Skip to main content
SpringerLink
Log in

Spherical Functions on p-Adic Homogeneous Spaces

  • Conference paper
Number Theory

Part of the book series: Developments in Mathematics ((DEVM,volume 15))

Abstract

In §1, after defining spherical functions on homogeneous spaces, we examine the case of symmetric forms as an enlightening example. In §2, we introduce a general formula of spherical functions using functional equations under suitable assumption. In §3, we study a certain mechanism of functional equations of spherical functions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Casselman, The unramified principal series of p-adic groups I. The spherical functions, Compositio Math. 40 (1980), 387–406.

    MATH  MathSciNet  Google Scholar 

  2. W. Casselman and J. Shalika, The unramified principal series of p-adic groups II. The Whittaker function, Compositio Math. 41 (1980), 207–231.

    MathSciNet  MATH  Google Scholar 

  3. D. A. Cox, Primes of the forms x 2 + ny 2, Wiley Interscience, 1989.

    Google Scholar 

  4. Y. Z. Flicker, On distinguished representations, J. reine angew. Mathematik 418 (1991), 139–172.

    Article  MATH  MathSciNet  Google Scholar 

  5. Y. Hironaka, Spherical functions of hermitian and symmetric forms I, II, III, Japan. J. Math. 14 (1988), 203–223; Japan. J. Math. 15 (1989), 15–51; Tôhoku Math. J. 40 (1988), 651–671.

    MATH  MathSciNet  Google Scholar 

  6. Y. Hironaka, Spherical functions and local densities on hermitian forms, J. Math. Soc. Japan 51 (1999), 553–581.

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. Hironaka, Local zeta functions on hermitian forms and its application to local densities, J. Number Theory 71 (1998), 40–64.

    Article  MATH  MathSciNet  Google Scholar 

  8. Y. Hironaka, Spherical functions on Sp 2 as a spherical homogeneous space Sp 2 × (Sp 1)2-space, J. Number Theory 112 (2005), 238–286.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Hironaka, Functional equations of spherical functions on p-adic homogeneous spaces, to appear in Abh. Math. Sem. Univ. Hamburg.

    Google Scholar 

  10. Y. Hironaka and F. Sato, Spherical functions and local densities of alternating forms, Amer. J. Math. 110 (1988), 473–512.

    MathSciNet  MATH  Google Scholar 

  11. Y. Hironaka and F. Sato, Local densities of alternating forms, J. Number Theory 33 (1989), 32–52.

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Hironaka and F. Sato, Eisenstein series on reductive symmetric spaces and representations of Heche algebras, J. Reine Angew. Math. 445 (1993), 45–108.

    MathSciNet  MATH  Google Scholar 

  13. J. Igusa, An introduction to Theory of Local Zeta Functions, AMS/IP Studies in Advanced Mathematics, vol.14, 2000.

    Google Scholar 

  14. H. Jacquet, Automorphic spectrum of symmetrc spaces, Proc. Sym. Pure Math. 61 (1997), 431–455.

    Google Scholar 

  15. S. Kato, A. Murase and T. Sugano, Whittaker-Shintani functions for orthogonal groups, Tohoku Math. J. 55 (2003), 1–64.

    MathSciNet  MATH  Google Scholar 

  16. I. G. Macdonald, Spherical functions on a group of p-adic type, Univ. Madras, 1971.

    Google Scholar 

  17. O. Offen, Relative spherical functions on p-adic symmetric spaces, Pacific J. Math. 215 (2004), 97–149.

    Article  MATH  MathSciNet  Google Scholar 

  18. V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Academic Press, 1994.

    Google Scholar 

  19. F. Sato, Eisenstein series on semisimple symmetric spaces of Chevalley groups, Advanced Studies in Pure Math. 7 (1985), 295–332.

    MATH  Google Scholar 

  20. F. Sato, On functional equations of zeta distributions, Advanced Studies in Pure Math. 15 (1989), 465–508.

    MATH  Google Scholar 

  21. F. Sato and Y. Hironaka, Local densities of representations of quadratic forms over p-adic integers’ the non-dyadic case, J. Number Theory 83 (2000), 106–136.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Education and Integrated Sciences, Weseda University, Nishi-Waseda, Tokyo, 169-8050, Japan

    Yumiko Hironaka

Authors
  1. Yumiko Hironaka
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Northwest University, Xi’an, P.R. China

    Wenpeng Zhang

  2. Nagoya University, Nagoya, Japan

    Yoshio Tanigawa

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer Science + Business Media, Inc.

About this paper

Cite this paper

Hironaka, Y. (2006). Spherical Functions on p-Adic Homogeneous Spaces. In: Zhang, W., Tanigawa, Y. (eds) Number Theory. Developments in Mathematics, vol 15. Springer, Boston, MA. https://doi.org/10.1007/0-387-30829-6_7

Download citation

Publish with us

Policies and ethics

Search

Navigation