Computational verification of the Birch and Swinnerton-Dyer conjecture for individual elliptic curves
Authors:
Grigor Grigorov, Andrei Jorza, Stefan Patrikis, William A. Stein and Corina Tarnita
Journal:
Math. Comp. 78 (2009), 2397-2425
MSC (2000):
Primary 11Y99
DOI:
https://doi.org/10.1090/S0025-5718-09-02253-4
Published electronically:
June 8, 2009
MathSciNet review:
2521294
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over of analytic ranks 0 and
. We apply our techniques to show that if
is a non-CM elliptic curve over
of conductor
and rank 0 or
, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the
-series is true for
, up to odd primes that divide either Tamagawa numbers of
or the degree of some rational cyclic isogeny with domain
. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or
, this completely verifies the full conjecture for these curves up to the primes excluded above.
- [ABC] B. Allombert, K. Belabas, H. Cohen, X. Roblot, and I. Zakharevitch, PARI/GP, http://pari.math.u-bordeaux.fr/.
- [ARS05] A. Agashe, K. A. Ribet, and W. A. Stein, The Manin constant, congruence primes, and the modular degree, Preprint, http://www.williamstein.org/papers/ manin-agashe/, With an appendix by J. Cremona (2005).
- [AS05] Amod Agashe and William Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero, Math. Comp. 74 (2005), no. 249, 455–484. With an appendix by J. Cremona and B. Mazur. MR 2085902, https://doi.org/10.1090/S0025-5718-04-01644-8
- [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over 𝐐: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, https://doi.org/10.1090/S0894-0347-01-00370-8
- [BCP97] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
- [BFH90] Daniel Bump, Solomon Friedberg, and Jeffrey Hoffstein, Nonvanishing theorems for 𝐿-functions of modular forms and their derivatives, Invent. Math. 102 (1990), no. 3, 543–618. MR 1074487, https://doi.org/10.1007/BF01233440
- [Cas62] J. W. S. Cassels, Arithmetic on curves of genus 1. III. The Tate-Šafarevič and Selmer groups, Proc. London Math. Soc. (3) 12 (1962), 259–296. MR 0163913, https://doi.org/10.1112/plms/s3-12.1.259
- [Cas65] 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 179169, https://doi.org/10.1515/crll.1965.217.180
- [Cha03] Byungchul Cha, Vanishing of Some Cohomology Groups and Bounds for the Shafarevich-Tate Groups of Elliptic Curves, Johns-Hopkins Ph.D. Thesis (2003).
- [Cha05] Byungchul Cha, Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves, J. Number Theory 111 (2005), no. 1, 154–178. MR 2124047, https://doi.org/10.1016/j.jnt.2004.08.009
- [CK] Alina Carmen Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, Canad. Math. Bull. 48 (2005), no. 1, 16–31. With an appendix by Ernst Kani. MR 2118760, https://doi.org/10.4153/CMB-2005-002-x
- [CM00] John E. Cremona and Barry Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28. MR 1758797
- [Coh93] Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
- [CPS06] J. E. Cremona, M. Prickett, and Samir Siksek, Height difference bounds for elliptic curves over number fields, J. Number Theory 116 (2006), no. 1, 42–68. MR 2197860, https://doi.org/10.1016/j.jnt.2005.03.001
- [Crea]
J. E. Cremona, Elliptic curves of conductor
, http://www.maths. nott.ac.uk/personal/ jec/ftp/data/.
- [Creb] -, mwrank (computer software), http://www.maths.nott. ac.uk/personal/jec/mwrank/
- [Cre97] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
- [Edi91] Bas Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 25–39. MR 1085254
- [Gri05] G. Grigorov, Kato's Euler System and the Main Conjecture, Harvard Ph.D. Thesis (2005).
- [Gro91] Benedict H. Gross, Kolyvagin’s work on modular elliptic curves, 𝐿-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. MR 1110395, https://doi.org/10.1017/CBO9780511526053.009
- [GZ86] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of 𝐿-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, https://doi.org/10.1007/BF01388809
- [Jor05] A. Jorza, The Birch and Swinnerton-Dyer Conjecture for Abelian Varieties over Number Fields, Harvard University Senior Thesis (2005).
