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Mathematics of Computation

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Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero
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by Amod Agashe and William Stein; with an Appendix by J. Cremona; B. Mazur PDF
Math. Comp. 74 (2005), 455-484 Request permission


This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank $0$ abelian varieties $A_f$ that are optimal quotients of $J_0(N)$ attached to newforms. We prove theorems about the ratio $L(A_f,1)/\Omega _{A_f}$, develop tools for computing with $A_f$, and gather data about certain arithmetic invariants of the nearly $20,000$ abelian varieties $A_f$ of level $\leq 2333$. Over half of these $A_f$ have analytic rank $0$, and for these we compute upper and lower bounds on the conjectural order of $\Sha (A_f)$. We find that there are at least $168$ such $A_f$ for which the Birch and Swinnerton-Dyer conjecture implies that $\Sha (A_f)$ is divisible by an odd prime, and we prove for $37$ of these that the odd part of the conjectural order of $\Sha (A_f)$ really divides $\#\Sha (A_f)$ by constructing nontrivial elements of $\Sha (A_f)$ using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.
  • Amod Agashé, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369–374 (English, with English and French summaries). MR 1678131, DOI 10.1016/S0764-4442(99)80173-6
  • A. Agashe and W. A. Stein, Appendix to Joan-C. Lario and René Schoof: Some computations with Hecke rings and deformation rings, to appear in J. Exp. Math.
  • A. Agashe and W. A. Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties, to appear in J. of Number Theory (2002).
  • A. Agashe and W. A. Stein, The Manin constant, congruence primes, and the modular degree, preprint, (2004).
  • 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).
  • B. J. Birch, Elliptic curves over $Q$: A progress report, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 396–400. MR 0314845
  • Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
  • 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 163913, DOI 10.1112/plms/s3-12.1.259
  • J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
  • J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28.
  • Brian Conrad and William A. Stein, Component groups of purely toric quotients, Math. Res. Lett. 8 (2001), no. 5-6, 745–766. MR 1879817, DOI 10.4310/MRL.2001.v8.n6.a5
  • C. Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $\bf {Q}$, Experiment. Math. 10 (2001), no. 2, 191–196.
  • Fred Diamond and John Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39–133. MR 1357209
  • 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, DOI 10.1007/978-1-4612-0457-2_{3}
  • M. Emerton, Optimal quotients of modular Jacobians. Preprint.
  • E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675–1697.
  • Jean-Marc Fontaine, Groupes finis commutatifs sur les vecteurs de Witt, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Ai, A1423–A1425 (French, with English summary). MR 374153
  • Benedict H. Gross, $L$-functions at the central critical point, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 527–535. MR 1265543
  • Alexander Grothendieck, Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 46–66 (French). MR 244269
  • Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
  • Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
  • Nicholas M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481–502. MR 604840, DOI 10.1007/BF01394256
  • V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253. MR 1036843
  • V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of SH over totally real fields, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 4, 851–876 (Russian); English transl., Math. USSR-Izv. 39 (1992), no. 1, 829–853. MR 1137589, DOI 10.1070/IM1992v039n01ABEH002228
  • D. R. Kohel and W. A. Stein, Component Groups of Quotients of $J_0(N)$, Proceedings of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, July 2–7, 2000 (Berlin), Springer, 2000.
  • Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
  • H. W. Lenstra Jr. and F. Oort, Abelian varieties having purely additive reduction, J. Pure Appl. Algebra 36 (1985), no. 3, 281–298. MR 790619, DOI 10.1016/0022-4049(85)90079-9
  • Joan-C. Lario and René Schoof, Some computations with Hecke rings and deformation rings, Experiment. Math. 11 (2002), no. 2, 303–311. With an appendix by Amod Agashe and William Stein. MR 1959271
  • Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
  • B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
  • B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
  • B. Mazur and J. Tate, Points of order $13$ on elliptic curves, Invent. Math. 22 (1973/74), 41–49. MR 347826, DOI 10.1007/BF01425572
  • J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150.
  • A. P. Ogg, Rational points on certain elliptic modular curves, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 221–231. MR 0337974
  • 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, DOI 10.2307/121064
  • Goro Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523–544. MR 318162, DOI 10.2969/jmsj/02530523
  • Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
  • Glenn Stevens, Arithmetic on modular curves, Progress in Mathematics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 1982. MR 670070
  • W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000).
  • W. A. Stein, An introduction to computing modular forms using modular symbols, to appear in an MSRI Proceedings (2002).
  • W. A. Stein, Shafarevich-Tate groups of nonsquare order, Proceedings of MCAV 2002, Progress of Mathematics (to appear).
  • Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280. MR 894516, DOI 10.1007/BFb0072985
  • John Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295. MR 0175892
  • J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1966 (reprinted in 1995), Exp. No. 306, 415–440.
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Additional Information
  • Amod Agashe
  • Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
  • Email:
  • William Stein
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 679996
  • Email:
  • J. Cremona
  • Affiliation: Division of Pure Mathematics, School of Mathematical Sciences, University of Nottingham, England
  • MR Author ID: 52705
  • ORCID: 0000-0002-7212-0162
  • Email:
  • B. Mazur
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts
  • MR Author ID: 121915
  • ORCID: 0000-0002-1748-2953
  • Email:
  • Received by editor(s): May 17, 2002
  • Received by editor(s) in revised form: June 9, 2003
  • Published electronically: May 18, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 455-484
  • MSC (2000): Primary 11G40; Secondary 11F11, 11G10, 14K15, 14H25, 14H40
  • DOI:
  • MathSciNet review: 2085902