STUDY OF ARITHMETIC PROPERTIES OF AUTOMORPHIC FORMS OF SEVERAL VARIABLES

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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.