an:03972989
Zbl 0603.12003
Gold, Robert; Madan, Manohar
Galois representations of Iwasawa modules
EN
Acta Arith. 46, 243-255 (1985).
00149252
1986
j
11R23 11R29
Galois representations; Iwasawa modules; \(\mathbb Z_p\)-extension; \(\mathbb Z_p\)-fields of CM-type; Kida's formula
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 [\textit{C. W. Curtis} and \textit{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 \(V=\Hom_{\mathbb Z_p}(A^-_ L, \mathbb Q_p/\mathbb Z_p)\otimes_{\mathbb Z_p} \mathbb Q_p;\) which gives an alternative unified proof of Theorems 4 and 5 by \textit{K. Iwasawa} [T??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 [\textit{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 \textit{J. G. D'Mello} gave another proof of Kida's formula in Manuscr. Math. 41, 75--107 (1983; Zbl 0516.12012).]
Teruo Takeuchi (Niigata)
Zbl 0469.20001; Zbl 0468.12004; Zbl 0455.12007; Zbl 0516.12012