Stark Units

斯塔克单位

• 批准号: 9624057
• 负责人: David Dummit
• 金额: $ 3.19万
• 依托单位: University of Vermont & State Agricultural College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9624057&HistoricalAwards=false
• 关键词: Stark; Units

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.

代数数论关注的是通过将代数数与有理数域相连而获得的域的研究。斯塔克猜想假设这样一个数域的解析定义的 L 级数的特殊值与该域的代数性质之间存在深刻的联系。特别是，这些猜想预测从 L 级数获得的复数的某些特定元素实际上是代数元素，即斯塔克单位。在数域的相对阿贝尔扩展的情况下，猜想进一步预测这些元素将生成数域的阿贝尔扩展，这些元素将由分析数据唯一定义，直至单位根的选择。斯塔克猜想是代数数论的核心问题之一。这项调查旨在审查有关这些猜想的一些问题，继续并扩展调查人员之前的工作。 这项研究属于数论的一般数学领域。 数论的历史根源在于对整数的研究，解决诸如一个整数能否被另一个整数整除等问题。它是数学最古老的分支之一，几个世纪以来纯粹出于美学原因而被人们所追求。然而，在过去的半个世纪中，它已成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。