Multiplicative Galois structure of algebraic number fields

代数数域的乘法伽罗瓦结构

• 批准号: RGPIN-2014-05185
• 负责人: Weiss, Alfred
• 金额: $ 1.31万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635511
• 关键词: Multiplicative; Galois; structure; algebraic; number

The natural numbers have associated mysteries. For example, it has long been known that the Riemann zeta function, which is built from them by analytic means, 'knows' about the distribution of the prime numbers, in the sense that this date is encoded in it.The study of natural numbers leads to algebraic numbers, which also have their zeta functions and the mysteries of what they 'know'. There seems to be much more structural information of this kind which is only gradually being unravelled.The algebraic numbers also have symmetries, and it turns out that that the zeta functions 'know' a lot about these. In particular, their values at negative integers suffice to predict a very detailed such connection, as conjectured by Iwasawa (1969). This classical Main Conjecture was proved by Wiles (1990) and in a stronger 'equivariant' form by us (2011).The detailed connections concern arithmetic objects, like ideal class groups and unit groups, and are give in terms of algebraic objects, like cohomology groups and K-theory, which provide the links between the analytic and arithmetic sides. The present research is now more concerned with getting a better grasp of these algebraic objects, in hopes of finding arithmetic applications of our new understanding.

自然数有相关的奥秘。例如，人们早就知道，通过解析手段建立的黎曼 zeta 函数“知道”素数的分布，也就是说该日期被编码在其中。对自然数的研究导致了代数数，代数数也有它们的 zeta 函数和它们“知道”的奥秘。似乎有更多的此类结构信息正在逐渐被解开。代数数也具有对称性，事实证明 zeta 函数“了解”很多关于这些的信息。特别是，正如 Iwasawa (1969) 所推测的那样，它们的负整数值足以预测非常详细的这种联系。这个经典的主猜想由 Wiles (1990) 证明，并由 us (2011) 以更强的“等变”形式证明。详细的联系涉及算术对象，如理想类群和单位群，并以代数对象的形式给出，如上同调群和 K 理论，它们提供了分析和算术方面之间的联系。目前的研究现在更关注于更好地掌握这些代数对象，希望找到我们新理解的算术应用。