Mathematische Annalen

, Volume 354, Issue 2, pp 529–587 | Cite as

Zeta elements in the K-theory of Drinfeld modular varieties

  • Satoshi KondoEmail author
  • Seidai Yasuda


Beilinson (Contemp Math 55:1–34, 1986) constructs special elements in the second K-group of an elliptic modular curve, and shows that the image under the regulator map is related to the special values of the L-functions of elliptic modular forms. In this paper, we give an analogue of this result in the context of Drinfeld modular varieties.

Mathematics Subject Classification (2000)

Primary 11G09 Secondary 11F52 11F67 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramenko P., Brown K.S.: Buildings. Theory and applications. Graduate Texts in Mathematics, vol. 248. Springer, Berlin (2008)Google Scholar
  2. 2.
    Beilinson, A.A.: Higher regulators of modular curves. Applications of algebraic K-theory to algebraic geometry and number theory. Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I. Contemp. Math., vol. 55, pp. 1–34 (1986)Google Scholar
  3. 3.
    Blum, A., Stuhler, U.: Drinfeld modules and elliptic sheaves. In: Narasimhan, M.S. (ed.) Vector Bundles on Curves—New Directions. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME) held in Cetraro (Cosenza), Italy, June 19–27, 1995. Lect. Notes Math., vol. 1649, pp. 110–188. Springer, Berlin (1997)Google Scholar
  4. 4.
    Borel A.: Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35, 233–259 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bosch S., Guntzer U., Remmert R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften, 261. Springer Verlag, Berlin (1984)Google Scholar
  6. 6.
    Bruhat F., Tits J.: Groupes reductifs sur un corps local. Publ. Math., Inst. Hautes Étud. Sci. 41, 5–251 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Coates J., Wiles A.: On p-adic L-functions and elliptic units. J. Aust. Math. Soc. Ser. A 26, 1–25 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cogdell, J.W.: Lectures on L-functions, converse theorems, and functoriality for GL n. Lectures on Automorphic L-Functions, Fields Inst. Monogr., vol. 20, pp. 1–96 (2004)Google Scholar
  9. 9.
    Deligne, P., Husemöller, D.H.: Survey of Drinfel’d modules. Current trends in arithmetical algebraic geometry. Proc. Summer Res. Conf., Arcata/Calif. 1985. Contemp. Math., vol. 67, pp. 25–91 (1987)Google Scholar
  10. 10.
    Drinfel’d, V.G.: Elliptic modules. Math. USSR, Sb. 23, 561–592 (1974); translation from Mat. Sb., n. Ser. 94(136), 594–627 (1974)Google Scholar
  11. 11.
    Fresnel J., van der Put M.: Rigid analytic geometry and its applications. Progress in Mathematics, vol. 218. Birkhäuser, Boston (2004)Google Scholar
  12. 12.
    Garland H.: p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. Math. 97(2), 375–423 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Genestier, A.: Drinfeld’s symmetric spaces. (Espaces symétriques de Drinfeld.) Astérisque 234. Société Mathématique de France, Paris (1996)Google Scholar
  14. 14.
    Gillet H.: Riemann-Roch theorems for higher algebraic K-theory. Adv. Math. 40, 203–289 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Godement R., Jacquet H.: Zeta functions of simple algebras. Lecture Notes in Mathematics, vol. 260. Springer-Verlag, Berlin-Heidelberg-New York (1972)Google Scholar
  16. 16.
    Grigorov, G.T.: Kato’s Euler systems and the main conjecture. Thesis, Harvard University (2005)Google Scholar
  17. 17.
    Gross B., Rosen M.: Fourier series and the special values of L-functions. Adv. Math. 69(1), 1–31 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Grothendieck, A.: Éléments de géométrie algébrique. IV: Étude locale des schemas et des morphismes de schemas. (Seconde partie.). (French) Publ. Math., Inst. Hautes Étud. Sci. 24, 1–231 (1965)zbMATHGoogle Scholar
  19. 19.
    Harder G.: Chevalley groups over function fields and automorphic forms. Ann. Math. 100(2), 249–306 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Jacquet H., Piatetski-Shapiro I.I., Shalika J.: Conducteur des représentations du groupe linéaire. Math. Ann. 256, 199–214 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jacquet H., Shalika J.A.: On Euler products and the classification of automorphic representations. I. Am. J. Math. 103, 499–558 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Kato, K.: p-adic Hodge theory and values of zeta functions of modular forms. Berthelot, P., et al. (eds.) Cohomologies p-adiques et applications arithmétiques (III). Astérisque 295, pp. 117–290. Société Mathématique de France, Paris (2004)Google Scholar
  23. 23.
    Kondo S.: Kronecker limit formula for Drinfeld modules. J. Number Theory 104(2), 373– 377 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kondo, S., Yasuda, S.: Distributions and Euler systems for the general linear group. PreprintGoogle Scholar
  25. 25.
    Kondo, S., Yasuda, S.: Modular symbols for the Bruhat–Tits building of PGL over a nonarchimedean local field. PreprintGoogle Scholar
  26. 26.
    Kondo, S., Yasuda, S.: On the second rational K-group of an elliptic curve over global fields of positive characteristic. Proc. Lond. Math. Soc. 102(3), 1053–1098 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kondo, S., Yasuda, S.: Local L and epsilon factors in Hecke eigenvalues. Preprint, IPMU10-0107 (2010)Google Scholar
  28. 28.
    Laumon G.: Cohomology of Drinfeld modular varieties. Part 1: geometry, counting of points and local harmonic analysis Cambridge Studies in Advanced Mathematics, vol. 41. Cambridge University Press, Cambridge (1996)Google Scholar
  29. 29.
    Schneider P., Stuhler U.: The cohomology of p-adic symmetric spaces. Invent. Math. 105(1), 47–122 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Shalika J.A.: The multiplicity one theorem for GL n. Ann. Math. 100(2), 171–193 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Waldhausen F.: Algebraic K-theory of generalized free products, I. Ann. Math. (2) 108(1), 135–204 (1978)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute for the Physics and Mathematics of the UniverseUniversity of TokyoKashiwaJapan
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations