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Distribution of units of an algebraic number field from the viewpoint of class field theory and analytic number theory

从类域论和解析数论的角度看代数数域的单位分布

基本信息

批准号：
13640049
负责人：
KITAOKA Yoshiyuki
金额：
$ 1.54万
依托单位：
Meijo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640049/
关键词：
class field analytic number theory unit distribution 整数論代数体単数の分布密度

项目摘要

The aim o f this research is to study the distribution of units of an algebraic number field. There maybe several viewpoints. Ours is based on the class field theory and the analytic number theory The reason for the class field theory is the following: For an integral ideal A of an algebraic number field F, we can associate the unique abelian extension of conductor Aover F, and the extension degree is the product of the lass number of F and the residual index of residue classes represented by units in the residue class group modulo A. The class number is studied very well. But there is nothing about residual indices. As a matter of fact, almost nobody knew how to formulate the vague problem "distribution of units". So we adopted the viewpoint that the distribution of values of residual indices is nothing but the distribution of units, and we studied it using methods in the analytic number theory. At the beginning, we studied real quadratic fields and real cubic fields with negative discriminant, in detail.. The results were published in Nagoya Math. J. and J. of Number Theory. The next problem was its generalization to any algebraic number field. I completed it in the case of prime ideals. To treat more general ideals, it is necessary to study algebraic number fields in detail. For the time being, we are trying the rational prime number case and in the case that the rank of the unit group is almost over. When the rank of unit group is greater than one, the situation is much more complicated and we are collecting more information.

研究代数数域的单位分布问题。可能有几种观点。类域论的理由是：对于代数数域F的整理想A，我们可以将F上的导体A的唯一阿贝尔扩张联系起来，扩张度是F的类数与模A的剩余类群中单位所表示的剩余类的剩余指数的乘积。班级号码研究得很好。但是没有关于剩余指数的内容。事实上，几乎没有人知道如何制定模糊的问题“单位分配”。因此，我们采用了剩余指数值的分布就是单位分布的观点，并利用解析数论的方法对其进行了研究。首先，我们详细研究了具有负判别式的真实的二次域和真实的三次域。结果发表在名古屋数学杂志和数论杂志上。下一个问题是它的推广到任何代数数域。我在素理想的情况下完成了它。为了处理更一般的理想，有必要详细研究代数数域。目前，我们正在尝试有理数素数的情况下，在单位群的秩几乎结束的情况下。当单元组的秩大于1时，情况要复杂得多，我们正在收集更多的信息。