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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
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Abstract: We describe theorems and computational methods for verifying the Birch and Swinnerton-Dyer conjectural formula for specific elliptic curves over  $ \mathbb{Q}$ of analytic ranks 0 and $ 1$. We apply our techniques to show that if $ E$ is a non-CM elliptic curve over  $ \mathbb{Q}$ of conductor $ \leq 1000$ and rank 0 or $ 1$, then the Birch and Swinnerton-Dyer conjectural formula for the leading coefficient of the $ L$-series is true for $ E$, up to odd primes that divide either Tamagawa numbers of $ E$ or the degree of some rational cyclic isogeny with domain $ E$. Since the rank part of the Birch and Swinnerton-Dyer conjecture is a theorem for curves of analytic rank 0 or $ 1$, this completely verifies the full conjecture for these curves up to the primes excluded above.


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  • [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] A. Agashe and W. 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 (electronic), With an appendix by J. Cremona and B. Mazur. MR 2085902
  • [BCDT01] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over $ \mathbb{Q}$: Wild $ 3$-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843-939 (electronic). MR 2002d:11058
  • [BCP97] W. Bosma, J. Cannon, and C. 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
  • [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 (92a:11058)
  • [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 29:1212
  • [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 31:3420
  • [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] -, Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves, J. Number Theory 111 (2005), 154-178. MR 2124047 (2006g:11130)
  • [CK] Alina Carmen Cojocaru and Ernst Kani, On the surjectivity of the Galois representations associated to non-CM elliptic curves, Canad. Math. Bull. 48 (2005), 16-31. MR 2118760 (2005k:11109)
  • [CM00] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13-28. MR 1758797
  • [Coh93] H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin, 1993. MR 94i:11105
  • [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 (2006k:11121)
  • [Crea] J. E. Cremona, Elliptic curves of conductor $ \leq 25000$, http://www.maths. nott.ac.uk/personal/ jec/ftp/data/.
  • [Creb] -, mwrank (computer software), http://www.maths.nott. ac.uk/personal/jec/mwrank/
  • [Cre97] -, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997, http://www.maths.nott.ac.uk/personal/jec/book/. MR 1628193 (99e:11068)
  • [Edi91] B. Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989), Birkhäuser Boston, Boston, MA, 1991, pp. 25-39. MR 92a:11066
  • [Gri05] G. Grigorov, Kato's Euler System and the Main Conjecture, Harvard Ph.D. Thesis (2005).
  • [Gro91] B. H. Gross, Kolyvagin's work on modular elliptic curves, $ L$-functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge, 1991, pp. 235-256. MR 1110395 (93c:11039)
  • [GZ86] B. Gross and D. Zagier, Heegner points and derivatives of $ {L}$-series, Invent. Math. 84 (1986), no. 2, 225-320. MR 87j:11057
  • [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 (2004), no. 295, ix, 117-290, Cohomologies $ p$-adiques et applications arithmétiques. III. MR 2104361
  • [Kol88] V. A. Kolyvagin, Finiteness of $ {E}(\mathbf{Q})$ and $ {\mbox{\textcyr{Sh}}}({E},\mathbf{Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522-540, 670-671. MR 89m:11056
  • [Kol90] -, Euler systems, The Grothendieck Festschrift, Vol. II, Birkhäuser Boston, Boston, MA, 1990, pp. 435-483. MR 92g:11109
  • [Kol91] V. A. Kolyvagin, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves, Proceedings of the International Congress of Mathematicians, Vols. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp. 429-436. MR 1159231 (93c:11046)
  • [Lan91] S. Lang, Number theory. III. Diophantine geometry, Springer-Verlag, Berlin, 1991. MR 93a:11048
  • [Man72] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66. MR 47:3396
  • [Mat03] Kazuo Matsuno, Finite $ \Lambda$-submodules of Selmer groups of abelian varieties over cyclotomic $ \mathbb{Z}\sb p$-extensions, J. Number Theory 99 (2003), no. 2, 415-443. MR 1969183 (2004c:11098)
  • [Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129-162. MR 482230 (80h:14022)
  • [McC91] W. G. McCallum, Kolyvagin's work on Shafarevich-Tate groups, $ L$-functions and arithmetic (Durham, 1989), Cambridge Univ. Press, Cambridge, 1991, pp. 295-316. MR 92m:11062
  • [Mil86] J. S. Milne, Arithmetic duality theorems, Academic Press Inc., Boston, Mass., 1986. MR 881804 (88e:14028)
  • [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 (92e:11050)
  • [MR04] Barry Mazur and Karl Rubin, Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799, viii+96. MR 2031496 (2005b:11179)
  • [PS99] B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109-1149. MR 2000m:11048
  • [Rub98] K. Rubin, Euler systems and modular elliptic curves, Galois representations in arithmetic algebraic geometry (Durham, 1996), Cambridge Univ. Press, Cambridge, 1998, pp. 351-367. MR 2001a:11106
  • [Ser72] J-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259-331. MR 0387283 (52:8126)
  • [Ser98] -, Abelian $ \ell$-adic representations and elliptic curves, A K Peters Ltd., Wellesley, MA, 1998, With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original. MR 0263823 (41:8422)
  • [Sil92] J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1992, Corrected reprint of the 1986 original. MR 1329092 (95m:11054)
  • [Sage] W. A. Stein, Sage: Open Source Mathematics Software, http://www.sagemath.org.
  • [Ste02] W. A. Stein, There are genus one curves over $ \mathbf{Q}$ of every odd index, J. Reine Angew. Math. 547 (2002), 139-147. MR 1900139 (2003c:11059)
  • [SW08] W. A. Stein and C. Wuthrich, Computations About Tate-Shafarevich Groups Using Iwasawa Theory, in preparation (2008).
  • [Sto05] M. Stoll, Explicit $ 3$-descent in Magma http://www.faculty.iu-bremen. de/stoll/magma/explicit-3descent/.
  • [Wal85] J.-L. Waldspurger, Sur les valeurs de certaines fonctions $ L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173-242. MR 783511 (87g:11061b)
  • [Wil95] A. J. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443-551. MR 1333035 (96d:11071)
  • [Wil00] -, The Birch and Swinnerton-Dyer Conjecture, http://www.claymath.org/prize_problems/birchsd.htm.
  • [Zha04] Shou-Wu Zhang, Gross-Zagier formula for $ \rm 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

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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.

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