Properties of mapping class groups related to Galois representations

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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.