On the ideal class groups of the maximal cyclotomic extensions of algebraic number fields.(English)Zbl 1332.11096

Let $$k_0\subset \mathbb{C}$$ be a number field. Let $$k_{\infty}$$ be the maximal cyclotomic extension of $$k_0$$ and $$C_{\infty}$$ its class group; $$C_{\infty}$$ is a direct sum of its $$p$$-primary components $$C_{\infty}(p)$$ for all prime number $$p$$: $$C_{\infty}=\bigoplus_pC_{\infty}(p)$$. Let $$k_1/k_0$$ be the subextension of $$k_{\infty}/k_0$$ obtained by adjoining a primitive $$4$$-th root of unity and all primitive $$l$$-th root of unity, where $$l$$ is an odd prime number. Put $$\mathfrak{g}=\operatorname{Gal}(k_{\infty}/k_1)$$, then $$\mathfrak{g}$$ acts naturally on $$C_{\infty}$$. The purpose of the paper is to investigate the structure of $$C_{\infty}$$ as a $$\mathfrak{g}$$-module.
From now on suppose that $$k_0$$ is totally real and $$p$$ is an odd prime number. Let $$k_{\infty}^{+}$$ denote the maximal totally real subfield of $$k_{\infty}$$. The complex conjugation $$\rho$$, which is a generator of $$\operatorname{Gal}(k_{\infty}/k_{\infty}^{+})$$, acts on $$C_{\infty}(p)$$. Let $$C_{\infty}(p)^{\pm}=\{c\in C_{\infty}(p), \rho(c)=\pm c\}$$. Then $$C_{\infty}(p)=C_{\infty}(p)^{+} \oplus C_{\infty}(p)^{-}$$, as discrete $$\mathfrak{g}$$-modules. According to a result of M. Kurihara [J. Eur. Math. Soc. (JEMS) 1, No. 1, 35–49 (1999; Zbl 0949.11055)], $$C_{\infty}(p)^{+}=\{0\}$$. Therefore $$C_{\infty}(p)=C_{\infty}(p)^{-}$$. Let $$W(p)$$ be the group of all $$p$$-power roots of unity. For a pro-$$p$$ $$\mathfrak{g}$$-module $$X$$, $$\operatorname{Hom}(X, W(p))$$ denotes the set of continuous homomorphisms from $$X$$ to $$W(p)$$. The group $$\mathfrak{g}$$ acts on $$\operatorname{Hom}(X, W(p))$$ by: $$\sigma(f)(x)=\sigma(f(\sigma^{-1}(x))$$, so that $$\operatorname{Hom}(X, W(p))$$ is a discrete $$\mathfrak{g}$$-module. Let $$\mathcal{A}_p$$ be the completed group algebra of $$\mathfrak{g}$$ over $$\mathbb{Z}_p$$ (the ring of $$p$$-adic integers), which is a pro-$$p$$ $$\mathfrak{g}$$-module. Let $$\mathfrak{C}_p=\operatorname{Hom}(\mathcal{A}_p, W(p))$$. The main result of the paper is: as discrete $$\mathfrak{g}$$ modules, $$C_{\infty}(p)\simeq \bigoplus_{N=1}^{\infty}\mathfrak{C}_p$$. The proof uses a characterization of the pro-$$p$$ $$\mathfrak{g}$$ module $$\prod_{N=1}^{\infty}\mathcal{A}_p$$ in terms of the solvability of embedding problems.

MSC:

 11R18 Cyclotomic extensions 11R23 Iwasawa theory

Zbl 0949.11055
Full Text:

References:

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