Study on the arithmetic theory of automorphic forms of several variables

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The p-adic theory of modular forms initiated by J.-P. Serre and H.P.F.Swinnerton-Dyer was established in the case of one variable by several people. I tried to generalize the theory to the case of several variables, for example, Siegel modular forms and Hermitian modular forms, and obtained several results. More specifically, I established a constructing method by taking a p-adic limit of a sequence of ordinary modular forms. Concerning the mod p theory, I studied the structure of the graded ring of Hermitian modular forms and determined the structure in some cases. Moreover, I generalized some congruence properties of Siegel modular forms originated with Ramanujan. This result was applied to show the existence of cusp forms of several variables. I studied the p-adic theory and mod p theory of modular forms of several variables and produced fruits. Moreover, the case of vector-valued was considered. In particular, the p-adic theory of the theta operator, which is a kind of differential operator on modular forms, was developed.