Skip to main content
Log in

Rational points of abelian varieties with values in towers of number fields

  • 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. Artin, M.: Auto-duality of the Jacobian. Mineographed notes. Bowdoin Summer Conference in Algebraic Geometry, 1967.

  2. [GT] Artin, M.: Grothendieck topologies. Mimeographed notes. Harvard University, 1962.

  3. [SGAA] Artin, M., Grothendieck, A.: Séminaire de géometrie algébrique 1963–64. Cohomologie, étale des schémas Mimeographed notes. Institut des Hautes Etudes Scientifiques, Paris.

  4. Artin, M., Mazur, B.: Etale homotopy. Lecture Notes in Math. no. 178. Berlin-Heidelberg-New York: Springer 1969.

    Google Scholar 

  5. Artin, M., Mazur, B.: Flat arithmetic duality (in preparation).

  6. Artin, M., Verdier, J.L.: Etale arithmetic duality. Proceedings of the summer conference in Algebraic Geometry held at Woodshole, Mass. 1965.

  7. Birch, B.J., Swinnerton-Dyer, H.P.F.: Notes on elliptic curves I. J. Reine Angew. Math.212, 7–25 (1963); 11218, 79–108 (1965).

    Google Scholar 

  8. Burnside, W.: The theory of groups (2nd Ed.). Cambridge University Press 1911.

  9. Cassels, J. W. S.: On a diophantine equation. Acta Arithmetica6, 47–52 (1960).

    Google Scholar 

  10. Cassels, J. W. S.: Diophantine equations with special reference to elliptic curves. J. London Math. Soc.41, 193–291 (1966).

    Google Scholar 

  11. Cassels, J. W. S.: Arithmetic on curves of genus one (IV). J. Reine Angew. Math.211, 95–112 (1962).

    Google Scholar 

  12. Cassels, J. W. S., Fröhlich, A. (eds.): Algebraic number theory. London-New York: Academic Press 1967.

    Google Scholar 

  13. Cassels, J. W. S., Sansone G.: Sur le probleme de M. Werner Mnich. Acta Arithmetica7, 187–190 (1961/62).

    Google Scholar 

  14. Deligne, P.: Variétés abéliennes ordinaires sur un corps fini. Inventiones math.8, 238–243 (1969).

    Google Scholar 

  15. [SGAD] Demazure, M., Grothendieck, A.: Schémas en groupes. Séminaire I. H. E. S., 1963–64. Lecture Notes in Math. nos. 151–153. Berlin-Heidelberg-New York: Springer 1970.

    Google Scholar 

  16. Eichler, M.: Quaternäre, quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math.5, 355–366 (1954).

    Google Scholar 

  17. Greenberg, M. J.: Schemata over local rings. Ann. of Math.73, no. 3, 624–648 (1961); II, Ann. of Math.78, no. 2, 256–266 (1963).

    Google Scholar 

  18. Greenberg, M. J.: Pro-algebraic structure on the rational subgroup, of ap-adic abelian variety. Ph. D. thesis. Princeton University 1959.

  19. Grothendieck, A.: Sur quelques points d'algèbre homologique. Tohoku Math. J.9, 119–221 (1957).

    Google Scholar 

  20. [GB III] Grothendieck, A.: Le groupe de Brauer III: exemples et compléments (a continuation of Bourbaki exposés: 290, 297). Published in Dix exposés sur la cohomologie des schémas. Amsterdam: North-Holland Pub. Cie. 1968.

    Google Scholar 

  21. Grothendieck, A.: Techniques de déscente et théorèmes d'existence en géometrie algébrique. Séminaire Bourbaki, 12, exp. 195 (1959–60). New York-Amsterdam: Benjamin Inc. 1966.

    Google Scholar 

  22. [EGA] Grothendieck, A.: Redigé avec la collaboration de J. Dieudonné, Éléments de géométrie algébrique. Publications Mathematiques, I. H. E. S., 4, 8, 11, 17, 20, 24, 28, 32 Paris (1961–68).

  23. [SGA] Grothendieck A.: Revêtements étales et groupes fondamentaux. Séminaires de Géométrie Algébrique à l'I. H. E. S. (60–61). Lecture Notes in Math. no. 224. Berlin-Heidelberg-New York: Springer 1971.

    Google Scholar 

  24. Hardy, G. H., Littlewood, J. E.: Some problems of partitic numerorum III. Acta Math.44, 1–70 (1923); reprinted in: G. H. Hardy, Collected papers, vol. 1, Oxford (1966), 561–630.

    Google Scholar 

  25. Hasse, H.: Existenz separabler zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über elliptischen Funktionenkörpern der Charkteristikp. J. Reine Angew. Math.172, 2, 77–85 (1934).

    Google Scholar 

  26. Hasse, H., Witt, E.: Zyklischer unverzweigter Erweiterungskörper vom Primzahlgradep über einem Funktionenkörper der Charakteristikp. Monatshefte für Math. u. Physik43, 477–492 (1936).

    Google Scholar 

  27. Honda, T.: Isogeny classes of abelian varieties over finite fields. J. Math. Soc. Japan20, 83–95 (1968).

    Google Scholar 

  28. Igusa, J.: Kroneckerian model of fields of elliptic modular functions. Amer. J. Math.81, 561–577 (1959).

    Google Scholar 

  29. Iwasawa, K.: On some properties of Γ-finite modules. Ann. of Math.70, no. 2, 291–312 (1959).

    Google Scholar 

  30. Iwasawa, K.: On Γ-extensions of number fields. Bull. Amer. Math. Soc.65, no. 4, 183–226 (1959).

    Google Scholar 

  31. Iwasawa, K.: On the theory of cyclotomic fields. Ann. of Math.70, no. 3, 530–561 (1959).

    Google Scholar 

  32. Iwasawa, K., Sims, C. C.: Computation of invariants in the theory of cyclotomic fields. J. of the Math. Soc. of Japan18, no. 1, 86–96 (1966).

    Google Scholar 

  33. Kubota, T., Leopoldt, H. W.: Einep-adische Theorie der Zetawerte (Teil I) J. Reine Angew. Math.213, 228–239 (1964).

    Google Scholar 

  34. Lang, S.: Algebraic numbers. Reading, Mass: Addison-Wesley 1964.

    Google Scholar 

  35. Lang, S.: Algebraic groups over finite fields. Amer. J. Math.78, no. 3, 555–563 (1956).

    Google Scholar 

  36. Ligozat, G.: FonctionL des courbes modulaires. Mimeo. notes. Séminaire Delange-Pisot-Poitou, 1969/70, no. 9. Version to appear Courbes modulaires de genre 1.

  37. Manin, Ju.: Cyclotomic fields and modular curves [in Russian]. Uspekhi Mat. Nauk. Tom XXVI6, (162), 7–71 (1971). Translation to appear in Russian Math. Surveys. London Math. Society.

    Google Scholar 

  38. Mazur, B.: Rational points of Abelian varieties with values in towers of number fields. Mimeo. notes, Harvard U. 1969.

  39. Mazur, B.: Arithmétique des courbes elliptiques sur les corps cyclotomiques. Mimeographed notes by J. F. Boutot of a course given at Orsay 1970, distributed by I. H. E. S. Paris.

  40. Mazur, B.: Local flat duality Amer. Journal of Math.92, 343–361 (1970).

    Google Scholar 

  41. Mazur, B., Roberts, L.: Local Euler characteristics. Inventiones math.9, 201–234 (1970).

    Google Scholar 

  42. Mazur, B., Swinnerton-Dyer H. P. F.: Thep-adicL-series of an elliptic curve (in preparation).

  43. Milne, J. S.: Extensions of abelian varieties defined over a finite field. Inventiones math.5, 63–84 (1968).

    Google Scholar 

  44. Mumford, D.: Lectures on curves on an algebraic surface (with the assistance of G. M. Bergman). Ann. of Math. Studies59, Princeton, 1966.

  45. Mumford, D.: Geometric invariant theory. Ergebnisse Math., Bd. 34. Berlin-Heidelerg-New York: Springer 1965.

    Google Scholar 

  46. Mumford, D., Oort, F.: Deformations and lifting of finite commutative group schemes. Inventiones math.5, 317–334 (1968).

    Google Scholar 

  47. Néron, A.: Modèles mimimaux des variétés abéliennes sur les corps locaux et globaux. Publications Mathematiques, I. H. E. S., no. 21 (1964).

  48. Ogg, A.: Elliptic curves and wild ramification. Amer. J. of. Math.89, 1–21 (1967).

    Google Scholar 

  49. Oort, F.: Commutative group schemes. Lecture Notes in Mathematics no. 15. Berlin-Heidelberg-New York: Springer 1966.

    Google Scholar 

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

    Google Scholar 

  51. Raynaud, M.: Passage au quotient par une relation d'équivalence plate. Proc. of a Conference on Local Fields. Berlin-Heidelberg-New York: Springer 1967.

    Google Scholar 

  52. Serre, J.-P.: Classes de corps cyclotomiques. Séminie bourbaki no. 174 (1958). New York-Amsterdam: W. A. Benjamin, Inc. 1966.

    Google Scholar 

  53. [CG] Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Mathematics no. 5. Berlin-Heidelberg-New York: Springer 1964.

    Google Scholar 

  54. [CL] Serre, J.-P.: Corps locaux. Paris: Hermann 1962.

    Google Scholar 

  55. Serre, J.-P.: Sur les corps locaux à corps résidue algébriquement clos, 2. Bull. Soc. Math. France89, 105–154 (1961).

    Google Scholar 

  56. Serre, J.-P.: Corps locaux et isogénies. Séminaire Bourbaki, exposé 185, 1958–59.

  57. Serre, J.-P.: Groupes proalgébriques, I.H.E.S., Publication Mathematique no. 7 (1960).

  58. Serre, J.-P.: Groupes de Liel-adiques attachés aux courbes elliptiques. Colloque de Clermont-Ferrand (1964). Mimeographed notes published by I.H.E.S.

  59. [LG] Serre, J.-P.: Lie algebra and Lie groups. Lectures at Harvard University, 1964. New York-Amsterdam: W. A. Benjamin Inc. 1965.

    Google Scholar 

  60. Serre, J.-P.: Abelianl-adic representations and elliptics curves. Lectures at McGill University. New York-Amsterdam: W. A. Benjamin Inc. 1968.

    Google Scholar 

  61. Serre, J.-P.: Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Inventiones math15, 259–331 (1972).

    Google Scholar 

  62. Serre, J.-P.: Groupesp-divisibles. Séminaire Bourbaki, exp. 318 (1966–67). New York-Amsterdam: W. A. Benjamin Inc. 1966.

    Google Scholar 

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

    Google Scholar 

  64. Shimura, G.: Correspondances modulaires et les fonctions zêta de courbes algébriques. J. Math. Soc. Japan10, 1–28 (1958).

    Google Scholar 

  65. Shimura, G., Taniyama, Y.: Complex multiplication of abelian varietie and its applications to number theory. Publ. Math. Soc., Japan, no. 6, Tokyo 1961.

  66. Swinnerton-Dyer, P.: The conjectures of Birch and Swinnerton-Dyer and of Tate. Proceeding of a conference on Local Fields, NUFFIC Summer School helds at Driebergen in 1966, p. 132–157. Berlin-Heidelberg-New York: Springer 1967.

    Google Scholar 

  67. Tate, J.: Duality theorems in Galois cohomology over number fields. Proc. Intern. Congress Math., at Stockholm, 1962, 288–295. Institute Mittag-Leffler Djursholm, Sweden (1963).

    Google Scholar 

  68. Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki, exp. 306 (1966). New York-Amsterdam: W. A. Benjamin Inc. 1966.

    Google Scholar 

  69. Tate, J.:p-divisible groups. Proceedings of a Conference on Local Fields, NUFFIC Summer School held at Driebergen, p. 158–183 (1966). Berlin-Heidelberg-New York: Springer 1967.

    Google Scholar 

  70. Tate, J.: Classes d'isogénie des variétés abéliennes sur un corps fini, (d'après T. Honda). Séminaire Bourbaki, exp. 352 (1968–69).

  71. Tate, J.: Endomorphisms of abelian varieties over finite fields. Inventiones math.2, 134–144 (1966).

    Google Scholar 

  72. Weil, A.: Varietés abéliennes et courbes algébriques. Paris: Hermann 1948.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Most of the work for this paper was done at the Institut Des Hautes Etudes Scientifiques, whose generosity and hospitality I greatly appreciate. It was also partially supported by a grant from the National Science Foundation-GP-31359.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mazur, B. Rational points of abelian varieties with values in towers of number fields. Invent Math 18, 183–266 (1972). https://doi.org/10.1007/BF01389815

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389815

Keywords

Navigation