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

模和拟模解的微分方程，出现在我们的工作与唐Zagier已被调查。特别感兴趣的是模块化的解决方案的重量五分之一的整数，这是密切相关的著名的罗杰斯-拉马努金功能，和quasimodular形式原来是“极值”的意义上，我们重新定义。后者exteremal quasimodular形式进一步研究了联合工作与小池。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一些有趣的观察，但不能给出证明。作为拟模形式的一个应用，我们给出了模群上尖点形式的傅立叶系数对于素数是“普通的”的一个条件。这一点和超奇异多项式的连接可能是一些兴趣。我们的研究也涉及所谓的多重zeta值。特别是，当我们仔细研究双zeta值的双洗牌关系时，我们自然会得到全模群上模形式的周期多项式。为了理解这种联系，我们定义并研究了二重爱森斯坦级数，并计算了它们的傅里叶系数。作为一个应用，我们已经找到了几个公式的傅里叶系数的拉马努金tau函数，系数的重量12尖点形式被称为判别函数或雅可比的δ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.