Geometry of modular varieties and congruence, P-adic theory of Siegel modular forms

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By studying arithmetic geometry of Siegel modular varieties, we solved the congruence problem of Siegel modular forms, and showed that weights of p-adicSiegel modular forms are determined as p-adic numbers. Further, we constructed a basictheory of arithmetic vector-valued Siegel modular forms and vector-valued p-adic Siegel modular forms with natural p-adic operators. Moreover, we studied the ring structure ofSiegel modular forms over rings in which 6 is invertible, and decided this structure in the degree 2 case.