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2006 Fiscal Year Final Research Report Summary

Research of modular and quasimodular forms arising in various areas of mathematics and their application

Research Project

Project/Area Number 15340014
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKYUSHU UNIVERCITY

Principal Investigator

KANEKO Masanobu  Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (70202017)

Co-Investigator(Kenkyū-buntansha) KOIKE Masao  Kyushu University, Faculty of Mathematics, Professor, Professor, 大学院数理学研究院, 教授 (20022733)
NAGATOMO Kiyokazu  Osaka University, Graduate School of Information Science and Technology, Associate Professor, 大学院情報科学研究科, 助教授 (90172543)
TAKATA Toshie  Niigata University, Faculty of Science, Associate Professor, 自然科学系, 助教授 (40253398)
ASAKURA Masanori  Kyushu University, Faculty of Mathematics, Research Associate, 大学院数理学研究院, 助手 (60322286)
Project Period (FY) 2003 – 2006
Keywordsmodular forms / quasimodular forms / period polynomials / Fourier coefficients
Research Abstract

Modular and quasimodular solutions of a differential equation that arose in our work with Don Zagier has been investigated. Of particular interest are modular solutions of weight fifth of integers, which are closely connected to the famous Rogers-Ramanujan functions, and quasimodular forms which turned out to be "extremal" in the sense we defined anew. The latter exteremal quasimodular forms were further studied in a joint work with Koike. We have given explicit formulas for them in case of depth one and two and found the differential equations they satisfy. We have made several interesting observations on the Fourier coefficients of extremal quasimodular forms of depth less than five, but could not give a proof. Also, as an application of quasimodular forms, we gave a condition for Fourier coefficients of cusp forms on the modular group being "ordinary" for a prime in terms of certain polynomials. A connection of this and the supersingular polynomials may be of some interest.
Our study also concerns so called multiple zeta values. In particular, when we look closely into the double shuffle relations of the double zeta values, we are naturally led to the period polynomials of modular forms on the full modular group. To understand the connection, we have defined and studied the double Eisenstein series and computed their Fourier coefficients. As an application, we have found several formulas for the Fouries coefficients of the Ramanujan tau function, the coefficients of weight 12 cusp form known as the discriminant function or Jacobi's delta function.

  • Research Products

    (12 results)

All 2006 2005 2004

All Journal Article (12 results)

  • [Journal Article] Derivation and double shuffle relations for multiple zeta values2006

    • Author(s)
      K.Ihara, M.Kaneko, D.Zagier
    • Journal Title

      Compositio Mathematicae 142-02

      Pages: 307-338

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On extremal quasimodular forms2006

    • Author(s)
      M.Kaneko, M.Koike
    • Journal Title

      Kyushu Journal of Mathematics 60-2

      Pages: 457-470

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Spherical designs attached to extremal lattices and the modulo p property of Fourier coefficients of extremal modular forms2006

    • Author(s)
      E.Bannai, M.Koike, M.Shinohara, M.Tagami
    • Journal Title

      Moscow Mathematical Journal 6

      Pages: 225-264

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Surjectivity of p-adic regulators on K2 of Tate curves2006

    • Author(s)
      M.Asakura
    • Journal Title

      Inventiones Mathematicae 165-2

      Pages: 267-324

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Derivation and double shuffle relations for multiple zeta values2006

    • Author(s)
      K.Ihara, M.Kaneko, D.Zagier
    • Journal Title

      Compositio Math. vol.142-02

      Pages: 307-338

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On extremal quasimodular forms2006

    • Author(s)
      M.Kaneko, M.Koike
    • Journal Title

      Kyushu J.Math. vol.60-2

      Pages: 457-470

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Spherical designs attached to extremal lattices and the modulo p property of Fourier coefficients of extremal modular forms2006

    • Author(s)
      E.Bannai, M.Koike, M.Shinohara, M.Tagami
    • Journal Title

      Moscow Math. J. 6

      Pages: 225-264

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Surjectivity of p-adic regulators on K2 of Tate curves.2006

    • Author(s)
      M.Asakura
    • Journal Title

      Invent.Math. 165, no.2

      Pages: 267-324

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Conformal field theories associated to regular chiral vertex operator algebras I2005

    • Author(s)
      K.Nagatomo, A.Tsuchiya
    • Journal Title

      Duke Mathematical Journal 128

      Pages: 393-471

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Conformal field theories associated to regular chiral vertex operator algebras I2005

    • Author(s)
      K.Nagatomo, A.Tsuchiya
    • Journal Title

      Duke Mathematical Journal 128-3

      Pages: 393-471

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Reshetikhin-Turaev invariants of Seifert 3-manifolds2004

    • Author(s)
      S.K.Hansen, T.Takata
    • Journal Title

      Journal of Knot Theory and Ramifications 13-5

      Pages: 617-668

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Reshetikhin--Turaev invariants of Seifert 3-manifolds2004

    • Author(s)
      S.K.Hansen, T.Takata
    • Journal Title

      J.Knot Theory Ramifications 13, no.5

      Pages: 617-668

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

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Published: 2008-05-27  

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