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On the ideal class groups of real abelian number fields. (English) Zbl 0665.12003
This paper presents a new relationship between the ideal class group $$A$$ and the unit group $$E$$ of a real abelian field $$K$$. In fact, the author finds annihilators of the $$p$$-class group $$(A)_ p$$ related to the structure of $$W=E/C$$ ($$C=$$ circular units), and in this way obtains smooth statements about relations between annihilators of $$(A)_ p$$ and $$(W)_ p$$.
Let $$m$$ denote the conductor of $$K$$, let $$\zeta_ n$$ be a primitive $$n$$th root of unity, and put $$\Delta =\text{Gal}(K/\mathbb Q)$$. Define the group $$C$$ of circular units of $$K$$ to consist of all units of the form $$f(1)$$, where $f(X)=\pm \prod^{j}_{i=1}\prod^{m-1}_{k=1}(X^ i-\zeta^ k_ m)^{a_{ik}} \in K(X)$ with $$a_{ik}\in\mathbb Z$$ and $$j\geq 1$$ (the author points out that this definition differs from the known and may produce a larger group). Consider first a field $$L=K(\zeta_ q)$$, where $$q$$ is a sufficiently large prime that splits completely in $$K$$. Starting with a unit $$\delta =f(1)\in C$$ the author observes that a generating automorphism of $$L/K$$ maps certain elements $$\alpha\in L$$ to $$f(\zeta_ q)\alpha$$. On studying the prime factorization of $$\alpha$$ he infers, back to $$K$$, a decomposition $(N_{L/K} (\alpha)) = R^ b \prod_{\sigma\in\Delta} \sigma^{-1} (Q)^{r_\sigma}, \tag{$$*$$}$ valid under certain conditions, where $$R$$ and $$Q$$ are ideals of $$K$$, $$Q$$ a prime above $$q$$, and $$b$$ is a divisor of $$q-1$$. The exponents $$r_{\sigma}$$ have a deep connection to the units of $$K$$. Here a crucial role is played by a local-global theorem depending on Chebotarev’s density theorem.
From (*) it follows that if $$R^ b$$ is principal then $$\sum_{\sigma \in \Delta}r_{\sigma}\sigma^{-1}$$ annihilates the ideal class $${\mathcal C}$$ containing $$Q$$. By use of the properties of $$r_{\sigma}$$ this implies the following results: If $$b=p^ n$$ is the exponent of $$(A)_ p$$ and if $$\delta$$ satisfies, for all $$\sigma\in \Delta$$, a congruence $$\sigma (\delta)\equiv \delta^{c_{\sigma}}\bmod E^{p^ n}$$ with $$p\nmid c_{\sigma}$$, then $$\sum_{\sigma \in \Delta}c_{\sigma}\sigma^{-1}$$, multiplied by a certain integer $$2d$$, annihilates all classes $${\mathcal C}\in (A)_ p$$ containing an ideal $$Q$$. If fact, $$d$$ depends on $$| \delta |$$ in a simple way.
Suppose that $$p\nmid [K:\mathbb Q]$$ and $$\chi$$ is a non-trivial $$p$$-adic valued character of $$\Delta$$, with the corresponding idempotent $$e_{\chi}$$ of $$\mathbb Z[\Delta]$$. Then, by taking $$c_{\sigma}=\chi (\sigma)$$ one proves that $$e_{\chi}(A)_ p$$ is annihilated by the exact exponent $$p^ a$$ of $$e_{\chi}(W)_ p$$. As corollary it follows that if $$K$$ is a subfield of $$\mathbb Q(\zeta_ p+\zeta_ p^{-1})$$, then every annihilator of $$(W)_ p$$ annihilates $$(A)_ p$$. Finally, a generalization of the above ideas leads to the beautiful theorem:
If $$p\nmid [K:\mathbb Q]$$ and if $$\theta\in\mathbb Z[\Delta]$$ annhilates $$(W)_ p$$, then $$2\theta$$ annihilates $$(A)_ p$$.

MSC:
 11R18 Cyclotomic extensions 11R29 Class numbers, class groups, discriminants 11G16 Elliptic and modular units 11R27 Units and factorization
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