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Value Distribution Theeory and Algebraic Geometry

值分布理论与代数几何

基本信息

批准号：
08304007
负责人：
KOBAYASHI Ryoichi
金额：
$ 1.98万
依托单位：
Nagoya University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至无数据
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08304007/
关键词：
holomorphic curve value distribution theory Diophantine approximation Radon transform Abeliam variety second main conjecture Tening on the bagasilhmic derivative Vojta's dictionary ディオファントス近似論第2主要予想

项目摘要

The ultimate purpose of this project is to establish the geometry which fully explain the similarity between value distribution theory of holomorphic curves in projective algebraic varieties and the diophantine approximations on arithmetically defined projective varicties. The major difficulty in this research is the lack of definition of the notion of differentiation (in the direction of Spec Z of rational points of arithmetic varieties. Through the investigation under this project, we arrived at the fundamental idea which may be described as follows. Instead of trying to define the notion of differentiation of rational points in the direction of Spec Z itself, we try to find a system of functional equations among value-distrivution-theoretic functions (e.g., Weil functions). Then, using Vojta's dictionary, we translate the functional equations to equations in diophantine approximations. These equations will "define" the diffrentials of rational points. In the course of justifying this idea, we obtained the following results : (1) We invented the Radon transform transforming a holomorphic curve in higher dimensional varieties into a system of meromnorphic functions. (2) We can include the group structure of Abelian varieties into the framework of Radon transform, if we replace "rational functions" by holomorphic maps into hyperbolic Riemann surfaces. As a result, we can show that the second main conjecture holds for holomorphic curves in Abelian varieties. (3) We showed that the asymptotic behavior of the Weil functions of jets of a given holomorphic curve is the same for all jets, if a curve under consideration is not contained in a special proper algebraic subset, which is the obstruction to the second main conjecture. This is considered to be system of functional equations we are looking for.

本项目的最终目的是建立一个几何学，充分解释射影代数簇中全纯曲线的值分布理论与算术定义的射影簇上的丢番图逼近之间的相似性。在这项研究中的主要困难是缺乏定义的概念分化（在规范Z方向的合理点的算术品种。通过本项目的调查研究，我们得出了如下基本思想。我们不是试图在Spec Z本身的方向上定义有理点的微分的概念，而是试图在值分布理论函数中找到一个函数方程组（例如，Weil函数）。然后，使用Vojta的字典，我们翻译的功能方程方程的丢番图近似。这些方程将“定义”有理点的微分。在证明这一思想的过程中，我们得到了如下结果：（1）我们发明了Radon变换，将高维簇中的全纯曲线变换为亚纯函数系。(2)在双曲Riemann曲面中，用全纯映射代替有理函数，可以将Abel簇的群结构纳入Radon变换的框架中。结果表明，第二个主要猜想对阿贝尔簇中的全纯曲线成立。(3)我们证明了，如果一条全纯曲线不包含在一个特殊的真代数子集中，那么对于所有的喷流，它的Weil函数的渐近行为是相同的，这是第二个主要猜想的障碍。这被认为是我们要寻找的函数方程组。