How to do a -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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we describe an algorithm that reduces the computation of the (full) -Selmer group of an elliptic curve
over a number field to standard number field computations such as determining the (
-torsion of) the
-class group and a basis of the
-units modulo
th powers for a suitable set
of primes. In particular, we give a result reducing this set
of `bad primes' to a very small set, which in many cases only contains the primes above
. As of today, this provides a feasible algorithm for performing a full
-descent on an elliptic curve over
, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of
is favorable, simplifications are possible and
-descents for larger
are accessible even today. To demonstrate how the method works, several worked examples are included.
- 1. M. F. Atiyah and C. T. C. Wall, Cohomology of groups, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 94–115. MR 0219512
- 2. Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
- 3. N. R. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, CWI Tract, vol. 133, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 2002. Dissertation, University of Leiden, Leiden, 1999. MR 1916903
- 4. J. W. S. Cassels, Second descents for elliptic curves, J. Reine Angew. Math. 494 (1998), 101–127. Dedicated to Martin Kneser on the occasion of his 70th birthday. MR 1604468, https://doi.org/10.1515/crll.1998.001
- 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 109136, https://doi.org/10.1515/crll.1959.202.52
- 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 0000031
- 7. Yen-Mei J. Chen, The Selmer groups and the ambiguous ideal class groups of cubic fields, Bull. Austral. Math. Soc. 54 (1996), no. 2, 267–274. MR 1411536, https://doi.org/10.1017/S000497270001772X
- 8. Yen-Mei J. Chen, The Selmer groups of elliptic curves and the ideal class groups of quadratic fields, Comm. Algebra 25 (1997), no. 7, 2157–2167. MR 1451686, https://doi.org/10.1080/00927879708825980
- 9. J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- 10. John E. Cremona and Barry Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28. MR 1758797
- 11. Matt DeLong, A formula for the Selmer group of a rational three-isogeny, Acta Arith. 105 (2002), no. 2, 119–131. MR 1932762, https://doi.org/10.4064/aa105-2-2
- 12. Z. Djabri, Edward F. Schaefer, and N. P. Smart, Computing the 𝑝-Selmer group of an elliptic curve, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5583–5597. MR 1694286, https://doi.org/10.1090/S0002-9947-00-02535-6
- 13.
T. Dokchitser, Deformations on
-divisible groups and
-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. Victor Flynn and Joseph L. Wetherell, Finding rational points on bielliptic genus 2 curves, Manuscripta Math. 100 (1999), no. 4, 519–533. MR 1734798, https://doi.org/10.1007/s002290050215
- 16. Gerhard Frey, Die Klassengruppen quadratischer und kubischer Zahlkörper und die Selmergruppen gewisser elliptischer Kurven, Manuscripta Math. 16 (1975), no. 4, 333–362 (German, with English summary). MR 379504, https://doi.org/10.1007/BF01323464
- 17. M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger, KANT V4, J. Symbolic Comput. 24 (1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR 1484479, https://doi.org/10.1006/jsco.1996.0126
- 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), no. 4, 385–404. MR 1414518, https://doi.org/10.4064/aa-77-4-385-404
- 20. J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
- 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. Jan Nekovář, Class numbers of quadratic fields and Shimura’s correspondence, Math. Ann. 287 (1990), no. 4, 577–594. MR 1066816, https://doi.org/10.1007/BF01446915
- 23. PARI homepage: http://www.parigp-home.de/
- 24. Bjorn Poonen and Edward F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line, J. Reine Angew. Math. 488 (1997), 141–188. MR 1465369
- 25. Karl Rubin, The one-variable main conjecture for elliptic curves with complex multiplication, 𝐿-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 353–371. MR 1110401, https://doi.org/10.1017/CBO9780511526053.015
- 26. Karl Rubin, Descents on elliptic curves with complex multiplication, Séminaire de Théorie des Nombres, Paris 1985–86, Progr. Math., vol. 71, Birkhäuser Boston, Boston, MA, 1987, pp. 165–173. MR 1017911, https://doi.org/10.1007/978-1-4757-4267-1_12
- 27. Philippe Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), no. 3, 294–317 (French). MR 846960, https://doi.org/10.1016/0022-314X(86)90075-2
- 28. Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447–471. MR 1612262, https://doi.org/10.1007/s002080050156
- 29. Edward F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114. MR 1370197, https://doi.org/10.1006/jnth.1996.0006
- 30. Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210
- 31. D. Simon, Équations dans les corps de nombres et discriminants minimaux, Thèse, Bordeaux, 1998.
- 32. Denis Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7–17. MR 1916919, https://doi.org/10.1112/S1461157000000668
- 33. Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, https://doi.org/10.4064/aa98-3-4
- 34. Jaap Top, Descent by 3-isogeny and 3-rank of quadratic fields, Advances in number theory (Kingston, ON, 1991) Oxford Sci. Publ., Oxford Univ. Press, New York, 1993, pp. 303–317. MR 1368429
- 35. A. Weil, Sur un théorème de Mordell, Bull. Sci. Math. (2) 54 (1930), 182-191.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G05, 14H25, 14H52, 14Q05
Retrieve articles in all journals with MSC (2000): 11G05, 14H25, 14H52, 14Q05
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