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Modular forms and Dedekind symbols as topological invariants

Research Project

Project/Area Number 16540081
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionTsuda College

Principal Investigator

FUKUHARA Shinji  Tsuda College, Faculty of Liberal Arts, Professor, 学芸学部, 教授 (20011687)

Co-Investigator(Kenkyū-buntansha) MIYAZAWA Haruko  Tsuda College, Institute of Math. and Comp. Sci., Research Fellow, 数学・計算機科学研究所, 研究員 (40266276)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
Keywordstopological invariant / knot / manifold / Dedekind sum / modular form / Hecke operator / period polynomial / 尖点形式 / 周期積分 / 絡み目
Research Abstract

The head investigator has been studying relationship between Dedekind symbols and knot (manifold) invariants. For example, he showed that Conway polynomials of two-bridge knot are given using Dedekind symbols in his paper "Explicit formulae for two-bridge knot polynomials, J. Aust. Math. Soc. 78 (2005), 149-166". Dedekind symbols with polynomial reciprocity laws are especially important. He found explicit formulas for Dedekind symbols with polynomial reciprocity laws and presented the result in his paper "Dedekind symbols with reciprocity laws, Math. Ann. 329 (2004), 315-334".
It is known that there is natural correspondences between Dedekind symbols, modular forms and period polynomials. He found that Hecke operators on Dedekind symbols can be defined so that they are compatible with Hecke operators on modular forms and period polynomials. As a corollary, he obtained explicit formulas for Hecke matrices of cusp forms. The results are published in the papers "Hecke operators on weighted Dedekind symbols, J. reine angew. Math. 593 (2006) 1-29" and "Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials, J. reine angew. Math. (in print) ".
The investigator Miyazawa studied how knot invariants change when knots are moved locally. She gave a talk on this subject at the meeting for "knots and low dimensional manifolds" and published a joint paper "Classification of n-component Brunnian links up to C_n-move, Topology Appl. 153 (2006) 1643-1650" with Akira Yasuhara.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (17 results)

All 2006 2005 2004 Other

All Journal Article (17 results)

  • [Journal Article] A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law2006

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. Number Theory 117

      Pages: 87-105

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Hecke operators on weighted Dedekind symbols2006

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. reine angew. Math. 593

      Pages: 1-29

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Classification of n-component Brunnian links up to C_n-move2006

    • Author(s)
      Haruko Aida Miyazawa
    • Journal Title

      Topology Appl. 153

      Pages: 1643-1650

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Hecke operators on weighted Dedekind symbols2006

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. reine angew, Math. 593

      Pages: 1-29

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Classification of n-component Brunnian links up to C_n-move2006

    • Author(s)
      Haruko Aida Miyazawa, Akira Yasuhara
    • Journal Title

      Topology Appl. 153

      Pages: 1643-1650

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law2006

    • Author(s)
      Shinji Fukuhara, Noriko Yui
    • Journal Title

      J.Number Theory 117

      Pages: 87-105

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Explicit formulae for two-bridge knot polynomials2005

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. Aust. Math. Soc. 78

      Pages: 149-166

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] The Dedekind symbol associated with the Eisenstein series of weight two2005

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      Arch. Math. 85

      Pages: 128-140

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Explicit formulae for two-bridge knot polynomials2005

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J.Aust.Math.Soc. 78

      Pages: 149-166

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The Dedekind symbol associated with the Eisenstein series of weight two2005

    • Author(s)
      Shinji Fukuhara, Noriko Yui
    • Journal Title

      Arch.Math. 85

      Pages: 128-140

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Dedekind symbols with polynomial reciprocity laws2004

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      Math. Ann. 329

      Pages: 315-334

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Elliptic Apostol sums and their reciprocity laws2004

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      Trans. Amer. Math. Soc. 356

      Pages: 4237-4254

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Dedekind symbols with polynomial reciprocity laws2004

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      Math.Ann. 329

      Pages: 315-334

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Elliptic Apostol sums and their reciprocity laws2004

    • Author(s)
      Shinji Fukuhara, Noriko Yui
    • Journal Title

      Trans.Amer.Math.Soc. 356

      Pages: 4237-4254

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. reine angew. Math. (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Explicit formulas for Hecke operators on cusp forms, Dedekind symbols and period polynomials

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. reine angew. Math.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Explicit formulas for Hecke operators on cups forms, Dedekind symbols and period polynomials

    • Author(s)
      Shinji Fukuhara
    • Journal Title

      J. reine angew, Math. (印刷中)

    • Related Report
      2006 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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