Stark Units

代数数论研究的是将一个代数数与有理数域相连而得到的域。斯塔克猜想假设了这样一个数域的解析定义的L-级数的特殊值与该数域的代数性质之间的深刻联系。特别是，这些猜想预言从L-级数得到的复数的某些特定元素实际上是代数元素，斯塔克单位。在数域的相对阿贝尔扩张的情况下，猜想进一步预言，这些元素，这将是唯一定义的分析数据，直到选择一个单位根，产生阿贝尔扩张的数域。斯塔克猜想是代数数论的核心问题之一。这项调查涉及审查与这些证据有关的一些问题，继续和扩大调查人员先前的工作。 本文的研究属于数论中的一般数学领域福尔斯。 数论有其历史根源，在研究整个数字，解决这样的问题，如那些处理整除一个整数由另一个。它是数学中最古老的分支之一，几个世纪以来，人们纯粹出于美学的原因而追求它。然而，在过去的半个世纪，它已成为一个不可或缺的工具，在各种 在诸如数据传输和处理以及通信系统的领域中的应用。

Algebraic number theory concern the study of the fields obtained by adjoining an algebraic number to the field of rational numbers. The Stark Conjectures postulate a deep connection between special values of the analytically defined L-series of such a number field and algebraic properties of the field. In particular, these Conjectures predict that certain specific elements of the complex numbers obtained from the L-series are actually algebraic elements, the Stark units. In the case of a relative abelian extension of number fields, the Conjectures further predict that these elements, which would be uniquely defined by the analytic data up to a choice of a root of unity, generate abelian extensions of the number field. The Stark Conjectures are among the central questions in algebraic number theory. This investigation is concerned with examining a number of questions regarding these conjectures, continuing and extending prior work of the investigators. This research falls into the general mathematical field of Number Theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.