L-Functions and Tamagawa Numbers of Motives

  • Spencer Bloch
  • Kazuya Kato
Part of the Progress in Mathematics book series (MBC)


The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values “modulo Q* multiples” in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.


Exact Sequence Commutative Diagram Elliptic Curf Abelian Variety Riemann Zeta Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Be1]
    Beilinson, A., Higher regulators and values of L-functions, J. Soviet Math. 30 (1985), 2036–2070.CrossRefMATHGoogle Scholar
  2. [Be2]
    Beilinson, A., Height pairings between algebraic cycles, in Current Trends in Arithmetical Algebraic Geometry, Contemporary Math. 67 (1987), 1–24.MathSciNetCrossRefMATHGoogle Scholar
  3. [B11]
    Bloch, S., A note on height pamngs, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture, Inv. Math. 58 (1980), 65–76.MathSciNetCrossRefMATHGoogle Scholar
  4. [B12]
    Bloch, S., Algebraic K-theory and zeta Junctions and elliptic curves, Proc. Int. Congress Helsinki (1978).Google Scholar
  5. [B13]
    Bloch, S., Height pairings for algebraic cycles, J. Pure and Appl. Algebra 34 (1984).Google Scholar
  6. [B14]
    Bloch, S., Lectures on Algebraic Cycles, Duke Univ. Math Series, 1981.Google Scholar
  7. [Col]
    Coleman, R., Division values in local fields, Inv. Math. 53 (1979), 91–116.MathSciNetCrossRefMATHGoogle Scholar
  8. [Co2]
    Coleman, R., Local units modulo circular units, Proc. Amer. Math. Soc. 89 (1983), 1–7.MathSciNetCrossRefMATHGoogle Scholar
  9. [Del]
    Deligne, P., La conjecture de Weil I, Publ. Math. THES 43 (1981), 273–307.MathSciNetCrossRefMATHGoogle Scholar
  10. [De2]
    Deligne, P., Valeurs de functions L et periods d’intégrales, Proc. Symp. Pure Math. 33 (1979) AMS, 313–346.MathSciNetCrossRefGoogle Scholar
  11. [De3]
    Deligne, P., La conjecture de Weil II, Publ. Math. IHES 52 (1981).Google Scholar
  12. [De4]
    Deligne, P., in preparation.Google Scholar
  13. [De5]
    Deligne, P., Letter to C. Soulé.Google Scholar
  14. [DW]
    Deninger, C., On the Belinson conjecture for elliptic curves with complex multiplication in Belinson’s Conjectures on Special Values of L-functions, Perspectives in Math., Academic Press 1988.Google Scholar
  15. [Fa]
    Faltings, G., Crystalline cohomology and p-adic Galois representations, (preprint).Google Scholar
  16. [Fo]
    Fontaine, J.M., Sur certains types de représentations p-adiques du groupe de Galois d’un corps local: construction d’un anneau de Barsotti-Tate, Ann. of Math. 115 (1982), 529–577.MathSciNetCrossRefMATHGoogle Scholar
  17. [FL]
    Fontaine, J.M. and Laffaille, G., Construction de représentations p-adiques, Ann. Sci. ENS. 15 (1982), 547–608.MathSciNetMATHGoogle Scholar
  18. [FM]
    Fontaine, J.M. and Messing, W., P-adic periods and p-adic étale cohomology, in Current Trends in Arithmetical Algebraic Geometry, Cont. Math. 67 (1987) AMS, 179–207.MathSciNetCrossRefMATHGoogle Scholar
  19. [Gros]
    Gros, M., Régulateurs syntomiques, preprint (1987).Google Scholar
  20. [Grot]
    Grothendieck, A. Formule de Lefchetz et rationalité des fonctions L, Sém. Bourbaki, exposé 279 (1964).Google Scholar
  21. [Ih]
    Ihara, Y. Profinite braid groups, Galois representations, and complex multiplications, Ann. of Math. 123 (1986), 43–106.MathSciNetCrossRefMATHGoogle Scholar
  22. [Ja]
    Jannsen, U., On the l-adic cohomology of varieties over number fields and its Galois cohomology, preprint (1987).MATHGoogle Scholar
  23. [Ka]
    Kato, K., The explicit reciprocity law and the cohomology of Fontaine-Messing, to appear in Bull. Math. Soc. France.Google Scholar
  24. [MW]
    Mazur, B. and Wiles, A., Class fields of abelian extensions of Q, Inv. Math. 76 (1984), 179–330.MathSciNetCrossRefMATHGoogle Scholar
  25. [Onl]
    Ono, T., Arithmetic of algebraic tori, Ann. of Mth. 74 (1961), 101–139.MathSciNetCrossRefMATHGoogle Scholar
  26. [On2]
    Ono, T., On the Tamagawa number of algebraic tori, Ann. of Math. 78 (1963), 47–73.MathSciNetCrossRefMATHGoogle Scholar
  27. [Se]
    Serre, J.-R, Cohomologie Galoisienne, LMN No. 5, Springer Verlag, 1973.CrossRefMATHGoogle Scholar
  28. [dS]
    de Shalit, E., Iwasawa Theory of Elliptic Curves with Complex Multiplication, Academic Press, 1987.MATHGoogle Scholar
  29. [So1]
    Soulé, C., k-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Inv. Math. 55 (1979), 251–295.CrossRefMATHGoogle Scholar
  30. [So2]
    C. Soulé, “On higher p-adic regulators”, in Algebraic K-theory, Evanston, 1980, LNM 854, 371–401, Springer Verlag (1981).Google Scholar
  31. [So3]
    Soulé, C., The rank of étale cohomology of varieties over p-adic number fields, Comp. Math. 53 (1984), 113–131.MATHGoogle Scholar
  32. [So4]
    Soulé, C., Opérations en k-théorie algébrique, Canadian J. Math., Vol. XXXVII, (1985).Google Scholar
  33. [So5]
    Soulé, C., Eléments cyclotomiques en K-théorie, IHES preprint, 1985.Google Scholar
  34. [So6]
    Soulé, C., p-adic K-theory of elliptic curves, Duke Math. J. 54 (1987), 249–269.MathSciNetCrossRefMATHGoogle Scholar
  35. [Tal]
    Tate, J., Duality theorems in Galois cohomology over number fields, Proc. ICM Stockholm 1962, 288–295 Institute Mittag-Leffler (1963).Google Scholar
  36. [Ta2]
    Tate, J., p-divisible groups, in Proc. of a Conf. On Local Fields, Driebergen 1966, 153–183, Springer Verlag (1967).Google Scholar
  37. [Ta3]
    Tate, J. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Sem. Bourbaki exposé 306, 1965–66, in Dix Exposés sur la Cohomologie des Schémas, North Holland (1968).Google Scholar
  38. [Wa]
    Washington, L., Introduction to cyclotomic fields, Graduate Texts in Math., Vol. 83, Springer (1982).CrossRefMATHGoogle Scholar
  39. [We]
    Weil, A. Adeles and algebraic groups, Progress in Math. 23, Birkhauser, (1982).CrossRefMATHGoogle Scholar
  40. [Ya]
    Yager, R., A Kummer criterion for imaginary quadratic fields, Comp. Math. 47 (1982), 31–42.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2007

Authors and Affiliations

  • Spencer Bloch
    • 1
  • Kazuya Kato
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of TokyoHongo 113, TokyoJapan

Personalised recommendations