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

在我们与Don Zagier的工作中出现的微分方程的模解和准模解已经被研究。特别有趣的是整数权值的五分之一的模解，它与著名的罗杰斯-拉马努金函数密切相关，而拟模形式在我们重新定义的意义上是“极值”的。后一种外部准模形式在与Koike的联合工作中进一步研究。在深度一和深度二的情况下，我们给出了它们的显式公式，并找到了它们满足的微分方程。我们对深度小于5的极值拟模形式的傅里叶系数做了一些有趣的观察，但不能给出证明。同时，作为准模形式的一个应用，我们给出了模群上的尖峰形式的傅里叶系数对于某些多项式的素数是“普通”的一个条件。它和超奇异多项式的联系可能会引起一些兴趣。我们的研究还涉及所谓的多重ζ值。特别是，当我们仔细研究双zeta值的双洗牌关系时，我们很自然地得到了全模群上模形式的周期多项式。为了理解这种联系，我们定义并研究了双爱森斯坦级数，并计算了它们的傅里叶系数。作为一个应用，我们已经找到了拉马努金tau函数的Fouries系数的几个公式，权重为12个尖点形式的系数被称为判别函数或雅可比函数。

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.