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Study on elliptic cohomology theory

Research Project

Project/Area Number 08404002
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionKYOTO UNIVERSITY

Principal Investigator

NISHIDA Goro  KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (00027377)

Co-Investigator(Kenkyū-buntansha) MARUYAMA Masaki  KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (50025459)
SHIMIZU Yuji  KYOTO UNIVERSITY,Faculty of Science, Instructor, 大学院・理学研究科, 講師 (80187468)
YOSHIDA Hiroyuki  KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (40108973)
FUKAYA Kenji  KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (30165261)
KONO Akira  KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (00093237)
Project Period (FY) 1996 – 1997
Project Status Completed (Fiscal Year 1997)
Budget Amount *help
¥12,300,000 (Direct Cost: ¥12,300,000)
Fiscal Year 1997: ¥4,900,000 (Direct Cost: ¥4,900,000)
Fiscal Year 1996: ¥7,400,000 (Direct Cost: ¥7,400,000)
KeywordsElliptic cohomology / Homotopy theory / Elliptic curve / Modular-form / Formal group low / Hopf algebra / Classifying space / Elliptic genera / ホモトピー論 / 楕円曲線 / ホップ代数 / 楕円種数
Research Abstract

The elliptic cohomology theory was defined by Landweber using a formal group low associated with elliptic curves to express the elliptic genera in terms of homotopy theory. Our research group studied relations between the homotopy theory and the theory of elliptic curves and modular forms via the elliptic cohomology theory. As an explicit result, we showed that the ring of level n modular forms can be identified with the elliptic cohomology of the classifying space of the cyclic group of order n. When n tends to infinity, these elliptic cohomology rings approximates the formal Hopf.algebra which represents the formal group low of the elliptic cohomology theory. This means that it may be possible to construct the elliptic cohomology theory from the Galois theory of higher level modular forms. As other related result, we showed that there is a close relation between the representation of general linear group acting on the cohomology group of m-times product of the classifying spaces of the cyclic group of order n and the Galois representation of a splitting field of the multiplication-by-n sequence of the height n formal group low of Lubin-Tate. As a fortune program, it is interesting to study the relation of the representation of general linear group to the Eichler-Shirura cohomology.

Report

(3 results)
  • 1997 Annual Research Report   Final Research Report Summary
  • 1996 Annual Research Report
  • Research Products

    (24 results)

All Other

All Publications (24 results)

  • [Publications] K.Kordzaya, G.Nishida: "A duality theorem in Hopf algebras and its application to Morava K theory of BZ/p" J.Math.Kyoto Univ.36. 771-778 (1996)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] S.Morimoto, G.Nishida: "Elliptic cohomology of classifying spaces of cyclic groups ans higher level modular forms" J.Math.Kyoto Univ.37. 701-715 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] K.Fukaya: "Anti Self Dual equations on 4-manifolds with degenerate metric" Geometric Analysis and Functional Analysis. 1-60 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] K.Fukaya, Y.Oh: "Zero loop open string in the cotangent bundle and Morse homotopy" Asian Journal of Mathematics. 1. 96-180 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] H.Yoshida: "On absolute CM-periods" Symposia Pure Math.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Y.Shimizu, T.Suzuki, K.Ueno: "Knizhnik-Zamolodchikov-Bemard equations of higher genera" Proceedings of Taniguchi Symposium 1997.

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] G.Nishida: "A duality theorem in Hopf algebras and its application to Morava K theory of BZ/p" J.Math.Kyoto Univ.36. 1-8 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] A.Kono: "A Remark on the homotopy type of certain gauge groups" J.Math.Kyoto Univ.36. 115-121 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] K.Fukaya: "Morse Theory and Chem Simons Perturbation theory" Commun.Math.Phys. 181. 37-90 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] H.Yoshida: "On absolute CM-periods" Lecture Note Series, RIMS,Kyoto Univ.965. 87-133 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] M.Maruyama: "Construction of moduli spaces of stable sheave via Simpson's idea" Proc.of Taniguchi Symp."Moduli of Vector Bundles". 147-188 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] Y.Shimizu: "KZB equations of higher genera" Proc.of Symp."Algebraic geometry". 100-107 (1996)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1997 Final Research Report Summary
  • [Publications] K.Kordzaya, G.Nishida: "A duality theorem in Hopf algebras and its application to Morava K theory of BZ/p" J.Math.Kyoto Univ.36. 771-778 (1996)

    • Related Report
      1997 Annual Research Report
  • [Publications] S.Morimoto, G.Nishida: "Elliptic cohomology of classifying spaces of cyclic groups ans higher level modular forms" J.Math.Kyoto Univ.37. 701-715 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Fukaya: "Anti Self Dual equations on 4-manifolds with degenerate metric" Geometric Analysis and Functional Analysis. 1-60 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] K.Fukaya, Y.Oh: "Zero loop open string in the cotangent bundle and Morse homotopy" Asian Journal of Mathematics. 1. 96-180 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] H.Yoshida: "On absolute CM-periods" Symposia Pure Math.

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Shimizu, T.Suzuki, K.Ueno: "Knizhnik-Zamolodchikov-Bernard equations of higher genera" Proceedings of Taniguchi Symposium 1997.

    • Related Report
      1997 Annual Research Report
  • [Publications] G.Nishida: "A duality theorem in Hopf algebras and its application to Morava K theory of BZ/p" J.Math.Kyoto Univ.36. 1-8 (1997)

    • Related Report
      1996 Annual Research Report
  • [Publications] A.Kono: "A Remark on the homotopy type of certain gauge groups" J.Math.Kyoto Univ.36. 115-121 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] K.Fukaya: "Morse Theory and Chern Simons Perturbation theory" Commun.Math.Phys. 181. 37-90 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] H.Yoshida: "On absolute CM-periods" 数理解析研究所講究録. 965. 87-133 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] M.Maruyama: "Construction of moduli spaces of stable sheave via Simpson's idea" Proc.of Taniguchi Symp.“Moduli of Vector Bundles". 147-188 (1996)

    • Related Report
      1996 Annual Research Report
  • [Publications] Y.Shimizu: "KZB equations of higher genera" 代数幾何学シンポジウム報告集. 100-107 (1996)

    • Related Report
      1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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