Ideal Class Groups of Iwasawa-Theoretical Abelian Extensions Over the Rational Field

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field Q, are contained in the complex field C. Let P be the set of all prime numbers. For any algebraic number field F, let CF denote the ideal class group of F and, writing F+ for the maximal real subfield of F, let C F − denote the kernel of the norm map from CF to the ideal class group of F+; for each l ∈ P, let CF(l) denote the l-class group of F, that is, the l-primary component of CF, and let C F − ( l ) denote the l-primary component of C F − . Furthermore, for each l ∈ P, we denote by Zl the ring of l-adic integers.