Number theory of algebraic varieties

本研究的目的是研究定义在代数整数环上的代数簇的数论。为了开始这项研究，重要的是用一个更自然的模型来取代原来的品种，并进行两栖转化。如果给定的簇在整数环上具有相对维度1，则经典的极小模型理论为我们提供了正则模型。Kawamata试图将极小模型理论推广到高维情形，并在相对维为2且簇具有半稳定约简的情况下取得了成功。在证明过程中，使用了新发展的复数上代数3-折叠理论。证明的困难在于，在复数上非常有用的Kodaira型消去定理在正特征上是错误的，具有半稳定约化的簇的奇异纤维是正规的交叉簇。相反，kawam…更多的ATA考虑了将正常杂交品种平滑为具有半稳定还原的品种，并与索菲亚大学的Yoshinori Nankawa一起发展了对数变形理论。特别地，他们证明了退化的Calabi-Yau簇的光滑化的存在性。上同调理论是研究代数簇的一个重要工具。Saito研究了L上同调群行列式上的一维Galois表示。在常系数的情况下，他得到了相应的二次扩张的描述。在变系数的情况下，他证明了它们是由Jacobi和确定的代数Hecke特征标来描述的。Zeta函数是一个分析对象，它附属于整数环上的一个代数簇。有几个神秘的猜想将Zeta函数和代数簇的数论联系在一起。黑川研究了多个Zeta函数和多个三角函数，得到了Selberg Zeta函数的Gamma因子和Zeta函数的特殊值的公式。较少

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