On a Quotient of the Unramified Iwasawa Module over an Abelian Number Field, Ii

Let p be an odd prime number, k an imaginary abelian field containing a primitive p-th root of unity, and k∞/k the cyclotomic Zp-extension. Denote by L/k∞ the maximal unramified pro–p abelian extension, and by L′ the maximal intermediate field of L/k∞ in which all prime divisors of k∞ over p split completely. Let N/k∞ (resp. N ′/k∞) be the pro–p abelian extension generated by all p-power roots of all units (resp. p-units) of k∞. In the previous paper, we proved that the Zp-torsion subgroup of the odd part of the Galois group Gal(N ∩ L/k∞) is isomorphic, over the group ring Zp[Gal(k/Q)], to a certain standard subquotient of the even part of the ideal class group of k∞. In this paper, we prove that the same holds also for the Galois group Gal(N ′∩L′/k∞).