Advertisement

The modular degree, congruence primes, and multiplicity one

  • Amod Agashe
  • Kenneth A. Ribet
  • William A. Stein
Chapter

Abstract

The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.

Key words

elliptic curves abelian varieties modular degree congruence primes multiplicity one 

Notes

Acknowledgments

The authors are grateful to M. Baker, F. Calegari, B. Conrad, J. Cremona, G. Frey, H. W. Lenstra, Jr. and B. Noohi for discussions and advice regarding this paper. We would especially like to thank B. Conrad for the material in the appendix and for his suggestions concerning a number of technical facts that are inputs to our arguments. The first author is also grateful to the Max-Planck-Institut für Mathematik for its hospitality during a visit when he partly worked on this paper.

References

  1. AK70.
    A. Altman and Steven Kleiman,Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin, 1970.Google Scholar
  2. ARS06.
    A. Agashe, K. Ribet and W. Stein, The Manin Constant, Pure and Applied Mathematics Quarterly, Special issue: In honor of John H. Coates (2006), to appear.Google Scholar
  3. AS05.
    A. Agashe and W. A. 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. Google Scholar
  4. AU96.
    A. Abbes and E. Ullmo,À propos de la conjecture de Manin pour les courbes elliptiques modulaires, Compositio Math. 103 (1996), no. 3, 269–286.Google Scholar
  5. 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).Google Scholar
  6. BCDT01.
    C. Breuil, B. Conrad, F. Diamond, and R. Taylor,On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939 (electronic).Google Scholar
  7. BLR90.
    S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990.Google Scholar
  8. Con00.
    B. Conrad,Grothendieck duality and base change, Lecture Notes in Mathematics, Vol. 1750, Springer-Verlag, Berlin, 2000.Google Scholar
  9. Con07.
    B. Conrad, Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu 6 (2007), no. 2, 209–278.Google Scholar
  10. CK04.
    A. C. Cojocaru and E. Kani, The modular degree and the congruence number of a weight 2 cusp form, Acta Arith. 114 (2004), no. 2, 159–167.Google Scholar
  11. Cre97.
    J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997, available athttp://www.maths.nott.ac.uk/personal/jec/book/.
  12. DI95.
    F. Diamond and J. Im,Modular forms and modular curves, Seminar on Fermat’s Last Theorem, Providence, RI, 1995, pp. 39–133.Google Scholar
  13. Dia97.
    F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), no. 2, 379–391.Google Scholar
  14. DR73.
    P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Berlin), Springer, 1973, Lecture Notes in Math., Vol. 349 pp. 143–316. Google Scholar
  15. Fre97.
    G. Frey,On ternary equations of Fermat type and relations with elliptic curves, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 527–548.Google Scholar
  16. FM99.
    G. Frey and M. Müller, Arithmetic of modular curves and applications, Algorithmic algebra and number theory (Heidelberg, 1997), Springer, Berlin, 1999, pp. 11–48.Google Scholar
  17. Kil02.
    L. J. P. Kilford,Some non-Gorenstein Hecke algebras attached to spaces of modular forms, J. Number Theory 97 (2002), no. 1, 157–164.Google Scholar
  18. KM85.
    N. M. Katz and B. Mazur,Arithmetic Moduli of Elliptic Curves, Princeton University Press, Princeton, N.J., 1985.Google Scholar
  19. KW08.
    L. J. P. Kilford and Gabor Wiese, On the failure of the Gorenstein property for Hecke algebras of prime weight, Experiment. Math. 17 (2008), no. 1, 37–52. o (N)RMR2410114 (2009c:11075).Google Scholar
  20. Li75.
    W-C. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.Google Scholar
  21. Maz77.
    B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).Google Scholar
  22. Maz78.
    B. Mazur,Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.Google Scholar
  23. MR91.
    B. Mazur and K. A. Ribet,Two-dimensional representations in the arithmetic of modular curves, Astérisque (1991), no. 196–197, 6, 215–255 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).Google Scholar
  24. Mur99.
    M. R. Murty, Bounds for congruence primes, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Amer. Math. Soc., Providence, RI, 1999, pp. 177–192.Google Scholar
  25. Rib75.
    K. A. Ribet,Endomorphisms of semi-stable abelian varieties over number fields, Ann. Math. (2) 101 (1975), 555–562.Google Scholar
  26. Rib81.
    K. A. Ribet,Endomorphism algebras of abelian varieties attached to newforms of weight 2, Seminar on Number Theory, Paris 1979–80, Progr. Math., Vol. 12, Birkhäuser Boston, Mass., 1981, pp. 263–276.Google Scholar
  27. Rib83.
    K. A. Ribet, Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1, 193–205.Google Scholar
  28. Rib90.
    K. A. Ribet, On modular representations of \(\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476.Google Scholar
  29. RS01.
    K. A. Ribet and W. A. Stein,Lectures on Serre’s conjectures, Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., Vol. 9, Amer. Math. Soc., Providence, RI, 2001, pp. 143–232.Google Scholar
  30. Ser88.
    J-P. Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, Vol. 117, Springer-Verlag, New York, 1988, Translated from the French.Google Scholar
  31. Shi94.
    G. Shimura,Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1.Google Scholar
  32. S + 09.
    W. A. Stein et al., Sage Mathematics Software (Version 4.2), The Sage Development Team, 2009,http://www.sagemath.org.
  33. Stu87.
    J. Sturm,On the congruence of modular forms, Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280.Google Scholar
  34. Til97.
    J. Tilouine, Hecke algebras and the Gorenstein property, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 327–342.Google Scholar
  35. Wie07.
    G. Wiese,Multiplicities of Galois representations of weight one, Algebra Number Theory 1 (2007), no. 1, 67–85, With an appendix by Niko Naumann.Google Scholar
  36. Wil80.
    A. Wiles,Modular curves and the class group of Q(ζ p ), Invent. Math. 58 (1980), no. 1, 1–35.Google Scholar
  37. Wil95.
    A. J. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551.Google Scholar
  38. Zag85.
    D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), no. 3, 372–384.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Amod Agashe
    • 1
  • Kenneth A. Ribet
    • 2
  • William A. Stein
    • 3
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of MathematicsUC BerkeleyBerkeleyUSA
  3. 3.University of WashingtonSeattleUSA

Personalised recommendations