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

Abstract

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.