STUDY OF ARITHMETIC PROPERTIES OF AUTOMORPHIC FORMS OF SEVERAL VARIABLES

这个项目的主要目的之一是发展多个复变量的自同构形理论。近年来，这一领域通过几种方法得到了发展，如表示论和代数几何。在这个项目中，我们主要研究了Siegel模形式理论的算术方面，这是多复变量自同构形式的一个典型例子。在这个项目期间，我们取得了两个主要成果。第一个是关于有限域上的Siegel模形式理论。在单变量情况下，这是由Swinnerton-Dyer开发和完成的。我们的结果被认为是他的结果的推广。第二个是关于模形式的p-进理论，该理论最初是由J.-P.Serre研究的。我们研究了它的推广，得到了p-进Eisenstein级数的一个有趣公式。这两个结果都被接受为数学时代裂痕的推论。

One of the main purposes of this project is to develop the theory of automorphic forms of several complex variables. Recently this field is developed by several methods, for example, that of the representation theory, and algebraic geometry. In this project, we mainly studied the arithmetic aspect of the theory of Siegel modular forms, which is a typical example of automorphic forms of several complex variables. During this project term, we got two main results. The first one is concerned with the theory of Siegel modular forms over the finite fields. In the one variable case, this is developed and completed by Swinnerton-Dyer. Our result is regarded as a generalization of his result. The second one is related to the p-adic theory of modular forms, which is originally studied by J.-P. Serre. We studied the generalization and got an interesting formula of p-adic Eisenstein series. Both results were accepted for pulication in the Mathematisch Zeitschrift.