1997 Fiscal Year Final Research Report Summary
Study on elliptic cohomology theory
| Project/Area Number |
08404002
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
NISHIDA Goro KYOTO UNIVERSITY,Faculty of Science, Professor, 大学院・理学研究科, 教授 (00027377)
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| 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)
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| Project Period (FY) |
1996 – 1997
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| Keywords | Elliptic cohomology / Homotopy theory / Elliptic curve / Modular-form / Formal group low / Hopf algebra / Classifying space / Elliptic genera |
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| 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.
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