Let be a non-empty finite set of prime numbers and, for each in , let denote the ring of -adic integers. Let be an abelian extension over the rational field such that the Galois group of over some subfield of with finite degree is topologically isomorphic to the additive group of the direct product of for all in . We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers for which the -class group of is nontrivial. This result implies our conjecture in  that the set of prime numbers for which the -class group of is trivial has natural density in the set of all prime numbers.
Kuniaki HORIE. "Triviality in ideal class groups of Iwasawa-theoretical abelian number fields." J. Math. Soc. Japan 57 (3) 827 - 857, July, 2005. https://doi.org/10.2969/jmsj/1158241937