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Study of zeta functions related to Jacobi forms and Siegel modular forms

Research Project

Project/Area Number 12640045
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

ARAKAWA Tsuneo  Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (60097219)

Co-Investigator(Kenkyū-buntansha) ENDO Mikihiko  Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (40062616)
AOKI Noboru  Rikkyo University, Faculty of Science, Associate Professor, 理学部, 助教授 (30183130)
SATO Fumihiro  Rikkyo University, Faculty of Science, Professor, 理学部, 教授 (20120884)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2001: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2000: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsmodular forms / Jacobi forms / Selberg zeta functions / Maass wave forms / prime geodesic theorem / multiple L-values / prehomogeneous vector space / b-function / Selberg跡公式 / Fermat曲線 / Hodge予想 / Maass wave forms / Selberg zeta 関数 / Shimura 対応 / 多重L-値 / double shuffle relation / zeta regularization / converse theorem / 概均質ゼータ関数 / セルバーグゼータ関数 / モジュラー群 / 不定符号4元数環 / Weierstrass部分空間 / 弱球等質空間 / Gamma行列
Research Abstract

1. On Siegel modular forms : We formulated the converse theorem of Imai in such a manner that it can be applica* to not necessarily cuspidal Siegel modular forms of degree two. As an application we reconstructed the Saito-Kurokawa lifting for Siegel modular forms of degree two by using Duke-Imamogle's method (joint work with the coworker F.Sato and I.Makino).
2. Solution of the Hashimoto conjecture : We solved the conjecture presented by K.Hashimoto concerning the that series associated to definite quaternion algebras over Q by using Jacobi forms, the theorem of Deligne-Serre on modular forms of weight one (joint work with S.Boecherer).
3. The study of Selberg zeta functions : The head investigator studied the Shimura correspondence for Maass wave forms via Selberg trace formulas and Selberg zeta functions and reduced the problem of bijectivity of the correspondence to a simple relation of the associated two Selberg zeta functions. On the other hand he with the co-operation of S.Koyama and M.Nakasuji obtained explicit forms of the arithmetic Selberg zeta functions attached to indefinite quaternion aogebras over Q and the prime geodesic theorem of Brun-Titchmarsh type.
4. On multiple L-values : The head investigator and M.Kaneko formulated the (regularized) double shuffle relations for multiple L-values. We also determined the principal part at the pole s=0 of the associated multiple L-function by using the zeta regularization.
5. The study of prehomogeneous vector spaces : The coworker F.Sato and K.Sugiyama showed that the b-function also has the decomposition corresponding to that of the associated representations of the vector space.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (26 results)

All Other

All Publications (26 results)

  • [Publications] T.Arakawa, S.Boecherer: "Vanishing of certain spaces of elliptic modular forms and some applications"J.Reine Angew.Math. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, S.Koyama, M.Nakasuji: "Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem"Proc.Japan Acad.. 78,SerA,.7. 120-125 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, I.Maino, F.Sato: "Converse the for not necessarily cuspidal Siegel modular forms of degree two and Saito-Kurokawa lifting"Comment.Math.Univ.St.Pauli. 50. 197-234 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] F.Sato: "p-adic Green functions and zeta function"Comment.Math.Univ.St.Pauli. 51. 79-97 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] F.Sato, Y.Hironaka: "Local densities of representations of quadratic forms over p-adic integers ; the non-dyadic case"J.of Number Thery. 83. 106-136 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Aoki: "Hodge cycles on CM abelian varieties of Fermat type"Comment.Math.Univ.St.Pauli. 51. 99-129 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 荒川恒男, 伊吹山知義, 金子昌信: "ベルヌーイ数とゼータ関数"牧野書店. 243 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa and S.Boecherer: "Vanishing of certain spaces of elliptic modular forms and some applications"J. Reine Angew. Math. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, S.Koyama and M.Nakasuji: "Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem"Proc. Japan Acad.. 78, Ser.A, No.7. 120-125 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, I.Makino and F.Sato: "Converse theorem for not necessarily cuspidal Siegel modular forms of degree 2 and Saito-Kurokawa lifting"Comment.Math. Univ. St. Pauli. 50. 197-234 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] F.Sato: "p-adic Green functions and zeta functions"Comment.Math. Univ. St. Pauli. 51. 79-97 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] F.Sato and Y.Hironaka: "Local densities of representations of quadratic forms over p-adic integers ; the non-dyadic case"J of Number Theory. 83. 106-136 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] N.Aoki: "Hodge cycles ofn CM abelian varieties of Fermat type"Comment.Math. Univ. St. Pauli. 51. 99-129 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, T.Ibukiyama and M.Kaneko: "Bernoulli numbers and zeta functions"Makino shoten. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Arakawa, S.Boecherer: "Vanishing of certain spaces of elliptic modular forms and some applications,"J. Reine Angew. Math.. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Arakawa, S.Koyama, M.Nakasuji: "Arithmetic forms of Selberg zeta functions with applications to prime geodesic theorem"Proc. Japan Acad.. 78,SerA,.7. 120-125 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Sato: "p-adic Green functions and zeta function"Comment. Math. Univ. St. Pauli. 51. 79-97 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Aoki: "On the Tate-Shafarevich groups of semistable elliptic curves with a ratinal S-torsion"Acta Arithmetica. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] N.Aoki: "Hodge cycles on CM abelian varieties of Fermat type"Comment. Math. Univ. St. Pauli. 51. 99-129 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Arakawa, I.Makino, F.Sato: "Converse theorm for not necessarily cuspidal Siegel modular forms of degree 2 and Saito-Kurokawa lifting"Comment. Math. Univ. St. Pauli. 50. 197-234 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Arakawa: "SL(2, R)のSelberg跡公式と低ウェイト保型形式空間の次元などについて"京都大学数理解析研究所講究録. 1219. 51-61 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Arakawa: "Selberg zeta functions and the Shimura correspondence for Maass wave forms"Proc of Japanese-German seminar on explicit structures of modular forms and zeta functions. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] F.Sato: "Functional equations of prehomogeneous zeta functions and intertwining operators"Proc of Japanese-German seminar on explicit structures of modular forms and zeta functions. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 荒川 恒男, 伊吹山 知義, 金子 昌信: "ベルヌーイ数とゼータ関数"牧野書店. 243 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Arakawa: "Saito-Kurokawa liftings for odd weights"Comment.Math.Univ.St.Pauli. 49. 159-176 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] F.Sato and Y.Hironaka: "Local densities of representations of quadratic forms over p-adic integers : the non-dvadic case"J.of Number Theory. 83. 106-136 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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