- [Kat04] Kazuya Kato, 𝑝-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies 𝑝-adiques et applications arithmétiques. III. MR 2104361
- [Kol88] V. A. Kolyvagin, Finiteness of 𝐸(𝑄) and SH(𝐸,𝑄) for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523–541. MR 954295, https://doi.org/10.1070/IM1989v032n03ABEH000779
- [Kol90] V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
- [Kol91] Victor Alecsandrovich Kolyvagin, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 429–436. MR 1159231
- [Lan91] Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552
- [Man72] Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
- [Mat03] Kazuo Matsuno, Finite Λ-submodules of Selmer groups of abelian varieties over cyclotomic ℤ_{𝕡}-extensions, J. Number Theory 99 (2003), no. 2, 415–443. MR 1969183, https://doi.org/10.1016/S0022-314X(02)00078-1
- [Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, https://doi.org/10.1007/BF01390348
- [McC91] William G. McCallum, Kolyvagin’s work on Shafarevich-Tate groups, 𝐿-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 295–316. MR 1110398, https://doi.org/10.1017/CBO9780511526053.012
- [Mil86] J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
- [MM91] M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular 𝐿-series, Ann. of Math. (2) 133 (1991), no. 3, 447–475. MR 1109350, https://doi.org/10.2307/2944316
- [MR04] Barry Mazur and Karl Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, viii+96. MR 2031496, https://doi.org/10.1090/memo/0799
- [PS99] Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, https://doi.org/10.2307/121064
- [Rub98] Karl Rubin, Euler systems and modular elliptic curves, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 351–367. MR 1696493, https://doi.org/10.1017/CBO9780511662010.009
- [Ser72] Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, https://doi.org/10.1007/BF01405086
- [Ser98] Jean-Pierre Serre, Abelian 𝑙-adic representations and elliptic curves, McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0263823
- [Sil92] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1992. Corrected reprint of the 1986 original. MR 1329092
- [Sage] W. A. Stein, Sage: Open Source Mathematics Software, http://www.sagemath.org.
- [Ste02] William A. Stein, There are genus one curves over ℚ of every odd index, J. Reine Angew. Math. 547 (2002), 139–147. MR 1900139, https://doi.org/10.1515/crll.2002.047
- [SW08] W. A. Stein and C. Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, in preparation (2008).
- [Sto05]
M. Stoll, Explicit
-descent in Magma http://www.faculty.iu-bremen. de/stoll/magma/explicit-3descent/.
- [Wal85] J.-L. Waldspurger, Quelques propriétés arithmétiques de certaines formes automorphes sur 𝐺𝐿(2), Compositio Math. 54 (1985), no. 2, 121–171 (French). MR 783510
- [Wil95] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, https://doi.org/10.2307/2118559
- [Wil00] -, The Birch and Swinnerton-Dyer Conjecture, http://www.claymath.org/prize_problems/birchsd.htm.
- [Zha04] Shou-Wu Zhang, Gross-Zagier formula for 𝐺𝐿(2). II, Heegner points and Rankin 𝐿-series, Math. Sci. Res. Inst. Publ., vol. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 191–214. MR 2083213, https://doi.org/10.1017/CBO9780511756375.008
Retrieve articles in Mathematics of Computation with MSC (2000): 11Y99
Retrieve articles in all journals with MSC (2000): 11Y99
Additional Information
Grigor Grigorov
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Andrei Jorza
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544-1000
Stefan Patrikis
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Princeton, New Jersey 08544-1000
William A. Stein
Affiliation:
Department of Mathematics, University of Washington, Seattle, Box 354350, Seattle, Washington 98195-4350
Corina Tarnita
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
DOI:
https://doi.org/10.1090/S0025-5718-09-02253-4
Received by editor(s):
June 30, 2005
Received by editor(s) in revised form:
October 30, 2008
Published electronically:
June 8, 2009
Additional Notes:
This material is based upon work supported by the National Science Foundation under Grant No. 0400386.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.