Moduli space of algebraic curves and automorphic forms

代数曲线和自守形式的模空间

基本信息

批准号：
09640047
负责人：
ICHIKAWA Takashi
金额：
$ 1.73万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640047/
关键词：
Algebraic curve Period integral Theta function Soliton equation Moduli space Automophic form Teichmueller groupoid Galois action 一意化フュージング・ムーヴリーマン面ショットキー群 p進解析テ-タ関数

项目摘要

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.

1.我们将代数曲线的Shottky-Mumford统一化理论推广到基环由有理整数环上的形式幂级数组成的情况，并具体构造了该幂级数环上的任意简并曲线的变形。进一步给出了该变形的微分形式和周期积分的计算方法。 2．我们构造了局部场上无限亏格的解析曲线作为无限生成肖特基群的均匀化，并给出了这些曲线的微分形式和周期积分的表达式。作为其应用，我们证明了无限亏格的p进解析曲线的theta函数产生了孤子方程之一的KP方程的解。 3．利用1中的结果，我们证明了代数曲线模空间（Teichniueller空间上的自守函数）上有理整数环上的自守形式环的有限生成性，并用环描述了该环的结构属2和属3情况下的Siegel模形式。 4．使用1中的结果，我们比较了有理整数环上的形式几何类别中给定曲线的不同退化所附加的参数。作为其在格洛腾迪克猜想中的应用，我们构造了算术Teichmueller群群的自然基集，并描述了该群群上的Ahe Galois作用。