雅可比模形式及其微分算子

发布时间：2019-06-07 16:18

【摘要】：本论文研究雅可比模形式上的微分算子理论,我们得到了一种构造雅可比模形式上的微分算子的新的方法,并由此得到了西格尔-雅可比上半平面上的一系列的微分算子。在数论中,经典的雅可比形式是一种重要的自守形式。它们与椭圆模形式有着很深刻的联系,同时也可以作为二维西格尔模形式的傅里叶-雅可比系数出现。有很多的微分算子作用在雅可比形式上,其中包括Masss-Shimura类型的微分算子与其他的非全纯微分算子。在本文中,类比模形式的情形,我们将首先回忆这些算子的性质及其与全纯算子的联系,比如塞尔导数与Rankin-Cohen括号。然后,与经典情形类似,我们将得到关于埃尔米特雅可比模形式上的微分算子的一些结果。正如西格尔模形式是经典模形式的推广,西格尔雅可比形式同样是经典雅可比形式的高次数的类比。在西格尔模形式上,我们已经知道存在许多微分算子,特别是不变微分算子和将模形式的权提升2的Maass-Shimura算子。本文的一个主要目标就是得到西格尔雅可比形式上的微分算子,为此我们将用到联络的概念。我们将证明西格尔雅可比平面是凯勒的,从而存在一陈省身联络,通过计算其Christoffel系数,我们给出了此联络在矩阵形式下的表达式。借助此结果,我们得到了西格尔雅可比模形式上的微分,这是因为联络可以作用在流形的不变截面上。特别地,我们得到了经典情形下的微分算子的推广、一系列不变微分算子以及两个Maass-Shimura类型的微分算子。
[Abstract]:In this paper, we study the theory of differential operators in the form of Jacob modules, and we obtain a new method for constructing differential operators on the form of Jacob modules, from which we obtain a series of differential operators on the upper half plane of Sigel-Jakobi. In number theory, the classical Jacob form is an important form of self-defense. They are deeply related to the form of elliptical modules, and can also be used as Fourier-Jacob coefficients in the form of two-dimensional Siegelian modules. There are many differential operators acting on Jacob form, including Masss-Shimura type differential operators and other Holomorphic differential operators. In this paper, in the case of analogical modular form, we will first recall the properties of these operators and their relationship with Holomorphic operators, such as Searle derivatives and Rankin-Cohen parentheses. Then, similar to the classical case, we will obtain some results on the differential operators in the form of Hermitian James modules. Just as the Sieger module form is a generalization of the classical module form, the Siger Jacob form is also a high-frequency analogy of the classical Jacob form. In Sieger module form, we already know that there are many differential operators, especially invariant differential operators and Maass-Shimura operators that raise the weight of module form by 2. One of the main objectives of this paper is to obtain the differential operator in Siger Jalcombe form, for which we will use the concept of connection. We will prove that the Siger Jacob plane is Keller's, so that there is a Shiing-Shen Chern connection. by calculating its Christoffel coefficient, we give the expression of this connection in the form of matrix. With the help of this result, we obtain the differential in the form of Siger Jacob module, which is because the connection can act on the invariant cross section of the manifolds. In particular, we obtain the generalization of differential operators in classical cases, a series of invariant differential operators and two Maass-Shimura differential operators.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175.3

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