Properties of mapping class groups related to Galois representations

设κ_0是有限代数数域，κ_∞是与κ_0邻接所有单位根得到的域，L是κ_∞的最大未分支阿贝尔扩张。设κ_1是通过将ζ_4和ζ_p对所有奇素数p邻接到κ_0而得到的域，考虑Gal1(κ_∞/κ_0)的子群g=Gal(κ_∞/κ_1)。本文研究了半群(L/κ_∞)的结构和具有这种g-作用的理想类群C_∞.对于半群(L/κ_∞)，我们证明了它作为完备群代数Z^[[g]]上的模，同构于Z^^[[g]]的可数拷贝数的直积.另一方面，理想类群C_∞是离散g-模。设κ_0为全实数，p为奇素数。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是{0}。)一般地，对于一个PRO-pg-模X和所有p次方单位根的群W(P)，设Hom(X，W(P))表示从X到W(P)的连续同态集。那么这自然是一个离散的g-模。对于C_∞(P)^-，我们证明了它与Hom(Π^∞_&lt；N=1&gt；Z_p[[g]]，W(P))同构。(Z_p[[g]]：环Z_p上的完备群代数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.