## Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field.(English)Zbl 1011.11072

Author’s summary: Let $$P$$ be the set of all prime numbers. For each $$l \in P$$ let $$\mathbb Z_t$$ denote the ring of $$l$$-adic integers and, given any algebraic number filed $$F$$, let $$C_F(l)$$ denote the $$l$$-primary component of the ideal class group of $$F$$, and $$C^{-}_F(l)$$ the $$l$$-primary component of the kernel of the norm map from the ideal class group of $$F$$ to that of the maximal real subfield of $$F$$. Let $$S$$ be a non-empty finite subset of $$P$$. Denote by $$\mathbb Q$$ the rational field, by $$\mathbb C$$ the complex field, and by $$\mathbb Q^S$$ the abelian extension over $$\mathbb Q$$ in $$\mathbb C$$ such that the Galois group of $$\mathbb Q^S/ \mathbb Q$$ is topologically isomorphic to the additive group of the direct product $$\prod_{l \in S} \mathbb Z_t$$. Let $$K$$ be an imaginary finite extension over $$\mathbb Q^S$$ in $$\mathbb C$$ such that $$K/\mathbb Q$$ is an abelian extension. In this paper, the author first proves that each of certain arithmetic progressions contains at most finitely many $$l \in P$$ for which $$C^{-}_K (l)$$ are nontrivial, and that the natural density of $$P$$ of the set of all $$l \in P$$ with $$C^{-}_K(l)=1$$ is equal to $$1$$. These results have been shown by L. C. Washington for the basic case where $$|S |=1$$ [Math. Ann. 214, 177-193 (1975; Zbl 0302.12007)]. The author next devotes much part of the paper to proving that, if $$S$$ consists only of an odd prime $$p$$, then $$C_{\mathbb Q^S}(l)= 1$$ holds for every prime number $$l \geq \frac{3}{2} p^2 \log p$$ which is a primitive root modulo $$p^2$$.
In the last part of the paper, some additional results are given along with some related problems: among others, the last proposition states that, for any positive integer $$n$$, the class number of the cyclotomic field of $$3^n$$th roots of unity is relatively prime to every $$l\in P$$ with $$l\equiv 2$$ or $$5 \pmod 9$$; while, to the author, it seems very likely that the natural density in $$P$$ of the set of all $$l\in P$$ with $$C_K(l) =1$$ is equal to $$1$$, and it seems a particularly interesting problem to know (in the case of $$|S |$$) whether the ideal class group of $$\mathbb Q^S$$ is trivial, infinite, or of finite order greater than $$1$$.

### MSC:

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

### Keywords:

ideal class group; infinite abelian extension

Zbl 0302.12007
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