Skip to main content
SpringerLink
Log in

The normalization constant of a certain invariant measure on GL n ()

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract.

The ratio of the Tamagawa measure and a certain invariant measure on the group GL n () is computed, where is the adèle of a division algebra D over a global field. An explicit formula of the ratio is described in terms of the special values of the zeta function of D. This formula yields (i) an explicit lower bound of the Hermite–Rankin constant γ n,m (D) of D and (ii) an explicit asymptotic behavior of the distribution of rational points on Brauer–Severi variety.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Borel, A., Tits, J.: Groupes réductifs. Publ. Math. I.H.E.S. 27, 55–150 (1965)

    Google Scholar 

  2. Coulangeon, R., Watanabe, T.: Hermite constant and Voronoï theory over a quaternion skew field. Preprint, 2003

  3. Ichino, A.: A regularized Siegel-Weil formula for unitary groups. Math. Z. To appear

  4. Ikeda, T.: On the residue of the Eisenstein series and the Siegel-Weil formula. Comp. Math. 103, 183–218 (1996)

    MATH  Google Scholar 

  5. Knapp, A.: Representation Theory of Semisimple Groups. Princeton Univ. Press, Princeton, 1986

  6. Lai, K.F.: Tamagawa numbers of reductive algebraic groups. Comp. Math. 41, 153–188 (1980)

    MATH  Google Scholar 

  7. Oesterlé, J.: Nombres de Tamagawa et groupes unipotents en caractéristique p. Invent. Math. 78, 13–88 (1984)

    MathSciNet  Google Scholar 

  8. Reiner, I.: Maximal Orders. Academic Press, London et al. 1975

  9. Turner, S.: Zeta-functions of central simple algebras over global fields. An. Acad. Brasil Ciênc. 48, 171–186 (1976)

    MATH  Google Scholar 

  10. Vignéras, M.-F.: Arithmétique des algébres de quaternions. Lecture Note in Mathematics, no. 800, Springer-Verlag, Heidelberg, 1980

  11. Watanabe, T.: Fundamental Hermite constants of linear algebraic groups. J. Math. Soc. Japan 55, 1061–1080 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Watanabe, T.: The Hardy-Littlewood property of flag varieties. Nagoya Math. J. 170, 185–211 (2003)

    MATH  Google Scholar 

  13. Weil, A.: Basic Number Theory. Springer-Verlag, Berlin et al. 1974

  14. Weil, A.: Adeles and Algebraic Groups. Progr. Math. 23, Birkhäuser, 1982

Download references

Acknowledgments.

The second author was partly supported by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Science and Technology, Japan.

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Yoshihide Nakamura & Takao Watanabe

Authors
  1. Yoshihide Nakamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takao Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshihide Nakamura.

Additional information

Mathematics Subject Classification (2000): Primary 11R52, Secondary 11H50

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nakamura, Y., Watanabe, T. The normalization constant of a certain invariant measure on GL n (). manuscripta math. 115, 259–280 (2004). https://doi.org/10.1007/s00229-004-0503-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-004-0503-8

Keywords

Search

Navigation