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2004 Fiscal Year Final Research Report Summary

Study on automorphic forms on algebraic groups and associated zeta functions

Research Project

Project/Area Number 13440016
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKyoto Sangyo University

Principal Investigator

MURASE Atsushi  Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (40157772)

Co-Investigator(Kenkyū-buntansha) SUGANO Takashi  Kanazawa University, Faculty of Science, Professor, 理学部, 教授 (30183841)
ITO Masami  Kyoto Sangyo University, Faculty of Science, Professor, 理学部, 教授 (50065843)
NARITA Hiroaki  Kyoto Sangyo University, Faculty of Science, Guest researcher (Japan Society for the Promotion of Science, Postdoctoral Fellows (PD)), 理学部, 客員研究員(日本学術振興会特別研究員PD)
HIRANO Miki  Ehime University, Faculty of Science, Associated Professor, 理学部, 助教授 (80314946)
OHNO Yasuo  Kinki University, Faculty of Science, Lecturer, 理工学部, 講師 (70330230)
Project Period (FY) 2001 – 2004
KeywordsKudla lift / Automorphic forms on unitary groups / Siegel Weil formula / Periods of automorphic forms / Automorphic L-functions
Research Abstract

1. Metaplectic representations of unitary groups :
We studied metaplectic representations of unitary groups over local fields and gave their "universal" splitting, which are in particular useful in the study of theta lifting. As an application of this result, we gave an explicit character formula for metaplectic representations.
2. Fourier-Jacobi expansion of automorphic form on unitary groups of degree three :
We reformulated Shintani' s theory on Fourier-Jacobi expansion of automorphic forms on unitary groups of degree three in adelic language, and calculated explicit form for Fourier-Jacobi expansion of Eisenstein series and Kudla lifts, theta lifts from elliptic modular forms. As an application, we gave a criterion for the non-vanishing of Kudla lifts.
3. Siegel-Weil formula :
We studied a non-regularized Siegel-Weil formula in the case of the dual reductive pair (U(2,2), U(2, 1)).
4. Inner product formula for Kudla lifts :
Using the formula stated in 3, we gave an explicit formula for the Petersson norms of Kudla lifts in term of special values of automorphic L-functions. As an application, we gave a criterion for the non-vanishing of Kudla lifts different from the one given in 2. (The studies 2- 4 are joint works with Takashi Sugano).
5. Support for the Summer School of Number Theory :
We supported financially the Summer School of Number Theory held annually.
The themes were as follows : "Zeta functions" in 2001, "Prehomogeneous vector spaces" in 2002, "Iwasawa theory" in 2003 and "Fundamental groups and Galois representations" in 2004.

  • Research Products

    (6 results)

All 2004 2003 2002 2001

All Journal Article (6 results)

  • [Journal Article] n-Insetion on languages2004

    • Author(s)
      M.Itoh
    • Journal Title

      Aspects of Molecular Computingh, Lecture Notes in Computer Science 2950

      Pages: 213-218

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fourier expansion of holomorphic modular forms on classical Lie groups of tube type along the minimal parabolic subgroup2004

    • Author(s)
      H.Narita
    • Journal Title

      Abh.Math.Sem.Univ.Hamburg 74

      Pages: 253-279

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Whittaker-Shintani functions for orthogonal groups2003

    • Author(s)
      S.Kato
    • Journal Title

      Tohoku Mathematical Journal 55

      Pages: 1-64

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fourier-Jacobi expansion of Eisenstein series on unitary groups of degree three2002

    • Author(s)
      A.Murase
    • Journal Title

      J.Math.Sci.Univ.Tokyo 9

      Pages: 347-404

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On metaplectic representations of unitary groups : I. Splitting2001

    • Author(s)
      A.Murase
    • Journal Title

      Proceeding of the Japan Academy 77A

      Pages: 59-62

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On metaplectic representations of unitary groups : II. Character formula2001

    • Author(s)
      A.Murase
    • Journal Title

      Proceeding of the Japan Academy 77A

      Pages: 63-65

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

URL: 

Published: 2006-07-11  

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