002), 7-17 (electronic). MR 2003g:11060
  • 33. M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), 245-277. MR 2002b:11089
  • 34. J. Top, Descent by 3-isogeny and 3-rank of quadratic fields, in: Advances in number theory, Ed. F. Gouvea and N. Yui, Clarendon Press, Oxford, 1993, pp. 303-317. MR 97d:11167
  • 35. A. Weil, Sur un théorème de Mordell, Bull. Sci. Math. (2) 54 (1930), 182-191.

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Additional Information

Edward F. Schaefer
Affiliation: Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California 95053
Email: eschaefe@math.scu.edu

Michael Stoll
Affiliation: School of Engineering and Science, International University Bremen, P.O. Box 750561, 28725 Bremen, Germany
Email: m.stoll@iu-bremen.de

DOI: https://doi.org/10.1090/S0002-9947-03-03366-X
Keywords: Elliptic curve over number field, $p$-descent, Selmer group, Mordell-Weil rank, Shafarevich-Tate group
Received by editor(s): January 24, 2003
Published electronically: October 27, 2003
Additional Notes: We are indebted to Claus Fieker for his invaluable help in getting KANT to produce a basis for the group $A(S,5)^{(1)}$ needed in the example in Section 8.2. We thank John Cremona, Zafer Djabri, Everett Howe, Hendrik W. Lenstra Jr., Karl Rubin, Nigel Smart and Don Zagier for useful and interesting discussions. The first author was supported by National Security Agency grant MSPF-02Y-033
Article copyright: © Copyright 2003 American Mathematical Society