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 ${\mathbf{Z}}_p$ 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 ${\mathbf{Z}}_p$ 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.