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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

摘要研究了代数整数环上代数簇的数论问题。为了开始这项研究，重要的是用一个更自然的模型来代替原始的品种，通过一个双理性变换。如果给定的簇在整数环上的相对维数为1，那么经典的极小模型理论为我们提供了典范模型。Kawamata试图将极小模型理论推广到高维情形，并在相对维数为2且簇有半稳定约化的情形下取得了成功，在证明过程中，他运用了新发展的复数代数3-折理论。证明中的困难来自于在复数情形下非常有用的科代拉型消失定理在正性上是假的，半稳定约化簇的奇异纤维是正规交叉簇。相反，Kawam ...更多信息 ata考虑了将正态交叉簇光滑化为具有半稳定约化的簇，并与上智大学的Yoshinori Namikawa一起发展了对数变形理论。特别地，他们证明了退化的Calabi-Yau簇的光滑性的存在性。上同调理论是研究代数簇的重要工具。齐藤研究了L-adic上同调群行列式的1维伽罗瓦表示。在常系数的情况下，他得到了相应的二次扩张的描述。在情况下，变系数，他证明了他们所描述的代数Hecke字符所确定的雅可比和。zeta函数的分析对象，这是附加到一个代数品种的环的整数。zeta函数和代数簇的数论之间有几个神秘的联系。黑川研究了多重zeta函数和多重三角函数，并发现了塞尔伯格zeta函数的Gamma因子和zeta函数的特殊值的公式。少