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Galois representations of Iwasawa modules. (English) Zbl 0603.12003
Let $$p$$ be an odd prime. The composite of a finite extension of $$\mathbb Q$$ with the unique $$\mathbb Z_p$$-extension over $$\mathbb Q$$ is called a $$\mathbb Z_p$$-field. Let $$L/K$$ be a finite Galois $$p$$-extension of $$\mathbb Z_p$$-fields of CM-type. Let $$G=\text{Gal}(L/K)$$ and $$A^-_ K$$ (resp. $$A_ L^-)$$ be the minus part of the $$p$$-class group of $$K$$ (resp. $$L$$). Assume $$\mu (A^-_ K)=0.$$
The authors determine the structure of $$A^-_ L$$ as a $$\mathbb Z_p[G]$$-module in the case $$G$$ is cyclic of order $$p$$ and in the case $$G$$ is cyclic of order $$p^ 2$$; where in the latter case they use Reiner’s classification of $$\mathbb Z_p[G]$$ indecomposables [C. W. Curtis and I. Reiner, Methods of representation theory, with applications to finite groups and orders. Vol. I (1981; Zbl 0469.20001), pp. 730–742]. When $$G$$ is a cyclic $$p$$-group, the structure of the subgroup of elements of order dividing $$p$$ in $$A^-_ L$$ as an $$\mathbb F_p[G]$$-module is also determined.
Moreover, using the result in the case where $$G$$ is cyclic of order $$p$$, by induction they determine the $$p$$-representation of $$G$$ on $$\mathrm{GL}(V)$$ for a finite Galois $$p$$-extension $$L/K$$ of $$\mathbb Z_p$$-fields of CM-type, where $$V=\operatorname{Hom}_{\mathbb Z_p}(A^-_ L, \mathbb Q_p/\mathbb Z_p)\otimes_{\mathbb Z_p} \mathbb Q_p;$$ which gives an alternative unified proof of Theorems 4 and 5 by K. Iwasawa [Tôhoku Math. J., II. Ser. 33, 263–288 (1981; Zbl 0468.12004)]. While in that paper using this result Iwasawa gave a different proof of Kida’s formula [Y. Kida, J. Number Theory 12, 519–528 (1980; Zbl 0455.12007)], the authors use this formula in the proof. [The second named author with J. G. D’Mello gave another proof of Kida’s formula in Manuscr. Math. 41, 75–107 (1983; Zbl 0516.12012).]

##### MSC:
 11R23 Iwasawa theory 11R29 Class numbers, class groups, discriminants
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