Multiplicative Galois structure of algebraic number fields

自然数带来了相关的谜团。例如，人们很早就知道，黎曼-泽塔函数是通过分析方法从它们建立起来的，从这个意义上说，它知道素数的分布，因为这个日期是编码在其中的。 对自然数的研究引出了代数数，代数数也有它们的Zeta函数和它们“知道”的奥秘。这类结构性信息似乎要多得多，这些信息只是在逐渐被解开。 这些代数数也有对称性，事实证明，Zeta函数对此“了解”很多。特别是，它们在负整数处的值足以预测非常详细的这种联系，正如岩泽(1969)猜测的那样。这一经典的主要猜想由Wiles(1990)证明，并由我们(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，它们提供了分析边和算术边之间的联系。目前的研究更关注于更好地掌握这些代数对象，希望找到我们新的理解的算术应用。

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.