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

本研究的目的是研究代数数域的单位分布。可能有几种观点。我们是基于场理论和解析数论的类类场论的原因如下:积分理想的一个代数数域F,我们可以把独特的阿贝尔的延伸导体;F和扩展程度的产品是小姑娘的F和剩余残渣类指数由单位剩余类组模A类数是学习很好。但是没有关于剩余指标的。事实上，几乎没有人知道如何表述“单位分配”这个模糊的问题。因此，我们采用了残差指标值的分布只不过是单位分布的观点，并运用解析数论的方法对其进行了研究。首先，我们详细地研究了具有负判别式的实二次场和实三次场。研究结果发表在《名古屋数学》杂志上。数论的j和j。下一个问题是将它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，有必要详细地研究代数数域。目前，我们正在尝试有理数的情况，在单位群的秩几乎结束的情况下。当单位群的秩大于1时，情况就复杂得多，我们收集的信息也更多。

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.