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Researches on zeta functions of prehomogeneous vector spaces

Research Project

Project/Area Number 20540028
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionRikkyo University

Principal Investigator

SATO Fumihiro  Rikkyo University, 理学部, 教授 (20120884)

Co-Investigator(Renkei-kenkyūsha) HIRONAKA Yumiko  早稲田大学, 教育総合科学学術院, 教授 (10153652)
KOGISO Takeyoshi  城西大学, 理学部, 准教授 (20282296)
TANIGUCHI Takashi  神戸大学, 理学研究科, 講師 (60422391)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2008: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords数論 / 概均質ベクトル空間 / 保型超関数 / ゼータ関数 / クリフォード代数 / 関数等式 / 対称空間
Research Abstract

Theory of prehomogeneous vector spaces gives a systematic method of constructing zeta functions from polynomial invariants of prehomogeneous group actions. We investigated zeta functions of prehomogeneous vector spaces from the following 3 view points : (1) relations to Eisenstein-periods, (2) relations to the Koecher-Maass zeta functions of automorphic forms, (3) zeta functions of invariants of non-prehomogeneous group-actions. The most important result is the construction of zeta functions of non-degenerate quadratic mappings, which include zeta functions of certain non-prehomogeneous polynomials of degree 4.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (20 results)

All 2011 2010 2009 2008

All Journal Article (6 results) (of which Peer Reviewed: 4 results) Presentation (14 results)

  • [Journal Article] Spherical functions on U(2n)/U(n)×U(n) and hermitian Siegel series, "Geometry and Analysis of Automorphic Forms of Several Variables"2011

    • Author(s)
      Yumiko Hironaka
    • Journal Title

      Series on Number Theory and Its Applications World Scientific. (発行予定)

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Spherical functions on p-adic homogeneous spaces, "Algebraic and Analytic Aspects of Zeta Functions and L-functions-Lectures at the French-Japanese Winter School (Miura, 2008)-"2010

    • Author(s)
      Yumiko Hironaka,
    • Journal Title

      MSJ Memoirs Vol.21

      Pages: 50-72

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] 概均質ベクトル空間の保型超関数と付随するL関数2010

    • Author(s)
      佐藤文広,田村敬太
    • Journal Title

      数理解析研究所講究録 No.1715

      Pages: 52-63

    • Related Report
      2010 Final Research Report
  • [Journal Article] Linear independence of local densities of quadratic forms and its application to the theory of Siegel modular forms2009

    • Author(s)
      Siegfried Bocherer, Yumiko Hironaka, Fumihiro Sato
    • Journal Title

      Contemporary Mathematics Vol.493

      Pages: 51-82

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Linear independence of local desities of quadratic forms and its application to the theory of Siegel modular forms2009

    • Author(s)
      Siegfried Bocherer
    • Journal Title

      Comtemporary Mathematics 493

      Pages: 51-82

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Representations of Clifford algebras and quartic polynomials with local functional equations2008

    • Author(s)
      Fumihiro Sato, Takeyoshi Kogiso
    • Journal Title

      数理解析研究所講究録 No.1617

      Pages: 51-62

    • Related Report
      2010 Final Research Report
  • [Presentation] L-functions of automorphic distributions and prehomogeneous vector spaces of commutative parabolic type2010

    • Author(s)
      佐藤文広
    • Organizer
      仙台整数論研究集会
    • Place of Presentation
      東北大学
    • Year and Date
      2010-10-09
    • Related Report
      2010 Final Research Report
  • [Presentation] L-functions of automorphic distributions and prehomogeneous vector spaces of commutative parabolic type2010

    • Author(s)
      佐藤文広
    • Organizer
      仙台整数論研究集会
    • Place of Presentation
      東北大学理学研究科
    • Year and Date
      2010-10-09
    • Related Report
      2010 Annual Research Report
  • [Presentation] 二次写像のゼータ関数2010

    • Author(s)
      佐藤文広
    • Organizer
      藤井昭雄教授退職記念ゼータ関数研究集会
    • Place of Presentation
      立教大学
    • Year and Date
      2010-03-09
    • Related Report
      2010 Final Research Report
  • [Presentation] 概均質ベクトル空間の保型超関数とゼータ関数2010

    • Author(s)
      佐藤文広、田村敬太
    • Organizer
      RIMS研究集会『保型形式・保型表現およびそれに伴うL関数と周期の研究
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2010-01-19
    • Related Report
      2010 Final Research Report
  • [Presentation] 概均質ベクトル空間の保型超関数とゼータ関数2010

    • Author(s)
      佐藤文広
    • Organizer
      RIMS研究集会「保型形式・保型表現およびそれに伴うL関数と周期の研究」
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2010-01-19
    • Related Report
      2009 Annual Research Report
  • [Presentation] 二次写像のゼータ関数2009

    • Author(s)
      佐藤文広
    • Organizer
      研究会『概均質ベクトル空間の最近の展開』
    • Place of Presentation
      九州大学数理学研究科
    • Year and Date
      2009-12-24
    • Related Report
      2010 Final Research Report
  • [Presentation] Clifford環から得られる二次写像を不変にするLie環の決定2009

    • Author(s)
      佐藤文広
    • Organizer
      研究会『概均質ベクトル空間の最近の展開』
    • Place of Presentation
      九州大学数理学研究科
    • Year and Date
      2009-12-23
    • Related Report
      2010 Final Research Report
  • [Presentation] Clifford代数の表現から得られる局所関数等式を満たす多項式について2009

    • Author(s)
      佐藤文広、小木曽岳義
    • Organizer
      日本数学会秋季総合分科会(関数解析分科会)
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-27
    • Related Report
      2010 Final Research Report
  • [Presentation] Clifford 代数の表現から得られる局所関数等式を満たす多項式について2009

    • Author(s)
      佐藤文広
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      大阪大学
    • Year and Date
      2009-09-27
    • Related Report
      2009 Annual Research Report
  • [Presentation] Representations of Clifford Algebras and local functional equations of non-prehomogeneous type2009

    • Author(s)
      佐藤文広
    • Organizer
      織田孝幸教授還暦記念国際研究集会"Geometry and Analysis of Automorphic Forms of Several Variables"
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2009-09-15
    • Related Report
      2010 Final Research Report
  • [Presentation] Representations of Clifford algebras and local functional equations of non-prehomogeneous type2009

    • Author(s)
      佐藤文広
    • Organizer
      織田孝幸教授還暦記念国際研究集会
    • Place of Presentation
      東京大学数理科学研究科
    • Year and Date
      2009-09-15
    • Related Report
      2009 Annual Research Report
  • [Presentation] Representations of Clifford algebras and local functional equations2009

    • Author(s)
      佐藤文広、小木曽岳義
    • Organizer
      研究集会『群の表現と非可換調和解析の新展開』
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-06-02
    • Related Report
      2010 Final Research Report
  • [Presentation] Representations of Clifford algebras and local functional equations2009

    • Author(s)
      佐藤文広
    • Organizer
      RIMS研究集会「群の表現と非可換調和解析の新展開」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-06-02
    • Related Report
      2009 Annual Research Report
  • [Presentation] Representations of Clifford algebras and quartic polynomials with local functional equations, JSPS-RFBR Workshop "Harmonic analysis on homogeneous spaces and quantization"2008

    • Author(s)
      佐藤文広、小木曽岳義
    • Organizer
      Tambara Institute of Mathematical Sciences
    • Place of Presentation
      The University of Tokyo
    • Year and Date
      2008-08-25
    • Related Report
      2010 Final Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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