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

数学各领域中出现的模和拟模形式的研究及其应用

基本信息

批准号：
15340014
负责人：
KANEKO Masanobu
金额：
$ 5.57万
依托单位：
KYUSHU UNIVERCITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2006
项目状态：
已结题

项目摘要

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.

研究了我们与Don Zagier合作中出现的一个微分方程的模解和拟模解。特别令人感兴趣的是与著名的Rogers-Ramanujan函数密切相关的整数权重的五分之一的模解，以及在我们重新定义的意义上被证明是“极值”的准模形式。后者的外部拟模形式在与Koike的合作中被进一步研究。在深度为1和2的情况下，我们给出了它们的显式公式，并找到了它们所满足的微分方程。我们对深度小于5的极值拟模形式的傅立叶系数做了几个有趣的观察，但不能给出一个证明。同时，作为拟模形的一个应用，给出了模群上尖点形的傅立叶系数对素数来说是“普通”的一个条件。这与超奇异多项式之间的联系可能会有些有趣。我们的研究也涉及到所谓的多重Zeta值。特别是，当我们仔细研究双Zeta值的双重洗牌关系时，我们自然会得到完全模群上的模形式的周期多项式。为了理解这种联系，我们定义和研究了双重艾森斯坦级数，并计算了它们的傅里叶系数。作为应用，我们找到了Ramanujan tau函数的傅里叶系数的几个公式，这些系数的权12尖点形式称为判别函数或Jacobi的Delta函数。