Number theory of algebraic varieties

本研究的目的是探讨在代数整数环上定义的代数变量的数论。为了开始这项研究，重要的是用一种更自然的模式来取代原始的品种，通过两国的转变。如果给定的变量在整数环上的相对维数为1，则经典极小模型理论给出了正则模型。Kawamata试图将最小模型理论推广到更高维度的情况，并成功地在相对维数为2且变异具有半稳定约简的情况下进行了推广。在证明过程中，使用了新发展的复数上的代数3折理论。证明的困难在于，在复数上非常有用的Kodaira型消失定理在正特征上是假的。半稳定还原品种的单纤维为正交品种。与此相反，Kawam…More ata则考虑了正交品种平滑为半稳定约简的品种，并与上智大学的Yoshinori Namikawa一起发展了对数变形理论。特别地，他们证明了简并的Calabi-Yau变元平滑的存在性。上同调理论是研究代数变异的一个重要工具。斋藤研究了l进上同调群行列式上的一维伽罗瓦表示。在常系数情况下，他得到了相应的二次扩展的描述。在变系数的情况下，他证明了它们是由Jacobi和决定的代数Hecke特征来描述的。函数是一个解析对象，它附属于整数环上的代数变量。在ζ函数和代数变数理论之间存在着一些神秘的猜想。Kurokawa研究了多个zeta函数和多个三角函数，并发现了Selberg zeta函数的Gamma因子和zeta函数的特殊值的公式。少

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