Inventiones mathematicae

, Volume 76, Issue 2, pp 179–330 | Cite as

Class fields of abelian extensions of Q

  • B. Mazur
  • A. Wiles
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Artin, M.: Algebraization of formal moduli I. Global Analysis. Papers in honor of K. Kodaira Spencer, D.C., Iyanaga, S., (eds.) Univ. of Tokyo Press and Princeton Univ. Press, 21–71 1969Google Scholar
  2. 2.
    Atkin, A.O.L., Lehner, J.: Hecke operators onΓ 0(m). Math. Ann.185, 134–160 (1970)Google Scholar
  3. 3.
    Atkin, A.O.L., Li, W.: Twists of newforms and pseudo-eigenvalues ofW-operators. Invent. Math.43, 221–244 (1978)Google Scholar
  4. 4.
    Auslander, M., Buchsbaum, D.: Groups, Rings, Modules. New York, Evanston, San Francisco, London: Harper & Row 1974Google Scholar
  5. 5.
    Bayer, P., Neukirch, J.: On values of zeta functions andl-adic Euler characteristics. Invent Math.50, 35–64 (1978)Google Scholar
  6. 6.
    Casselman, W.: On representations ofGL 2 and the arithmetic of modular curves. (International Summer School on Modular functions, Antwerp 1972) Modular functions of one variable II. Lecture Notes in Mathematics Vol. 349, pp. 109–141. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  7. 7.
    Cassels, J.W.S., Fröhlich, A.: Algebraic number theory. London-New York: Academic Press 1967Google Scholar
  8. 8.
    Coates, J.:p-adicL-functions and Iwasawa's theory. In: Algebraic Number Fields, Fröhlich, A., (ed.) London-New York: Academic Press 1977Google Scholar
  9. 9.
    Coates, J.: The Work of Mazur and Wiles on Cyclotomic Fields. Séminaire Bourbaki No. 575, Lecture Notes in Mathematics Vol. 901. Berlin-Heidelberg-New York: Springer 1981Google Scholar
  10. 10.
    Coates, J.:K-theory and Iwasawa's analogue of the Jacobian, in AlgebraicK-theory II. Lecture Notes in Mathematics Vol. 342. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  11. 11.
    Coates, J., Lichtenbaum, S.: Onl-adic zeta functions. Ann. of Math.98, 498–550 (1973)Google Scholar
  12. 12.
    Coates, J., Sinnott, W.: An analogue of Stickelberger's theorem for the higherK-groups. Invent. Math.24, 149–161 (1974)Google Scholar
  13. 13.
    Deligne, P.: Formes modulaires et représentationsl-adiques, Séminaire Bourbaki 68/69 no. 355. Lecture Notes in Mathematics Vol. 179, pp. 136–172. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  14. 14.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publications Mathématiques I.H.E.S.,36, 75–109 (1969)Google Scholar
  15. 15.
    Deligne, P., Rapoport, M.: Schémas de modules de courbes elliptiques. Lecture Note in Mathematics Vol. 349. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  16. 16.
    Demazure, M.: Lectures onp-divisible groups. Lecture Notes in Mathematics Vol. 302. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  17. 17.
    Federer, L., Gross, B.: Regulators and Iwasawa Modules. Invent. Math.62, 443–457 (1981)Google Scholar
  18. 18.
    Ferrero, B., Greenberg, R.: On the behavior of thep-adicL-function ats=0. Invent. Math.50, 91–102 (1978)Google Scholar
  19. 19.
    Ferrero, B., Washington, L.: The Iwasawa invariantµ p vanishes for abelian number fields. Ann. of Math.109, 377–396 (1979)Google Scholar
  20. 20.
    Fontaine, J.-M.: Groupes finis commutatifs sur les vecteurs de Witt. C. R. Acad. Sc. Paris t.280, (serie A) 1423–1425 (1979)Google Scholar
  21. 21.
    Gras, G.: Classes d'idéaux des corps abéliens et nombres de Bernoulli généralisés. Ann. Inst. Fourier27, 1–66 (1977)Google Scholar
  22. 22.
    Greenberg, R.: On a certainl-adic representation. Invent. Math.,21, 198–205 (1973)Google Scholar
  23. 23.
    Greenberg, R.: Onp-adicL-functions and cyclotomic fields. Nagoya Math. J.56 61–77 (1974)Google Scholar
  24. 24.
    Greenberg, R.: Onp-adicL-functions and cyclotomic fields II. Nagoya Math. J.67, 139–158 (1977)Google Scholar
  25. 25.
    Greenberg, R.: On the structure of certain Galois groups. Invent. Math.47, 85–99 (1978)Google Scholar
  26. 26.
    Grothendieck, A.: Modéles de Néron et monodromie. SGA 7 I exposé IX. Lecture Notes in Mathematics, Vol. 288. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  27. 27.
    Hartshore, R.: Algebraic Geometry. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  28. 28.
    Igusa, J.: Kroneckerian model of fields of elliptic modular functions. Amer. J. Math.81, 561–577 (1959)Google Scholar
  29. 29.
    Igusa, J.: On the algebraic theory of elliptic modular functions. J. Math. Soc. Japan20, 96–106 (1968)Google Scholar
  30. 30.
    Iwasawa, K.: Onp-adicL-functions. Ann. Math.89, 198–205 (1969)Google Scholar
  31. 31.
    Iwasawa, K.: Lectures onp-adicL-functions. Princeton: Princeton Univ. Press and Univ. of Tokyo Press 1972Google Scholar
  32. 32.
    Iwasawa, K.: OnZ l-extensions of algebraic number fields. Ann. of Math.98, 246–326 (1973)Google Scholar
  33. 33.
    Kamienny, S.: OnJ 1(p) and the arithmetic of the kernel of the Eisenstein ideal. Harvard Ph.D. Thesis, 1980Google Scholar
  34. 34.
    Katz, N.:p-adic properties of modular schemes and modular forms, vol. III of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, Vol. 350, pp. 69–190. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  35. 35.
    Katz, N.: Higher congruences between modular forms. Ann. of Math.101, (no. 2) 332–367 (1975)Google Scholar
  36. 36.
    Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. To appear in Annals of Math. Studies, Princeton U. Press.Google Scholar
  37. 37.
    Knutson, D.: Algebraic spaces. Lecture Notes in Mathematics, Vol. 203. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  38. 38.
    Kubert, D.: Quadratic relations for generators of units in the modular function fields. Math. Ann.225, 1–20 (1977)Google Scholar
  39. 39.
    Kubert, D., Lang, S.: Modular Units. Berlin-Heidelberg-New York: Springer 1981Google Scholar
  40. 40.
    Lang, S.: Introduction to modular forms. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  41. 41.
    Lang, S.: Cyclotomic fields. Berlin-Heidelberg-New York: Springer 1978Google Scholar
  42. 42.
    Lang, S.: Cyclotomic fields II. Berlin-Heidelberg-New York: Springer 1980Google Scholar
  43. 43.
    Lang, S.: Units and class numbers in Number theory and algebraic Geometry. Lecture notes distributed in conjunction with the colloquium lectures given at the 85-th summer meeting of the A.M.S., University of Pittsburgh, Pittsburgh, Pennsylvania, August 17–20, 1981Google Scholar
  44. 44.
    Langlands, R.P.: Modular forms andl-adic representations. (International Summer School on Modular Functions, Antwerp, 1972) Modular functions of one variable II, Lecture Notes in Mathematics, Vol. 349, pp. 361–500, Berlin-Heidelberg-New York: Springer 1973Google Scholar
  45. 45.
    Li, W.: Newforms and functional equations. Math. Ann.212, 285–315 (1975)Google Scholar
  46. 46.
    Lichtenbaum, S.: On the values of zeta andL-functions I. Ann. of Math.96 (no. 2) 338–360 (1972)Google Scholar
  47. 47.
    Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. I.H.E.S.47, (1948)Google Scholar
  48. 48.
    Mazur, B., Tate, J.: Points of order 13 on elliptic curves. Invent. Math.22, 41–49 (1973)Google Scholar
  49. 49.
    Mazur, B.: Rational isogenies of prime degree. Invent. Math.44, 129–162 (1978)Google Scholar
  50. 50.
    Northcott, D.G.: Finite free resolutions. Cambridge Univ. Press, Cambridge-New York 1976Google Scholar
  51. 51.
    Oort, F.: Commutative group schemes. Lecture Notes in Mathematics Vol. 15. Berlin-Heidelberg-New York: Springer 1966Google Scholar
  52. 52.
    Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Ec. Norm. Sup. (serie 4)3, 1–21 (1970)Google Scholar
  53. 53.
    Raynaud, M.: Passage au quotient par une relation d'équivalence plate. Proc. of a Conference on Local Fields, NUFFIC Summer School held at Driebergen in 1966, pp. 133–157, Berlin-Heidelberg-New York: Springer 1967Google Scholar
  54. 54.
    Raynaud, M.: Spécialisation du foncteur de Picard. Publ. Math. I.H.E.S.38, 27–76 (1970)Google Scholar
  55. 35.
    Raynaud, M.: Schémas en groupes de type (p, ... p). Bull. Soc. Math. France102, 241–280 (1974)Google Scholar
  56. 56.
    Ribet, K.: A modular construction of unramifiedp-extensions ofQ(µ p). Invent. Math.34, 151–162 (1976)Google Scholar
  57. 57.
    Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique p. Symp. Int. de Top. Alg., Mexico, 1958Google Scholar
  58. 58.
    Serre, J-P.: Classes des corps cyclotomiques, Séminaire Bourbaki. Exp.174 (1958–9)Google Scholar
  59. 59.
    Shafarevitch, I.R.: Lectures on minimal models and birational transformations of two-dimensional schemes. Tata Institute of fundamental research: Bombay 1966Google Scholar
  60. 60.
    Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Publ. Math. Soc. Japan11, Tokyo-Princeton (1971)Google Scholar
  61. 61.
    Sinnott, W.: On the Stickelberger ideal and the circular units of a cyclotomic field. Ann. Math.108, 107–134 (1978)Google Scholar
  62. 62.
    Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math.62, 181–234 (1980)Google Scholar
  63. 63.
    Soulé, C.:K-theorie des anneaux d'entiers de corps de nombres et cohomologie étale. Invent. Math.55, 251–295 (1979)Google Scholar
  64. 64.
    Tate, J.:p-divisible groups. Proceedings of a conference on local fields (Driebergen 1966). Berlin-Heidelberg-New York: Springer 1967Google Scholar
  65. 65.
    Tate, J.: Relations between K2 and Galois cohomology. Invent. Math.36, 257–274 (1976)Google Scholar
  66. 66.
    Tate, J.: Number theoretic background. Proceedings of Symposia in Pure Mathematics33, (part II) 3–26 (1979)Google Scholar
  67. 67.
    Wiles, A.: Modular curves and the class group ofQ(ζ p). Invent. Math.58, 1–35 (1980)Google Scholar
  68. 68.
    Yu, J.: A cuspidal class number formula for the modular curvesX 1(N). Math. Ann.252, 197–216 (1980)Google Scholar
  69. 69.
    Serre, J-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math.88, 492–517 (1968)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • B. Mazur
    • 1
  • A. Wiles
    • 1
  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations