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On arithmetic theory of automorphic forms and special values of automorphic L-functions

Research Project

Project/Area Number 10640028
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionOsaka City University (1999)
Hiroshima University (1998)

Principal Investigator

FURUSAWA Masaaki  Osaka City University, Faculty of Science Professor, 理学部, 教授 (50294525)

Co-Investigator(Kenkyū-buntansha) KOMORI Youhei  Osaka City University, Faculty of Science Lecturer, 理学部, 講師 (70264794)
IMAYOSHI Yoichi  Osaka City University, Faculty of Science Professor, 理学部, 教授 (30091656)
KAMAE Tetsuro  Osaka City University, Faculty of Science Professor, 理学部, 教授 (80047258)
MATSUMOTO Keiji  Hokkaido University, Graduate School of Science Assistant Professor, 大学院・理学研究科, 助教授 (30229546)
MOCHIZUKI Takuro  Osaka City University, Faculty of Science Assistant, 理学部, 助手 (10315971)
都築 暢夫  広島大学, 理学部, 助手 (10253048)
木村 俊一  広島大学, 理学部, 講師 (10284150)
谷崎 俊之  広島大学, 理学部, 教授 (70142916)
隅広 秀康  広島大学, 理学部, 教授 (60068129)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,800,000 (Direct Cost: ¥3,800,000)
Fiscal Year 1999: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 1998: ¥2,500,000 (Direct Cost: ¥2,500,000)
KeywordsSiegel modular form / automorphic L-function / special value of L-function / Deligne's conjecture / relative trace formula / trace formula / ジーゲル保型形式 / 保型形式の持ち上げ / クルスターマン和
Research Abstract

We proved the fundamental lemma for the unit element in the Hecke algebra for two relative trace formulas for GSp(4). Our ultimate goal is to prove Bocherer's conjecture on the central critical values of the quadratic twists of the spinor L-functions associated to holomorphic Siegel eigen cusp forms of degree two. The announcements of the fundamental lemma have been published in C. R. Acad. Sci. Paris and the details of the proof will appear elsewhere.
In the course of the proof of the fundamental lemma, we evaluated certain matrix argument Kloosterman sums explicitly in terms of the classical GL(2) Kloosterman sums. We remark that our Kloosterman sum is a special case of the generalized Kloosterman sum which appears in the Fourier coefficients of the Poincare series for the Siegel modular group. Our result on the Kloosterman sum may be of some independent interest, since it is rare that such generalized Kloosterman is evaluated explicitly.
Our second conjectural trace formula is related to the quadratic base charge for GSp (4). Our result suggests that the Jacquet-Ye criterion for the quadratic base change for GL(2) generalizes to GSp(4). This clearly deserves some further investigation. Finally our result implies that it is important to study the whole L-packet when we study the special values of automorphic L-functions. It seems very interesting to clarify the relationship between the period part of the special value expected by our result and Deligne's conjecture.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (11 results)

All Other

All Publications (11 results)

  • [Publications] Masaaki FURUSAWA: "The fundamental lemma for the Bessel and Novodvorsky Subgroups of GSp (4)"C. R. Acad. Sci. Paris. 328. 105-110 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Masaaki FURUSAWA: "The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp (4) II"C. R. Acad. Sci. Paris. 331. 593-598 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] FURUSAWA M. and SHALIKA J.A.: "The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4)"C. R. Acad. Sci. Paris. 328. 105-110 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] FURUSAWA M. and SHALIKA J.A.: "The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4) II"C. R. Acad. Sci. Paris. 331. 593-598 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Furusawa-J.A.Shalika: "The fundamental lemra for the Bessel and Novodvorsky subgroups"C. R. Acad. Sci. Paris. 328. 105-110 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Masaaki Furusawa: "The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp (4)" C.R.Acad.Sci.Paris. 328. 105-110 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nobuo Tsuzuki: "The local index and the Swan conductor" Compositio Math.111. 245-288 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nobuo Tsuzuki: "Slope filtration of quasi-unipotent overconvergent F-isocrystals" Ann.Inst.Fourier, Grenoble. 48. 379-412 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Nobuo Tsuzuki: "Finite local monodromy of overconvergent unit-root F-isocrystals on a curve" Amer.J.Math.120. 1165-1190 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Keiji Matsumoto: "Intersection numbers for 1-forms associated with confluent hypergenometric functions" Funkcial.Ekvac.41. 291-308 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Keiji Matsumoto: "Intersection numbers for logarithmic k-forms" Osaka J.Math.35. 873-893 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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