Transactions of the American Mathematical Society

Author(s): Edward F. Schaefer; Michael Stoll
Journal: Trans. Amer. Math. Soc. 356 (2004), 1209-1231.
MSC (2000): Primary 11G05; Secondary 14H25, 14H52, 14Q05
Posted: October 27, 2003
Retrieve article in: PDF DVI PostScript

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, 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 -Selmer group of an elliptic curve, Trans. Amer. Math. Soc. 352 (2000), 5583-5597. MR 2001b:11047
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.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.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11G05, 14H25, 14H52, 14Q05

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: 10.1090/S0002-9947-03-03366-X
PII: S 0002-9947(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
Posted: 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
Copyright of article: Copyright 2003, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS