On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology

  • Uwe Jannsen
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 16)

Abstract

If X is a smooth, projective variety over a number field k, then the absolute Galois group Gk = Gal(/k) acts on the étale cohomology groups Hi(, ℚ/ℤ(n)), where = X Xk for an algebraic closure of k. In this paper I study some properties of these Gk-modules; in particular, I am interested in the corank of the Galois cohomology groups
$${H^v}\,({G_k},{H^i}(\bar X,\,{Q_\ell }/{Z_\ell }(n))).$$

Keywords

Filtration Tate Kato Dian E211 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bel]
    A. Beilinson, Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036–2070.MATHCrossRefGoogle Scholar
  2. [Be2]
    A. Beilinson, Height pairing between algebraic cycles, in “Current Trends in Arithmetical Algebraic Geometry,” Comtemporary Mathematics Vol. 67, Amer. Math. Soc., 1987, pp. 1–24.CrossRefGoogle Scholar
  3. [BO]
    P. Berthelot and A. Ogus, F-isocrystals and DeRham cohomology I, Invent. Math. 72 (1987), 15–199.MathSciNetGoogle Scholar
  4. [Bl]
    S. Bloch, Algebraic cycles and values of L-functions II, Duke Math. J. 52 (1985), 37+–397.MathSciNetCrossRefGoogle Scholar
  5. [Br]
    A. Brumer, Galois groups of extensions of number fields with given ramification, Mich. Math. J. 13 (1966), 33–40.MathSciNetMATHCrossRefGoogle Scholar
  6. [C]
    H. Carayol, Sur les représentations L-adiques associées aux formes modulaires de Hilbert, Ann. Sei. École Norm. Sup. (4) 19 (1986), 40+–468.MathSciNetGoogle Scholar
  7. [ChE]
    C. Chevalley and S. Eilenberg, Colomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.MathSciNetMATHCrossRefGoogle Scholar
  8. [Dl]
    P. Deligne, Théorie de Hodge I, in “Actes ICM Nice,” t. I, Gauthier-Villars, 1970, pp. 425–430.Google Scholar
  9. [D2]
    P. Deligne, Formes modulaires et représentations l-adiques, Sém. Bourbaki, exp. 355. Lecture Notes in Math. 179, Springer-Verlag, 1971.Google Scholar
  10. [D3]
    P. Delinge, Valeurs de fonctions L et périodes d’intégrales, Proc. Symp. Pure Math. 33 (1979), 313–346.Google Scholar
  11. [D4]
    P. Deligne, La conjecture de Weil II, Publ. Math. IHES 52 (1981).Google Scholar
  12. [D5]
    P. Deligne, letter to Soulé, January 20, 1985.Google Scholar
  13. [DF]
    W. Dwyer and E. Friedlander, Étale K-theory and arithmetic, Trans. Amer. Math. Soc. 292 (1985), 247–280.MathSciNetMATHGoogle Scholar
  14. [Fal]
    G. Faltings, p-adic Hodge theory, Jour. AMS 1 (1988).Google Scholar
  15. [Fa2]
    G. Faltings, Lecture at Wuppertal, 1987.Google Scholar
  16. [Fol]
    J. M. Fontaine, Modules Galoisiens, modules filtrés, et anneaux de Barsotti-Tate, Astérisque 65 (1979), 3–80.MathSciNetMATHGoogle Scholar
  17. [Fo2]
    J. M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local: constructions d’un anneau de Barsotti-Tate, Ann. Math. 115 (1982), 52–577.MathSciNetCrossRefGoogle Scholar
  18. [FL]
    J. M. Fontaine and G. LafFaille, Construction de représentations p-adiques, Ann. Sei. École Norm. Sup. (4) 15 (1982), 547–608.MathSciNetMATHGoogle Scholar
  19. [FM]
    J. M. Fontaine and W. Messing, p-adic periods and p-adic étale cohomology, in “Current Trends in Arithmetical Algebraic Geometry,” Comtemporary Mathematics Vol. 67, Amer. Math. Soc., 1987, pp. 17–207.Google Scholar
  20. [Gr]
    A. Grothendieck, Groupes de Barsotti-Tate et cristaux, in “Actes ICM Nice,” t. I, Gauthier-Villars, 1970, pp. 431–436.Google Scholar
  21. [Jl]
    U. Jannsen, Continuous étale cohomology, Math. Ann. 280 (1988), 207–245.MathSciNetMATHCrossRefGoogle Scholar
  22. [J2]
    U. Jannsen, Deligne homology, Hodge-D-conjecture, and motives, in “Beilinson’s Conjectures on Special Values of Z-functions,” Perspectives in Mathematics, vol. 4, Academic Press, 1988, pp. 305–372.Google Scholar
  23. J3] U. Jannsen, “Mixed Motives and Algebraic K-Theory,” in préparation.Google Scholar
  24. [KM]
    N. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.MathSciNetMATHCrossRefGoogle Scholar
  25. [L]
    M. Lazard, Groupes Analytiques p-Adiques, Publ. Math. IHES 26 (1965).Google Scholar
  26. [Mal]
    B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.MathSciNetMATHCrossRefGoogle Scholar
  27. [Ma2]
