On the Structure of Ideal Class Groups of CM - Fields dedicated to Professor K . Kato on his 50 th birthday

For a CM-field K which is abelian over a totally real number field k and a prime number p, we show that the structure of the χ-component AχK of the p-component of the class group of K is determined by Stickelberger elements (zeta values) (of fields containing K) for an odd character χ of Gal(K/k) satisfying certain conditions. This is a generalization of a theorem of Kolyvagin and Rubin. We define higher Stickelberger ideals using Stickelberger elements, and show that they are equal to the higher Fitting ideals. We also construct and study an Euler system of Gauss sum type for such fields. In the appendix, we determine the initial Fitting ideal of the non-Teichmüller component of the ideal class group of the cyclotomic Zp-extension of a general CM-field which is abelian over k.