## Journal of the Mathematical Society of Japan

### Triviality in ideal class groups of Iwasawa-theoretical abelian number fields

Kuniaki HORIE

#### Abstract

Let $S$ be a non-empty finite set of prime numbers and, for each $p$ in $S$, let $\bm{Z}_p$ denote the ring of $p$-adic integers. Let $F$ be an abelian extension over the rational field such that the Galois group of $F$ over some subfield of $F$ with finite degree is topologically isomorphic to the additive group of the direct product of $\bm{Z}_p$ for all $p$ in $S$. We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers $l$ for which the $l$-class group of $F$ is nontrivial. This result implies our conjecture in [3] that the set of prime numbers $l$ for which the $l$-class group of $F$ is trivial has natural density $1$ in the set of all prime numbers.

#### Article information

Source
J. Math. Soc. Japan, Volume 57, Number 3 (2005), 827-857.

Dates
First available in Project Euclid: 14 September 2006

https://projecteuclid.org/euclid.jmsj/1158241937

Digital Object Identifier
doi:10.2969/jmsj/1158241937

Mathematical Reviews number (MathSciNet)
MR2139736

Zentralblatt MATH identifier
1160.11357

Subjects
Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R23: Iwasawa theory 11R27: Units and factorization

#### Citation

HORIE, Kuniaki. Triviality in ideal class groups of Iwasawa-theoretical abelian number fields. J. Math. Soc. Japan 57 (2005), no. 3, 827--857. doi:10.2969/jmsj/1158241937. https://projecteuclid.org/euclid.jmsj/1158241937

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