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Green functions and relative trace formulas

Research Project

Project/Area Number 18540049
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

TSUZUKI Masao (MASAO Tsuzuki)  Sophia University, 理工学部, 准教授 (80296946)

Co-Investigator(Kenkyū-buntansha) GOMI Yasushi  上智大学, 理工学部, 講師 (50276515)
森山 知則  大阪大学, 理学研究科, 准教授 (80384171)
Co-Investigator(Renkei-kenkyūsha) MORIYAMA Tomonori  大阪大学, 理学研究科, 准教授 (80384171)
Project Period (FY) 2006 – 2009
Project Status Completed (Fiscal Year 2009)
Budget Amount *help
¥4,020,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥720,000)
Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Keywordsグリーン函数 / 相対跡公式 / 一般超幾何級数 / エル関数 / L函数 / 一般超幾何関数 / 指数和 / グリーン関数 / L関数 / ヘッケ環 / L-関数
Research Abstract

We develop a version of relative trace formula for symmetric spaces associated with indefinite unitary groups of arbitrary real rank and for orthogonal groups of real rank one. As an application, we prove two kinds of asymptotic formulas ; (1) It is proved that the Plancherel formula of the symmetric space for unitary group is approximated by certain sequence of discrete measures arising from the average of square norm of period integrals of automorphic forms on an arithmetic quotient of the unitary group. (2) We showed an asymptotic formula for certain average of period integrals of L2-normalized Laplace eigenfunctions on an arithmetic quotient of rank one orthogonal group ; the formula should be regarded as an analogue of Weyl's law for spectral counting function for multiplicity of Laplace eigenvalues.

Report

(6 results)
  • 2009 Annual Research Report   Final Research Report ( PDF )
  • 2008 Annual Research Report   Self-evaluation Report ( PDF )
  • 2007 Annual Research Report
  • 2006 Annual Research Report
  • Research Products

    (30 results)

All 2009 2008 2007 2006

All Journal Article (16 results) (of which Peer Reviewed: 8 results) Presentation (14 results)

  • [Journal Article] Spectral square means for period integrals of wave functions on real hyperbolic spaces2009

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      J. Number Theory 129

      Pages: 2287-2438

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Certain Rankin-Selberg integrals for Unitary Groups2009

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      Tohoku Math. J. 61

      Pages: 115-164

    • NAID

      110007031057

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] L-functions for $GSp(2)\times GL(2)$: archimedean theory and applications2009

    • Author(s)
      Tomonori Moriyama
    • Journal Title

      Canad. J. Math. 61

      Pages: 395-426

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Certain Rankin-Selberg integrals for unitary groups2009

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      Tohoku Math.J 61

      Pages: 115-164

    • NAID

      110007031057

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Spectral square means for period integrals of wave functions on real hyperbolic spaces2009

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      J.Number Theory 129

      Pages: 2387-2438

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The secondary spherical functions and automorphic Green currents for certain symmetric pairs2009

    • Author(s)
      T.Oda, M.Tsuzuki
    • Journal Title

      Pure Appl. Math. Q. Vol.5

      Pages: 977-1028

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Certain Rankin-Selberg integrals for unitary groups2009

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Tohoku Math. J. vol.61

      Pages: 115-164

    • NAID

      110007031057

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Limit formulas of period integrals for a certain symmetric pair2008

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      J. Funct. Ann. 255(5)

      Pages: 1139-1190

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Limit formulas of period integrals for a certain symmetric pair2008

    • Author(s)
      M.Tsuzuki
    • Journal Title

      J. Funct. Anal. Vol.255

      Pages: 1139-1190

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] The secondary spherical functions and Green currents associated with certain symmetric pairs2008

    • Author(s)
      T. Oda, M. Tsuzuki
    • Journal Title

      RIMS Kokyuroku Bessatsu B7

      Pages: 121-135

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Spectral square means for period integrals of wave functions2008

    • Author(s)
      M.Tsuzuki
    • Journal Title

      RIMS 研究集会「保型表現、保型形式とL関数の周辺」講究録 1617

      Pages: 11-17

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Limit formulas of period integrals for a certain symmet ric pair2008

    • Author(s)
      Masao Tsuzuki
    • Journal Title

      Journal of functional analysis 255

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] The Markov traces and the Fourier transforms2006

    • Author(s)
      Yasushi Gomi
    • Journal Title

      J. Algebra 303(2)

      Pages: 566-591

    • Related Report
      2009 Final Research Report
    • Peer Reviewed
  • [Journal Article] Green currents for modular cycles in arithmetic quotients of complex hyper balls2006

    • Author(s)
      M.Tsuzuki
    • Journal Title

      Pacific J. Math. Vol.227 No.2

      Pages: 311-359

    • Related Report
      2008 Self-evaluation Report
  • [Journal Article] Green currents for modular cycles in arithmetic quotients of complex hyperballs2006

    • Author(s)
      Tsuzuki, Masao
    • Journal Title

      Pacific J. Math. 227

      Pages: 311-359

    • Related Report
      2006 Annual Research Report
  • [Journal Article] The Markov traces and the Fourier transforms2006

    • Author(s)
      Gomi, Yasushi
    • Journal Title

      J. Algebra 303, no.2

      Pages: 566-591

    • Related Report
      2006 Annual Research Report
  • [Presentation] Spectral square means for central values of L-functions on SL_2(Z)2009

    • Author(s)
      Masao Tsuzuki
    • Organizer
      International conference "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
      2009-09-17
    • Related Report
      2009 Final Research Report
  • [Presentation] Limit period formulas for special cycles on real hyperbolic spaces2009

    • Author(s)
      Masao Tsuzuki
    • Place of Presentation
      数理解析研究所
    • Year and Date
      2009-01-20
    • Related Report
      2009 Final Research Report
  • [Presentation] Limit formulas for special cycles on real hyperbolic spaces2009

    • Author(s)
      都築正男
    • Organizer
      RIMS 研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2009-01-20
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Limit formulas for spacial cycles on real hyperbolic spaces2009

    • Author(s)
      都築 正男
    • Organizer
      RIMS研究集会「保型表現と保型L関数の数論的研究」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2009-01-20
    • Related Report
      2008 Annual Research Report
  • [Presentation] Spectral square means of period integ rals for wave forms on real hyp erbolic spaces2008

    • Author(s)
      Masao Tsuzuki
    • Organizer
      Summer school on spe ctral theory of automorphic forms
    • Place of Presentation
      at Postech Pohang
    • Year and Date
      2008-08-13
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] SPECTRAL SQUARE MEANS FOR PERIOD INTEGRALS OF WAVE FUNCTIONS2008

    • Author(s)
      Masao Tsuzuki
    • Place of Presentation
      数理解析研究所
    • Year and Date
      2008-01-21
    • Related Report
      2009 Final Research Report
  • [Presentation] Spectral square means for period integrals of wave functions2008

    • Author(s)
      都築正男
    • Organizer
      RIMS 研究集会「保型表現、保型形式とL関数の周辺」
    • Place of Presentation
      京都大学
    • Year and Date
      2008-01-21
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Spectral square means for period integrals of wave functions on real hyperbolic spaces2008

    • Author(s)
      Masao Tsuzuki
    • Organizer
      Summer school on spectral theory of automorphic forms
    • Place of Presentation
      at Pstech, Pohan.
    • Related Report
      2009 Final Research Report
  • [Presentation] Spectal zeta functions and square mean of periods of wave forms2007

    • Author(s)
      都築 正男
    • Organizer
      白馬保型形式オータムワークショップ
    • Place of Presentation
      白馬ハイマウントホテル
    • Year and Date
      2007-11-28
    • Related Report
      2007 Annual Research Report
  • [Presentation] Spectral zeta functions and spectral square means of periods of wave functions2007

    • Author(s)
      Masao Tsuzuki
    • Organizer
      The 10-th autum workshop on number theory
    • Place of Presentation
      Hakuba
    • Year and Date
      2007-11-24
    • Related Report
      2009 Final Research Report
  • [Presentation] SL_2(Z)上のMaass波動形式の閉測地線に沿った周期積分に付随するスペクトル型ゼータ関数について2007

    • Author(s)
      都築正男
    • Organizer
      福岡整数論研究集会
    • Place of Presentation
      九州大学
    • Year and Date
      2007-08-29
    • Related Report
      2009 Final Research Report
  • [Presentation] SL_2(Z)上のマース形式の閉測地線に沿った周期積分に付随するスペクトラル型ゼータ関数について2007

    • Author(s)
      都築正男
    • Organizer
      福岡数論研究集会
    • Place of Presentation
      九州大学
    • Year and Date
      2007-08-29
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] 対称空間に付随した算術商の上の保型グリーンカレントと保型熱核2007

    • Author(s)
      都築正男
    • Organizer
      2007 年度日本数学会代数学分科会特別講演
    • Place of Presentation
      埼玉大学
    • Year and Date
      2007-03-28
    • Related Report
      2008 Self-evaluation Report
  • [Presentation] Fourier coefficient of the Poincare dual form of modular divisor on U(n, 1)2006

    • Author(s)
      Masao Tsuzuki
    • Organizer
      (at the Erwin Schr\"{o}dinger International Institute for Mathematical Physics
    • Related Report
      2008 Self-evaluation Report

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Published: 2006-04-01   Modified: 2016-04-21  

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