Class fields of abelian extensions of Q
Class fields of abelian extensions of Q
复制标题
DOI：
10.1007/bf01388599
复制
发表时间：
1984-06
影响因子：
3.1
通讯作者：
B. Mazur;A. Wiles
中科院分区：
文献类型：
--
作者：
B. Mazur;A. Wiles
设p是一个奇素数。本文的目的是证明Kubota-Leopoldt的p-adic L-函数的零点等于作用在有限维Qv-向量空间上的某些”算术定义”算子的特征值。这些Qv-向量空间定义岩泽在限制某些组件的理想类组塔的分圆数域。岩泽的向量空间和p-adic L-函数之间的联系是由他证明的（关于域Q上素数p的”主要猜想”，参见[1]）。章正如读者将看到的，如果我们考虑到已经知道的东西--由于库默、斯蒂克尔伯格、岩泽、费雷罗-华盛顿的工作，我们已经知道了很多--我们的问题可以归结为构造Q的阿贝尔扩张的足够多的类域，同时密切关注伽罗瓦对类域的作用。广义地说，我们面临的问题属于显式类域理论（对于Q的阿贝尔扩张）。但是，“显式类场理论”的标签可能会产生错误的印象。首先，我们可以问，在什么意义上我们的建构是外显的？我们获得我们的
Letp be an odd prime number. The object of this paper is to show that the zeroes of the p-adic L-functions of Kubota-Leopoldt are equal to the eigenvalues of certain" arithmetically-defined" operators acting on finite-dimensional Qv-vector spaces. These Qv-vector spaces were defined by Iwasawa in terms of limits of certain components of the ideal class groups of towers of cyclotomic number fields. The connection between Iwasawa's vector spaces and the p-adic L-function was conjectured by him (the" main conjecture" for the prime p over the field Q, cf. Chap. 1, w 1).As the reader will see, if we take into account what is already known,-and much is known, thanks to the work of Kummer, Stickelberger, Iwasawa, Ferrero-Washington,-our problem can be reduced to the construction of enough classfields of abelian extensions of Q, while keeping close tabs on the action of Galois on the classfields. Broadly speaking, the problem we face comes under the rubric of explicit classfield theory (for abelian extensions of Q). But a wrong impression might arise from the label" explicit class field theory". First, we may ask in what sense is our construction explicit? We obtain our