    B. Mazur, Modular curves and elliptic curves, Lecture at Harvard Univ., Spring 1984.Google Scholar
  28. [MTT]
    B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.MathSciNetMATHCrossRefGoogle Scholar
  29. [Mil]
    J. S. Milne, “Etale Cohomology,” Princeton Univ. Press, 1980.Google Scholar
  30. [Mi2]
    J. S. Milne, “Arithmetic Duality Theorems,” Perspectives in Mathematics, Vol. 1, Academic Press, Boston, 1986.Google Scholar
  31. [Neu]
    O. Neumann, On p-closed algebraic number fields with restricted ramification, Math. USSR Izv. 9 (1975), 243–254.MATHCrossRefGoogle Scholar
  32. [Qui]
    D. Quillen, Higher algebraic K-theory, in “Proc. ICM Vancouver 1974,” Canadian Math. Congress, 1975.Google Scholar
  33. [RZ]
    M. Rapoport and Th. Zink, Uber die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math. 68 (1982), 21–101.MathSciNetMATHCrossRefGoogle Scholar
  34. [Sehl]
    P. Schneider, Uber gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181–205.MathSciNetMATHCrossRefGoogle Scholar
  35. [Sch2]
    P. Schneider, Iwasawa L-functions of varieties over algebraic number fields: A first approach, Invent. Math. 71 (1983), 251–293.MathSciNetMATHCrossRefGoogle Scholar
  36. [Sch3]
    P. Schneider, p-adic height pairings II, Invent. Math. 79 (1985), 32–374.CrossRefGoogle Scholar
  37. [Sel]
    J.-P. Serre, “Cohomologie Galoisienne,” Lecture Notes in Math. 5, Springer-Verlag, 1973.Google Scholar
  38. [Se2]
    J.-P. Serre, Sur les groupes de congruence de variétés abéliennes I, Izv. Akad. Nauk SSSR, Sér. Mat. 28 (1964), 3–20.MATHGoogle Scholar
  39. [Se3]
    J.-P. Serre, Sur les groupes de congruences des variétés abéliennes II, Izv. Akad. Nauk SSSR, Sér. Mat. 35 (1971), 731–737.MATHGoogle Scholar
  40. [Se4]
    J.-P. Serre, “Œuvres — Collected Papers,” Springer-Verlag, Berlin, 1986.Google Scholar
  41. [Soul]
    C. Soulé, K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), 251–295.MathSciNetMATHCrossRefGoogle Scholar
  42. [Sou2]
    C. Soulé, On higher p-adic regulators, in “Algebraic K-Theory, Evanston 1980,” Lecture Notes in Math. 854, Springer-Verlag, Berlin, 1981, pp. 372–401.CrossRefGoogle Scholar
  43. [Sou3]
    C. Soulé, Operations on étale K-theory, in “Algebraic K-Theory, Oberwolfach 1980,” Lecture Notes in Math. 966, Springer-Verlag, Berlin, 1982, pp. 271–303.Google Scholar
  44. [Sou4]
    C. Soulé, The rank of étale cohomology of varieties over p-adic or number fields, Comp. Math. 53 (1984), 113–131.MATHGoogle Scholar
  45. [Sou5]
    C. Soulé, p-adic K-theory of elliptic curves, Duke Math. J. 54 (1987), 24–269.CrossRefGoogle Scholar
  46. [Tal]
    J. Tate, Duality theorems in Galois cohomology over number fields, in “Proc. ICM, Stockholm 1962, ” Institut Mittag-Leffier, 1963, pp. 288–295.Google Scholar
  47. [Ta2]
    J. Tate, p-divisible groups, in “Proceedings of a Conference on Local Fields, Driebergen 1966, ” Springer-Verlag, 1967, pp. 153–183.Google Scholar
  48. [Ta3]
    J. Tate, Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.MathSciNetMATHCrossRefGoogle Scholar
  49. [Th]
    R. Thomason, Bott stability in algebraic K-theory, in “Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Part I,” Contemporary Math., Vol. 55, Amer. Math. Soc., 1986, pp. 38–406.Google Scholar
  50. Wak] R. Wake, Phantom points on abelian varieties, preprint.Google Scholar
  51. [Wal]
    L. Washington, “Introduction to Cyclotomic Fields,” Graduate Texts in Math., Vol. 83, Springer-Verlag, 1982.Google Scholar
  52. [Wa2]
    L. Washington, Zeroes of p-adic L-functions, in “Sém. de Théorie des Nombres, Paris 1980–81, ” Birkhäuser, 1982, pp. 337–357.Google Scholar
  53. [Wil]
    K. Wingberg, Duality theorems for T-extensions of algebraic number fields, Comp. Math. 55 (1985), 333–381.MathSciNetMATHGoogle Scholar
  54. [Wi2]
    K. Wingberg, On the étale K-theory of an elliptic curve with complex multi-plication for regular primes, preprint.Google Scholar
  55. [SGA 7 1]
    A. Grothendieck, et al., “Groupes de Monodromie en Géométrie Algébrique,” Lecture Notes in Math. 288, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1989

Authors and Affiliations

  • Uwe Jannsen
    • 1
  1. 1.FB MathematikUniversität RegensburgRegensburgFederal Republic of Germany

Personalised recommendations