Advertisement

Mathematische Annalen

, Volume 328, Issue 1–2, pp 87–119 | Cite as

Complexes of exact hermitian cubes and the Zagier conjecture

  • Yuichiro TakedaEmail author
Article

Abstract

In this paper, we present a new way of constructing a map from the higher Bloch group to the higher rational K-theory for an algebraic number field. The composition of it with the regulator map is expressed in terms of the polylogarithm function. To do this, we employ exact hermitian cubes and their Bott-Chern forms.

Keywords

Number Field Algebraic Number Algebraic Number Field Polylogarithm Function Bloch Group 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beilinson, A., Deligne, P.: Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs. In: Motives, Proc. Sym. in Pure Math. 55, American Mathematical Society, 1994, pp. 97–121Google Scholar
  2. 2.
    Bloch, S.: Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph series 11, Am. Math. Soc., Providence, 2000Google Scholar
  3. 3.
    Bloch, S.: Algebraic cycles and the Lie algebra of mixed Tate motives. J. Am. Math. Soc. 4, 771–791 (1991)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Borel, A.: Cohomologie de SL n et valeurs de fonctions zêta. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4, 613–636 (1977)zbMATHGoogle Scholar
  5. 5.
    Burgos, J.I.: Arithmetic Chow rings and Deligne-Beilinson cohomology. J. Algebraic Geom. 6, 335–377 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Burgos, J.I.: Hermitian vector bundles and characterisic classes, The arithmetic and geometry of algebraic cycles, CRM Proceedings and Lecture Notes. Am. Math. Soc., Providence, 2000, pp. 155–182Google Scholar
  7. 7.
    Burgos, J.I.: The regulators of Beilinson and Borel. CRM Monograph series 15, Am. Math. Soc., Providence, 2001Google Scholar
  8. 8.
    Burgos, J.I., Wang, S.: Higher Bott-Chern forms and Beilinson’s regulator. Invent. Math. 132, 261–305 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    de Jeu, R.: Zagier’s conjecture and wedge complexes in algebraic K-theory. Compositio Math. 96, 197–247 (1995)zbMATHGoogle Scholar
  10. 10.
    Deligne, P.: La déterminant de la cohomologie. Comtemp. Math. 67, 93–177 (1987)zbMATHGoogle Scholar
  11. 11.
    Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with hermitian metric. Ann. Math. 131, 163–238 (1990)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Goncharov, A.: Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114, 197–318 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Goncharov, A.: Chow polylogarithms and regulators. Math. Res. Lett. 2, 95–112 (1995)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Goncharov, A.: Periods and mixed motives. Preprint math.AG/0202154Google Scholar
  15. 15.
    Goncharov, A.: Polylogarithms, regulators, and Arakelov motivic complexes. Preprint math.AG/0207036Google Scholar
  16. 16.
    Levin, A.: Note on ℝ-Hodge Tate sheaves. Preprint MPI-2001-37, available at ‘‘http://www.mpim-bonn.mpg.de/html/preprints/preprints.html’‘Google Scholar
  17. 17.
    McCarthy, R.: A chain complex for the spectrum homology of the algebraic K-theory of an exact category, Algebraic K-theory. Fields Inst. Commun. 16, Amer. Math. Soc., Providence, 1997, pp. 199–220Google Scholar
  18. 18.
    Milnor, J.: Introduction to algebraic K-theory. Ann. Math. Studies 72, Princeton University Press, 1971Google Scholar
  19. 19.
    Suslin, A.A.: K 3 of a field, and the Bloch group. Trudy Mat. Inst. Steklov 183, 180–199 (1990)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Takeda, Y.: Higher arithmetic K-theory. Preprint math.AG/0204321Google Scholar
  21. 21.
    Zagier, D.: Hyperbolic manifolds and special values of Dedekind zeta functions. Invent. Math. 83, 285–302 (1986)Google Scholar
  22. 22.
    Zagier, D.: Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic algebraic geometry. Progress Math. 89, Birkhäuser, 1991, pp. 391–430Google Scholar
  23. 23.
    Zagier, D., Gangl, H.: Classical and elliptic polylogarithms and special values of L-series. In: The arithmetic and geometry of algebraic cycles (Gordon B.B., et.al.(eds.), Kluwer Academic Publishers, 2000, pp. 561–615Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityJapan

Personalised recommendations