Motivic structure of nilpotent completions of modular groups

通过对代数曲线、交换簇及其模空间的算术几何的研究，我们得到了如下结果。1.构造了基于椭圆模动机的Hecke算子理论，并作为应用，证明了多重模L-值的代数性。2.利用刚性分析的方法，给出了肖特基问题的解，即阿贝尔族成为雅可比族的条件。3.构造了p-进向量值Siegel模形式的基本理论。进一步，我们给出了Shimura的几乎全纯向量值Siegel模形式的p-进形式，并证明了它们在CM点处的值的代数性。4.利用算术肖特基一致化理论，证明了双曲三维流形的几何Zeta函数的特异值的算术性。

By studying arithmetic geometry of algebraic curves, abelian varieties and their moduli spaces, we obtained the following results. 1. We constructed a theory of Hecke operators on elliptic modular motives, and as its application, we showed the algebraicity of multiple modular L-values. 2. Using rigid analysis, we gave a solution to the Schottky problem, namely a condition that abelian varieties become Jacobi varieties. 3. We constructed a basic theory of p-adic vector-valued Siegel modular forms. Further, we gave p-adic versions of Shimura's nearly holomorphic vector-valued Siegel modular forms and showed the algebraicity of their values at CM points. 4. By using the arithmetic Schottky uniformization theory, we showed the arithmeticity of the special values for geometric zeta functions of hyperbolic 3-manifolds.