Integrality of Stickelberger elements

斯蒂克伯格元素的完整性

• 批准号: 399131371
• 负责人: Professor Dr. Andreas Nickel
• 金额: --
• 依托单位: Professur für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/399131371?language=en
• 关键词: Integrality; Stickelberger; elements

The class group of a number field is one of its most interesting arithmetic invariants. The analysis of its structure is a central theme of research in number theory for a long time. It is conjectured that certain analytic objects act on and indeed annihilate the class group, giving some constraints on its structure. In this project we like to investigate basic properties of the occurring analytic objects which are necessary to enable an action on the class group. These so-called Stickelberger elements should then be compared to p-adic L-series - an entirely different kind of analytic object. From this we like to deduce in as many cases as possible that the Stickelberger elements indeed annihilate the class group. If possible we even like to prove new cases of the considerably stronger "equivariant Tamagawa number conjecture" which also predicts a close relation between certain analytic and arithmetic objects.For this we have to study the functorial properties of Stickelberger elements and of the occurring algebraic structures. In particular, the behaviour in infinite towers of number fields will play a pivotal role in order to apply methods of Iwasawa theory. This should put us in a position to deduce new cases of the aforementioned conjectures. However, Stickelberger elements are only interesting for totally complex number fields. Examining appropriate conjectures for not necessarily totally complex number fields might then be a further topic of research.

数域的类组是它最有趣的算术不变量之一。对其结构的分析一直是数论研究的中心课题。据推测，某些分析对象作用于并实际上消灭了阶级群体，从而对其结构给予了一些约束。在这个项目中，我们想要研究发生的分析对象的基本属性，这是在类组上启用操作所必需的。然后，这些所谓的斯蒂克尔伯格元素应该被比作p级的L系列--一种完全不同的分析对象。从这一点上，我们愿意在尽可能多的情况下推断，斯蒂克尔伯格分子确实消灭了阶级群体。如果可能，我们甚至愿意证明更强的“等变玉马川数猜想”的新情况，该猜想也预测了某些分析对象和算术对象之间的密切关系。为此，我们必须研究Stickelberger元和所发生的代数结构的函数性。特别是，为了应用岩泽理论的方法，数域无限塔中的行为将发挥关键作用。这应该使我们能够推断出上述猜测的新案例。然而，Stickelberger元素只对完全复杂的数域感兴趣。然后，检查不一定是完全复数域的适当猜想可能是进一步研究的主题。