Properties of mapping class groups related to Galois representations

设κ 0是有限代数数域，κ ∞是与κ 0相连的所有单位根所构成的域，L是κ ∞的极大非分歧阿贝尔扩张。设κ_1是由对所有奇素数p，κ_4和κ_p与κ_0邻接而得到的域，并考虑Gal（κ_∞/κ_0）的子群g=Gal（κ_∞/κ_1）。本文研究了Gal（L/κ_∞）的结构和具有这种g-作用的κ_∞的理想类群C_∞，证明了Gal（L/κ_∞）作为完备群代数Z^^[[g]]上的模同构于Z^^[[g]]的可数个副本的直积.（Z^：有理整数环Z的profinite完备化。）另一方面，理想类群C ∞是离散g-模。设κ_0是全真实的，p是奇素数.设C_∞（p）^-表示C_∞（p）在复共轭作用下的负部分。（栗原的结果表明加号部分为{0}。）一般地，对于一个pro-pg-模X和所有p-幂单位根的群W（p），设Hom（X，W（p））表示从X到W（p）的连续同态的集合。那么这自然是一个离散g-模。对于C_∞（p）^-，我们证明了它同构于Hom（H ^∞_<N=1> Z_p[[g]]，W（p））.（Z_p[[g]]：p进整数环Z_p上的完全群代数g。

Let κ_0 be a finite algebraic number field, κ_∞ be the field obtained by adjoining to κ_0 all roots of unity, and L be the maximal unramified abelian extension of κ_∞. Let κ_1 be the field obtained by adjoining ζ_4 and ζ_p for all odd prime p to κ_0 and consider the subgroup g=Gal(κ_∞/κ_1) of Gal(κ_∞/κ_0). In this research, we have investigated the structure of Gal(L/κ_∞) and the ideal class group C_∞ of κ_∞ with this g-action.As for Gal(L/κ_∞), we have shown that it is, as modules over the completed group algebra Z^^^[[g]], isomorphic to the direct pruduct of countable number of copies of Z^^^[[g]]. (Z^^^: the profinite completion of the ring of rational integers Z.)On the other hand, the ideal class group C_∞ is a discrete g-module. Assume that κ_0 is totally real and p is an odd prime. Let C_∞ (p)^- denote the minus part of C_∞)(p) under the action of the complex conjugation. (A result of Kurihara indicates that the plus part is {0}.) In general, for a pro-p g-module X and the group W(p) of all p-powerth roots of unity, let Hom(X,W(p)) denote the set of continuous homomorphisms from X to W(p). Then this is naturally a discrete g-module. As for C_∞(p)^-, we have shown that it is isomorphic to Hom(Π^∞_<N=1> Z_p[[g]], W(p)). (Z_p[[g]] : the completed group algebra g over the ring of p-adic integers Z_p.