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 p n -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 .