Moduli space of algebraic curves and automorphic forms

1.将代数曲线上的Shottky-Mumford统一化理论推广到有理整数环上基环由形式幂级数构成的情形，并具体构造了幂级数环上任意退化曲线的变形.进一步给出了这种变形的微分形式和周期积分的计算方法.构造了局部域上无限亏格的解析曲线作为无限生成Schottky群的一致化，并给出了这些曲线的微分形式和周期积分的表达式.作为应用，我们证明了无穷亏格的p-adic解析曲线的theta函数生成孤立子方程之一的KP方程的解.利用文[1]的结果，证明了代数曲线模空间（Teichniueller空间上的自守函数）上的有理整数环上的自守形式环的有限生成性，并用亏格2和亏格3的Siegel模形式环描述了该环的结构.利用文献[1]中的结果，比较了有理整数环上形式几何范畴中给定曲线的不同退化所对应的参数。作为它在Grothendieck猜想中的应用，我们构造了算术Teichmueller广群的一个自然基集，并给出了该广群上Ahe Galois作用的一个刻画.

1. We extended the Shottky-Mumford unifonnization theory on algebraic curves to the case that the base ring consists of formal power series over the rational integer ring, and constructed concretely a deformation of any degenerate curve over this power series ring. Further, we gave a method to caluculate differential forms and period integrals of this deformation.2. We constructed analytic curves of infinite genus over local fields as uniformizations of infinitely generated Schottky groups, and gave an expression of differential forms and period integrals of these curves. As its application, we showed that the theta functions of p-adic analytic curves of infinite genus generate solutions of the KP equation which is one of soliton equations.3. Using the result in 1, we showed the finitely-generatedness of the ring of automorphic forms over the rational integer ring on the moduli space of algebraic curves ( : automorphic functions on the Teichniueller space), and described the structure of this ring by the ring of Siegel modular forms in the genus 2 and 3 cases.4. Using the result in 1, we compared the parameters attached to different degenerations of a given curve in the category of formal geometry over the rational integer ring. As its application to Grothendieck's conjecture, we constructed a natural base set of the arithmetic Teichmueller groupoid, and gave a description of Ahe Galois action on this groupoid.