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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 [TĂ´hoku 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).]

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants
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