Journal of the Mathematical Society of Japan

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

Kuniaki HORIE

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Let S be a non-empty finite set of prime numbers and, for each p in S , let 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 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

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

First available in Project Euclid: 14 September 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

abelian number field ideal class group Iwasawa theory class number formula


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.

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