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

通过研究Siegel模簇的算术几何，解决了Siegel模形式的同余问题，证明了p-adic Siegel模形式的权由p-adic数决定.进一步，我们构造了算术向量值Siegel模形式和向量值p-adic Siegel模形式的基本理论。此外，我们还研究了6可逆环上Siegel模形式的环结构，并在2次环上确定了这种结构。

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.