Around equivariant iwasawa theory

自然数与神秘有关。例如，人们早就知道，由它们构建的黎曼ζ函数，“知道”质数的分布，从某种意义上说，这些数据被编码在其中。对自然数的研究导致了代数数的研究，而代数数也有它们的ζ函数和它们所“知道”的奥秘。似乎有更多的这类结构性信息只是逐渐被揭示出来。代数数也有对称性，结果是，函数对这些有很多了解。特别是Iwasawa理论的主猜想预测了一个非常详细的这种联系，并由a . wiles在1990年证明。最近，在这些对称性的具体问题的推动下，主猜想的一个精炼的“等变”形式被公式化了。这个提议的主要目标是努力证明这个“等变”主猜想。调查这一尝试所引起的困难是本建议的次要目标。最后，有一些工作正在进行中，只是有些相关，我想继续下去，并希望完成。讨论了代数数域非阿贝尔扩展中投降核的群理论限制。

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.