Inventiones mathematicae

, Volume 44, Issue 2, pp 129–162 | Cite as

Rational isogenies of prime degree

  • B. Mazur
  • D. Goldfeld
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Atkin, A.O.L., Lehner, J.: Hecke operators on 161-1. Math. Ann.185, 134–160 (1970)Google Scholar
  2. 2.
    Artin, M.: The implicit function theorem in algebraic geometry. Algebraic Geometry: Papers presented at the Bombay Colloquium 1968, 13–34, Oxford University Press 1969Google Scholar
  3. 3.
    Baker, A.: On the class number of imaginary quadratic fields. Bull Amer. Math. Soc.77, 678–684 (1971)Google Scholar
  4. 4.
    Berkovic, B.G.: Rational points on the jacobians of modular curves (in Russian). Mat. Sbornik T.101 (143), No. 4 (12), 542–567 (1976)Google Scholar
  5. 5.
    Bourbaki, N.: Commutative algebra. Paris: Hermann 1972Google Scholar
  6. 6.
    Burgess, D.A.: On character sums andL-series II. Proc. London Math. Soc. (3)13, 524–536 (1963)Google Scholar
  7. 7.
    Deligne, P., Mumford, D.: The irreducibility of the space curves of given genus. Publ. Math. I.H.E.S.36, 75–109 (1969)Google Scholar
  8. 8.
    Deligne, P., Rapoprt, M.: Schémas de modules des courbes elliptiques. Vol. II of the Proceedings of the International Summer School on modular functions, Antwerp (1972). Lecture Notes in Mathematics349. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  9. 9.
    Dickson, L.E.: Linear groups with an exposition of the Galois field theory. Leipzig: Teubner 1901Google Scholar
  10. 10.
    Fricke, R.: Die elliptischen Funktionen und ihre Anwendungen. I. Leipzig-Berlin: Teubner 1922Google Scholar
  11. 11.
    Goldfeld, D.M.: A simple proof of Siegel's theorem, Proc. Nat. Acad. Sci. USA71, 1055–1055 (1974)Google Scholar
  12. 12.
    Katz, N.M.:p-adic properties of modular schemes and modular forms. Vol. III of the Proceedings of the Internation Summer School on modular functions, Antwerp (1972). Lecture Notes in Mathematics 350, 68–190. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  13. 13.
    Kubert, D.: Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3)33, 193–237 (1976)Google Scholar
  14. 14.
    Ligozat, G.: Courbes Modulaires de genre 1. Bull. Soc. Math. France, Mémoire43, 1–80 (1975)Google Scholar
  15. 15.
    Linnik, J.V., Vinogradov, A.I.: Hypoelliptic curves and the least prime quadratic residue. [in Russian] Dokl. Akad. Nauk CCCP168, 259–261 (1966). [Eng. transl.: Soviet Math. Dokl.7, 612–614 (1966)]Google Scholar
  16. 16.
    Manin, Y.: A uniform bound forp-torsion in elliptic curves [in Russian]. Izv. Akad. Nauk CCCP33, 459–465 (1969)Google Scholar
  17. 17.
    Manin, Y.: Parabolic points and zeta functions of modular curves [in Russian]. Izv. Akad. Nauk CCCP36, 19–65 (1972). [English transl.: Math. USSR Izv.6, 19–64 (1972)]Google Scholar
  18. 18.
    Mazur, B.: Rational points on modular curves. Proceedings of a conference on modular functions held in Bonn 1976. Lecture Notes in Math., 601, Berlin-Heidelberg-New York: Springer 1977Google Scholar
  19. 19.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. I.H.E.S.47 (1977)Google Scholar
  20. 20.
    Mazur, B.:p-adic analytic number theory of elliptic curves and abelian varieties overQ. Proc. of International Congress of Mathematicians at Vancouver, 1974, Vol. I, 369–377, Canadian Math. Soc. (1975)Google Scholar
  21. 21.
    Mazur, B., Serre, J.-P.: Points rationnels des courbes modulairesX 0(N). Séminaire Bourbakino469. Lecture Notes in Mathematics,514. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  22. 22.
    Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves. Inventiones math.25, 1–61 (1974)Google Scholar
  23. 23.
    Mazur, B., Tate, J.: Points of order 13 on elliptic curves. Inventiones math.22, 41–49 (1973)Google Scholar
  24. 24.
    Mazur, B., Vélu, J.: Courbes de Weil de conducteur 26. C.R. Acad. Sc. Paris275, Série A, 743–745Google Scholar
  25. 25.
    Ogg, A.: Rational points on certain elliptic modular curves. Proc. Symp. Pure Math.24, 221–231 (1973), AMS, ProvidenceGoogle Scholar
  26. 26.
    Ogg, A.: Diophantine equations and modular forms. Bull. Soc. Math. France102, 449–462 (1974)Google Scholar
  27. 27.
    Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Éc. Norm. Sup., série 4,3, 1–21 (1970)Google Scholar
  28. 28.
    Raynaud, M.: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics119, Berlin-Heidelberg-New York: Springer 1970Google Scholar
  29. 29.
    Raynaud, M.: Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S.38, 27–76 (1970)Google Scholar
  30. 30.
    Raynaud, M.: Passage au quotient par une relation d'équivalence plate. Proceedings of a conference on Local Fields, NUFFIC Summer School held at Driebergen in 1966, 133–157, Berlin-Heidelberg-New York: Springer 1967Google Scholar
  31. 31.
    Raynaud, M.: Schémas en groupes de type (p,...,p), Bull. Soc. Math. France,102, 241–280 (1974)Google Scholar
  32. 32.
    Serre, J.-P.: Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones math.15, 259–331 (1972)Google Scholar
  33. 33.
    Serre, J.-P.:p-torsion des courbes elliptiques (d'après Y. Manin). Séminaire Bourbakino380. Lecture Notes in Mathematics, 180, Berlin-Heidelberg-New York: Springer 1971Google Scholar
  34. 34.
    Serre, J.-P.: Groupes algébriques et corps de classes. Paris: Hermann 1959Google Scholar
  35. 35.
    Serre, J.-P., Tate, J.: Good reduction of abelian varieties, Ann. of Math.88, 492–517 (1968)Google Scholar
  36. 36.
    Siegel, C.L.: Über die Classenzahl quadratischer Zahlkörper. Acta Arith.1, 83–86 (1935). Also in Gesammelte Abhandlungen I, 406–409, Berlin-Heidelberg-New York: Springer 1966Google Scholar
  37. 37.
    Stark, H.M.: On complex quadratic fields with class-number equal to one. Trans. Amer. Math. Soc.122, 112–119 (1966)Google Scholar
  38. 38.
    Stark, H.M.: A complete determination of the complex quadratic fields of class-number one. Mich. Math. J.14, 1–27 (1967)Google Scholar
  39. 39.
    Swinnerton-Dyer, H.P.F., Birch, B.J.: Elliptic curves and modular functions. Modular functions of one variable IV (Proc. of the Int. Summer School, University of Antwerp, RUCA, 1972). Lecture Notes in Mathematics, 476, 2–31, Berlin-Heidelberg-New York: Springer 1975Google Scholar
  40. 40.
    Tate, J.: Algorithm for determining the Type of a Singular Fiber in an Elliptic Pencil, 33–53, Modular functions of one variable IV (Proc. of the Int. Summer School, University of Antwerp, RUCA, 1972). Lecture Notes in Mathematics, 476, Berlin-Heidelberg-New York: Springer 1975Google Scholar
  41. 41.
    Tate, J.: Classes d'isogénies des variétés abéliennes sur un corps fini (d'après T. Honda), Séminaire Bourbaki no. 352. Lecture Notes in Mathematics. 179 Berlin-Heidelberg-New York: Springer 1971Google Scholar
  42. 42.
    Tatuzawa, T.: On a theorem of Siegel. Japanese J. of math.21, 163–178 (1951)Google Scholar
  43. 43.
    Modular functions of one variable IV. (Ed. by B.J. Birch and W. Kuyk). Lecture Notes in Mathematics. 476 Berlin-Heidelberg-New York: Springer 1975Google Scholar
  44. 44.
    [EGA] Éléments de géométrie algébrique (par A. Grothendieck, rédigés avec la collaboration de J. Dieudonné) II. Étude globale élémentaire de quelques classes de morphismes. Publ. Math. I.H.E.S.8 (1961). IV Étude locale des schémas et des morphismes de schémas. Publ. Math. I.H.E.S.32 (1967)Google Scholar
  45. 45.
    [SGA 7II] Groupes de Monodromie en Géométrie Algébrique (dirigé par A. Grothendieck avec la collaboration de M. Raynaud et D.S. Rim). Lecture Notes in Mathematics 288, Berlin-Heidelberg-New York: Springer 1972Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • B. Mazur
    • 1
  • D. Goldfeld
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations