Skip to main content
Log in

Rational isogenies of prime degree

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Atkin, A.O.L., Lehner, J.: Hecke operators on 161-1. Math. Ann.185, 134–160 (1970)

    Google Scholar 

  2. Artin, M.: The implicit function theorem in algebraic geometry. Algebraic Geometry: Papers presented at the Bombay Colloquium 1968, 13–34, Oxford University Press 1969

  3. Baker, A.: On the class number of imaginary quadratic fields. Bull Amer. Math. Soc.77, 678–684 (1971)

    Google Scholar 

  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. Bourbaki, N.: Commutative algebra. Paris: Hermann 1972

    Google Scholar 

  6. Burgess, D.A.: On character sums andL-series II. Proc. London Math. Soc. (3)13, 524–536 (1963)

    Google Scholar 

  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. 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 1973

    Google Scholar 

  9. Dickson, L.E.: Linear groups with an exposition of the Galois field theory. Leipzig: Teubner 1901

    Google Scholar 

  10. Fricke, R.: Die elliptischen Funktionen und ihre Anwendungen. I. Leipzig-Berlin: Teubner 1922

    Google Scholar 

  11. Goldfeld, D.M.: A simple proof of Siegel's theorem, Proc. Nat. Acad. Sci. USA71, 1055–1055 (1974)

    Google Scholar 

  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 1973

    Google Scholar 

  13. Kubert, D.: Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3)33, 193–237 (1976)

    Google Scholar 

  14. Ligozat, G.: Courbes Modulaires de genre 1. Bull. Soc. Math. France, Mémoire43, 1–80 (1975)

    Google Scholar 

  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. Manin, Y.: A uniform bound forp-torsion in elliptic curves [in Russian]. Izv. Akad. Nauk CCCP33, 459–465 (1969)

    Google Scholar 

  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. 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 1977

    Google Scholar 

  19. Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. I.H.E.S.47 (1977)

  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)

  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 1976

    Google Scholar 

  22. Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves. Inventiones math.25, 1–61 (1974)

    Google Scholar 

  23. Mazur, B., Tate, J.: Points of order 13 on elliptic curves. Inventiones math.22, 41–49 (1973)

    Google Scholar 

  24. Mazur, B., Vélu, J.: Courbes de Weil de conducteur 26. C.R. Acad. Sc. Paris275, Série A, 743–745

  25. Ogg, A.: Rational points on certain elliptic modular curves. Proc. Symp. Pure Math.24, 221–231 (1973), AMS, Providence

    Google Scholar 

  26. Ogg, A.: Diophantine equations and modular forms. Bull. Soc. Math. France102, 449–462 (1974)

    Google Scholar 

  27. Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Éc. Norm. Sup., série 4,3, 1–21 (1970)

    Google Scholar 

  28. Raynaud, M.: Faisceaux amples sur les schémas en groupes et les espaces homogènes. Lecture Notes in Mathematics119, Berlin-Heidelberg-New York: Springer 1970

    Google Scholar 

  29. Raynaud, M.: Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S.38, 27–76 (1970)

    Google Scholar 

  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 1967

    Google Scholar 

  31. Raynaud, M.: Schémas en groupes de type (p,...,p), Bull. Soc. Math. France,102, 241–280 (1974)

    Google Scholar 

  32. Serre, J.-P.: Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones math.15, 259–331 (1972)

    Google Scholar 

  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 1971

    Google Scholar 

  34. Serre, J.-P.: Groupes algébriques et corps de classes. Paris: Hermann 1959

    Google Scholar 

  35. Serre, J.-P., Tate, J.: Good reduction of abelian varieties, Ann. of Math.88, 492–517 (1968)

    Google Scholar 

  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 1966

    Google Scholar 

  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. 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. 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 1975

    Google Scholar 

  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 1975

    Google Scholar 

  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 1971

    Google Scholar 

  42. Tatuzawa, T.: On a theorem of Siegel. Japanese J. of math.21, 163–178 (1951)

    Google Scholar 

  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 1975

    Google Scholar 

  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)

  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 1972

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mazur, B., Goldfeld, D. Rational isogenies of prime degree. Invent Math 44, 129–162 (1978). https://doi.org/10.1007/BF01390348

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01390348

Keywords

Navigation