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 $p$-descent on an elliptic curve
HTML articles powered by AMS MathViewer

by Edward F. Schaefer and Michael Stoll PDF

Trans. Amer. Math. Soc. 356 (2004), 1209-1231 Request permission

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

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
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
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
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, DOI 10.1515/crll.1998.001
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, DOI 10.1515/crll.1959.202.52
G. A. Miller, Groups which contain ten or eleven proper subgroups, Proc. Nat. Acad. Sci. U.S.A. 25 (1939), 540–543. MR 31, DOI 10.1073/pnas.25.10.540
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, DOI 10.1017/S000497270001772X
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, DOI 10.1080/00927879708825980
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
John E. Cremona and Barry Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28. MR 1758797, DOI 10.1080/10586458.2000.10504633
Matt DeLong, A formula for the Selmer group of a rational three-isogeny, Acta Arith. 105 (2002), no. 2, 119–131. MR 1932762, DOI 10.4064/aa105-2-2
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, DOI 10.1090/S0002-9947-00-02535-6
T. Dokchitser, Deformations on $p$-divisible groups and $p$-descent on elliptic curves, Ph.D. dissertation, Universiteit Utrecht, 2000.
T. Fisher, On 5 and 7 descents for elliptic curves, Ph.D. thesis, Cambridge, UK, 2000.
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, DOI 10.1007/s002290050215
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, DOI 10.1007/BF01323464
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, DOI 10.1006/jsco.1996.0126
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/ .)
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, DOI 10.4064/aa-77-4-385-404
J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
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.
Jan Nekovář, Class numbers of quadratic fields and Shimura’s correspondence, Math. Ann. 287 (1990), no. 4, 577–594. MR 1066816, DOI 10.1007/BF01446915
PARI homepage: http://www.parigp-home.de/
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
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, DOI 10.1017/CBO9780511526053.015
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, DOI 10.1007/978-1-4757-4267-1_{1}2
Philippe Satgé, Groupes de Selmer et corps cubiques, J. Number Theory 23 (1986), no. 3, 294–317 (French). MR 846960, DOI 10.1016/0022-314X(86)90075-2
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, DOI 10.1007/s002080050156
Edward F. Schaefer, Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79–114. MR 1370197, DOI 10.1006/jnth.1996.0006
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
D. Simon, Équations dans les corps de nombres et discriminants minimaux, Thèse, Bordeaux, 1998.
Denis Simon, Computing the rank of elliptic curves over number fields, LMS J. Comput. Math. 5 (2002), 7–17. MR 1916919, DOI 10.1112/S1461157000000668
Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245–277. MR 1829626, DOI 10.4064/aa98-3-4
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
A. Weil, Sur un théorème de Mordell, Bull. Sci. Math. (2) 54 (1930), 182–191.