• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

1996 Fiscal Year Final Research Report Summary

Arithmetic research of automorphic functions

Research Project

Project/Area Number 06452003
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

SAITO Hiroshi  Kyoto Univ, Human and Environmental studo.P., 大学院・人間・環境学研究科, 教授 (20025464)

Co-Investigator(Kenkyū-buntansha) YOSHINO Yuji  Kyoto, Univ, Integrated Human Studies A.P., 総合人間学部, 助教授 (00135302)
MORIMOTO Yoshinori  Kyoto Univ, Human and Environmented studies A.P., 総合人間学部, 助教授 (30115646)
YAMAUTI Masatoshi  Kyoto Univ, Integrated Huyman studies P., 総合人間学部, 教授 (30022651)
GYOJA Akihiko  Kyoto Univ, Integrated Huyman Studies A.P., 総合人間学部, 助教授 (50116026)
KATO Shinichi  Kyoto Univ, Integrated Huyman Studies A.P., 総合人間学部, 助教授 (90114438)
Project Period (FY) 1994 – 1996
KeywordsSiegel modular form / prehomogeneous vector space / zeta function / dimension formula / speherical homogeneous space / sphrical function / Hecke ring / Gauss sum
Research Abstract

In this research project, we mainly seek the application of the theory of prehomogeneous spaces to the theory of Siegel modular forms. It has been known long before that such zeta functions and their special vulues play important roles in the theory of Siegel modular forms. But It was generally thought that these special values were difficult to determine in general. But in our research, it was shown that those zeta function, which are zeta functions associated to the prehomogeneous vector space of the spaces of symmetric matrices, can be expressed by Rimeann's zeta functions and Dirichlet series associated to Eisenstein series of half integral weight. By this results the special values of zeta functions mentioned above can be easily obtained and are shown to be expressed by Bernoulli numbers. From this we formulated a conjecture on the explicit dimension formula of Siegel modular forms. The method used in this calculation of zeta functions associated to the space of symmetric matrices can be applied to the calculation of zeta functions of the other types of prehomogeneous vector spaces. By this method, we can reduce the calculation of global zeta functions of prehomogeneous vector spaces to that of local orbital zeta functions, and we hope this method allow us to compute those global zeta functions in many cases. Gyoja's results on Gauss sums of prehomogeneous vector spaces over finite fields and Hecke ring are important not only in the thoery of representations of reductive groups over finite fields but also in that of Siegel modular form. Kato's results on spehrical homogeneous spaces and spherical functions have many applications in the analytic theory of zeta functions of automorphic forms, especially in their integral representation and analytic continuation.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] 斎藤裕: "On a Classification of Prehomogeneous Vector Spaces over docal and Global Fields" Journal of Algebra. 187. 510-536 (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 加藤信一: "Hecke algebras and quontum general linear groups" J. Math. Kyoto Univ.(未定). (1997)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 山内正敏: "Modular forms with coefficients involving class numbers and congruence of eigen values of Hecke operators" Hokkaido Mathematical Journal. 25. 149-165 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 吉野雄二: "On the higher delta invariants of a Gorenstein localring" Proceedings of the AMS. (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 吉野雄二: "The theory of L-Complexes and week lifting of complexes" Journal of Algebra. (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 宇敷重広: "Parbolic fixed points of two dimensional complex dynamical systems" Complex Dynamics and Related Problems. 959. 168-180 (1996)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hiroshi, Saito: "On a Classification of Prehomogeneous vector spaces over focal Global Ficdls" Journal of Algebla. No.187. 510-536 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shiniti, Kato: "Hecke algeblas and Quontum generallinear" Journal of Kyoto Univ.(to appear). (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Masatoshi, Yamauti: "Modular forms with coetticients involving class numbers and corgruence of eigen valves of Hecke Operators" Hokkaido Mathematical Journal. No.25. 149-165

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yuji, Yoshino: "On the higher delta invariants of a Gorenstein localring" Proceedings of the A.M.S.(to appear). (1997)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yuji, Yoshino: "The theory of L-complexes and week litting of complexes" Journal of Algebla. (1996)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Shigehiro, Ushiki: "Parabolic tixed points of two dimensional complex dynamical systems" Complex Dynamics and Related Ploblems. No.959. 168-180 (1996)

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 1999-03-09  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi