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.

数域的类群是其最有趣的算术不变量之一。对其结构的分析长期以来一直是数论研究的中心主题。据推测，某些分析对象作用于甚至消灭了类群，对其结构给予了一些限制。在这个项目中，我们希望研究出现的分析对象的基本属性，这些属性是在类组上启用操作所必需的。这些所谓的 Stickelberger 元素应该与 p 进 L 级数进行比较 - 一种完全不同类型的分析对象。由此，我们希望在尽可能多的情况下推断出斯蒂克伯格元素确实消灭了阶级群体。如果可能的话，我们甚至想证明更强大的“等变玉川数猜想”的新案例，它也预测某些分析和算术对象之间的密切关系。为此，我们必须研究斯蒂克伯格元素和出现的代数结构的函子性质。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想的新案例。然而，斯蒂克伯格元素仅对完全复杂的数字字段感兴趣。检查不一定完全复杂的数域的适当猜想可能是进一步的研究主题。