How to do a 𝑝-descent on an elliptic curve

Schaefer, Edward; Stoll, Michael

doi:10.1090/S0002-9947-03-03366-X

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
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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 $\mathbb Q$ , but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of 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.