Primary components of the ideal class group of an Iwasawa-theoretical abelian number field

Abstract

Let S be a non-empty finite set of prime numbers, and 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 the p-adic integer rings for all p in S. Let m be a positive integer that is neither congruent to 2 modulo 4 nor divisible by any prime number outside S but divisible by all prime numbers in S. Let Ω denote the composite of pⁿ-th cyclotomic fields for all p in S and all positive integers n. In our earlier paper [3], it is shown that there exist only finitely many prime numbers l for which the l-class group of F is nontrivial and the m-th cyclotomic field contains the decomposition field of l in Ω. We shall prove more precise results providing us with an effective upper bound for a prime number l such that the l-class group of F is nontrivial and that the m-th cyclotomic field contains the decomposition field of l in Ω.