Around equivariant iwasawa theory

自然数具有相关的性质。例如，人们早就知道，由自然数构建的黎曼zeta函数“知道”素数的分布，因为这些数据被编码在其中。对自然数的研究导致代数数，它们也有它们的zeta函数和它们“知道”的神秘。似乎有更多的这种结构信息，只是逐渐地被解开 代数数也有对称性，而zeta函数对这些对称性“知道”很多。特别是岩泽理论的主要猜想预言了一个非常详细的这种联系，并在1990年被A.Wiles证明。 最近，主要猜想的一种改进的“等变”形式被提出，其动机是关于这些对称性的具体问题。这个提议的主要目标是努力证明这个“等变”的主要猜想。 调查的困难所提出的这一尝试是一个次要目标的建议。最后，有一些工作正在进行中，只有一些相关的，我想继续，并希望完成。这涉及到群论的限制内核的投降在一个unramified非阿贝尔扩展代数数域。

The natural numbers have associated mysteries.For example,it has long been known that the Riemann zeta function,which is built from them,'knows' the distribution of the prime numbers,in the sense that this data is encoded in it.  The study of natural numbers leads to algebraic numbers,which also have their zeta functions and the mystery 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 the zeta functions 'know'a lot about these.In particular,the Main Conjecture of Iwasawa theory predicted a very detailed such connection which was proved by A.Wiles in 1990.  More recently,a refined 'equivariant' form of the Main Conjecture has been formulated,motivated by concrete problems about these symmetries.The main goal of this proposal is to work toward a proof of this 'equivariant'Main Conjecture.  The investigation of the difficulties raised by this attempt are a secondary goal of the proposal.Finally,there is some work in progress,only somewhat related,which I would like to continue with,and hopefully complete.This concerns the group theoretic limitations on the kernel of capitulation in an unramified non-abelian extension of algebraic number fields.