Number theory of algebraic varieties

代数簇数论

基本信息

批准号：
03452003
负责人：
KAWAMATA Yujiro
金额：
$ 3.97万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1991
资助国家：
日本
起止时间：
1991 至 1992
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-03452003/
关键词：
algebraic varieties number theory semistable reduction minimal model zeta function log rithmic structure Hecke character Galois representation 数論的代数幾何ゼ-タ関数 P進Hodge理論岩沢理論形式群局所定数基本群

项目摘要

The purpose of this research was to investigate the number theory of algebraic varieties defined over a ring of algebraic integers. In order to start this investigation, it is important to replace the originalvariety by a more natural model by a birational transformation. If the given variety has relative dimension 1 over the ring of integers, then the classical minimal model theory provides us the canonical model. Kawamata tried to extend the minimal model theory to higher dimensional case, and succeeded in the case in which the relative dimension is 2 and the variety has semistable reduction.In the course of the proof, newly developed theory of algebraic 3-folds over the complex numbers was used. The difficulty in the proof came from the fact that the vanishing theorem of Kodaira type, which was very useful in the case over the complex numbers, is false in positive characteristic.The singular fiber of a variety with semistable reduction is a normal crossing variety. Conversely, Kawam … More ata considered the smoothing of normal crossing variety into a variety with semistable reduction, and developed the theory of logarithmic deformations with Yoshinori Namikawa at Sophia University. In particular, they proved the existence of a smoothing of a degenerate Calabi-Yau variety.The cohomology theory is an important tool in the investigaition of algebraic varieties. Saito investigated the 1 dimensional Galois representations on the determinant of L-adic cohomology groups. In the case of constant coefficients, he obtained the description of the corresponding quadratic extensions. In the case of variable coefficients, he proved that they are described by the algebraic Hecke characters determined by the Jacobi sums.The zeta functions an analytic object which is attached to an algebraic variety over the ring of integers. There are several mysterious conjectures connecting the zeta functions and the number theory of algebraic varieties. Kurokawa investigated multiple zeta funcitons and multiple trigonometric functions, and found formulas of the Gamma factor of the Selberg zeta functions and of the special values of the zeta functions. Less

本研究的目的是研究在代数整数环上定义的代数簇的数论。为了开始这项研究，重要的是通过给定的双有理变换用更自然的模型替换原始簇。品种在整数环上的相对维数为1，那么经典最小模型理论为我们提供了规范模型，川俣尝试将最小模型理论扩展到更高维的情况，并在相对维数为1的情况下取得了成功。 2且簇具有半稳定约简。证明过程中采用了新发展的复数代数三倍理论，证明的难点在于小平型消失定理。在复数的情况下有用，在正特性中是假的。具有半稳定还原的品种的奇异纤维是离线的正常杂交品种，Kawam … More ata 考虑了将正常杂交品种平滑化为具有半稳定的品种。半稳定约简，并与上智大学的 Yoshinori Namikawa 一起发展了对数变形理论。特别是，他们证明了简并 Calabi-Yau 簇的存在性。上同调理论是研究代数簇的重要工具。研究了常数系数情况下L-进上同调群行列式的一维伽罗瓦表示，得到了相应的描述。在变量系数的情况下，他证明了它们是由雅可比和确定的代数赫克特征描述的。zeta 函数是一个解析对象，它附加在整数环上的代数簇上。连接 zeta 函数和代数簇数论的猜想研究了多重 zeta 函数和多重三角函数，并找到了 Gamma 因子的公式。 Selberg zeta 函数和 zeta 函数的特殊值。