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

抄録

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 \cite{H2} 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.