2006 Fiscal Year Final Research Report Summary
Research of modular and quasimodular forms arising in various areas of mathematics and their application
| Project/Area Number |
15340014
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYUSHU UNIVERCITY |
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Principal Investigator |
KANEKO Masanobu Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (70202017)
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| 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)
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| Project Period (FY) |
2003 – 2006
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| Keywords | modular forms / quasimodular forms / period polynomials / Fourier coefficients |
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| 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.
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