On the Manin constants of modular elliptic curves

  • Bas Edixhoven
Part of the Progress in Mathematics book series (PM, volume 89)


For M a positive integer, let X 0 (M) Q be the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order M and let J 0(M)Q be the jacobian of X 0(M)Q. An elliptic curve E over Q is said to be modular if it is an isogeny factor (isogenics over Q) of some J 0(M)Q; the smallest M for which this happens is then called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of E. A modular elliptic curve of level M is called strong if there exists a closed immersion EJ 0(M)Q. It follows from the multiplicity one principle for modular forms that such an immersion is unique up to sign.


Elliptic Curve Elliptic Curf Stable Model Springer Lecture Note Modular Curve 
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  1. [1]
    B.J. Birch and H.P.F. Swinnerton-Dyer. Elliptic curves and modular functions. Modular functions of one variable IV, 2–32. Springer Lecture Notes in Mathematics 476 (1975).Google Scholar
  2. [2]
    H. Carayol. Sur les représentations l-adiques associées aux formes modulaires de Hubert. Ann. scient. Éc. Norm. Sup., série 4, t. 19, 409–468 (1986).Google Scholar
  3. [3]
    P. Deligne and M. Rapoport. Les schémas de modules des courbes elliptiques. Modular functions of one variable II, 143–316. Springer Lecture Notes in Mathematics 349.Google Scholar
  4. [4]
    S.J. Edixhoven. Minimal resolution and stable reduction of X 0 (N). Preprint Utrecht November 1986, accepted for publication in the Annales de l’Institut Fourier.Google Scholar
  5. [5]
    S.J. Edixhoven. Stable models of modular curves and applications. Thesis Utrecht June 1989, to be published.Google Scholar
  6. [6]
    N.M. Katz and B. Mazur. Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108, Princeton University Press (1985).MATHGoogle Scholar
  7. [7]
    B. Mazur. Rational isogenics of prime degree. Invent. Math. 44, 129–162 (1978).MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    B. Mazur. Courbes elliptiques et symboles modulaires. Séminaire Bourbaki juin 1972. Springer Lecture Notes in Mathematics 317.Google Scholar
  9. [9]
    B. Mazur and H.P.F. Swinnerton-Dyer. Arithmetic of Weil curves. Invent. Math. 25, 1–16 (1974).MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    J.-F. Mestre and J. Oesterlé. Letter to B. Gross, May 1985.Google Scholar
  11. [11]
    F. Oort. Commutative group schemes. Springer Lecture Notes in Mathematics 15 (1966).MATHGoogle Scholar
  12. [12]
    M. Raynaud. Spécialisation du fondeur de Picard. Publications Mathématiques de l’I.H.E.S. 38 (1970).Google Scholar
  13. [13]
    M. Raynaud. Schémas en groupes de type (p,…,p). Bull. Soc. Math. France 102, 241–280 (1974).MathSciNetMATHGoogle Scholar
  14. [14]
    G. Stevens. Stickelberger elements and modular parametrizations of elliptic curves. To be published.Google Scholar
  15. [15]
    J. Tate. Algorithm for determining the type of a singular fibre in an elliptic pencil. In Modular Functions of One Variable IV, 33–52. Springer Lecture Notes in Mathematics 476 (1975).Google Scholar
  16. [16]
    M. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. of Number Theory 15, 199–202 (1982).MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    D. Zagier. Modular parametrizations of elliptic curves. Canad. Math. Bull. 28(3), 372–384 (1985).MathSciNetMATHCrossRefGoogle Scholar

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© Springer Science+Business Media New York 1991

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  • Bas Edixhoven

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