几何中的联络在几类自守型导数研究中的应用

In this program, we plan to study the derivative operators of several kinds of automorphic forms using the device of connections in differential geometry. This is a new method in the research of the derivatives of automorphic forms and will produces new results. By computing the connection coefficients, we will determine the Levi-Civita connections associated to the invariant metrics in.the Siegel upper half plane, generalized upper half plane and Siegel-Jacobi plane,respectively, and further we will compute the expressions of the differential forms under the respective Levi-Civita connections, and thus get the non-holomorphic derivative operators of the Siegel modular forms, the Maass forms and the Siegel-Jacobi-Maass modular forms. Moreover, we will calculate the Rankin-Cohen Bracket of these operatos, and wish to show a conjecture of Bump on the derivatives of Siegel modular forms by it.. To get the holomorphic derivative operators of these automorphic forms, we need to construct a "modular connection" in the respective plane, whose condition is weaker than that of a connection in geometry. In addition, using connections, we will also study the derivatives of theta series and differential equations they satisfy。 This research is signification because theta series are important device in the research of abelian varieties.

在本项目中，我们计划使用微分几何中的联络这一工具来研究几种自守型的导数算子问题。这是研究自守型导数的一种全新方法，并会产生全新的结果。通过计算联络系数，我们首先决定Siegel上半平面、一般化上半平面、Siegel-Jacobi平面上对应于不变度量的Levi-Civita联络，然后再计算微分形式在这种联络下的表达式，从而得到相应的Siegel模形式，Maass形以及Siegel-Jacobi-Maass模形式的非全纯导数算子。然后计算这些算子的Rankin-Cohen Bracket，并希望以此来证明Bump关于Siegel模形式导数的一个猜想。为了给出这些自守型的全纯导数算子，我们需要在对应的空间上构造比联络更弱的所谓的"模联络"。我们还将用联络来研究Theta级数的导数以及它满足的微分方程。因为Theta级数是研究阿贝尔簇的重要工具, 研究其导数以及它所满足的微分方程具有重要意义。