539 On the Structure of Ideal Class Groups of CM Fields dedicated to Professor K Kato on his 50th birthday Masato Kurihara Received November 30 2002 Revised June 30 2003 Abstract 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 component A K 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 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 Teichmu ller component of the ideal class group of the cyclotomic Zp extension of a general CM field which is abelian over k 0 Introduction It is well known that the cyclotomic units give a typical example of Euler systems Euler systems of this type were systematically investigated by Kato 8 Perrin Riou 14 and in the book by Rubin 18 In this paper we propose to study Euler systems of Gauss sum type which are not Euler systems in the sense of 18 We construct an Euler system in the multiplicative groups of CM fields which is a generalization of the Euler system of Gauss sums and generalize a structure theorem of Kolyvagin and Rubin for the minus class groups of imaginary abelian fields to general CM fields The aim of this paper is to prove the structure theorem Theorem 0 1 below and we do not pursue general results on the Euler systems of Gauss sum type in this paper One of very deep and remarkable works of Kato is his construction Documenta Mathematica Extra Volume Kato 2003 539 563 540 Masato Kurihara of the Euler system which lies in H 1 T for a Zp representation T associated to a modular form We remark that we do not have an Euler system of Gauss sum type in H 1 T but fixing n 0 we can find an Euler system of Gauss sum type in H 1 T pn which will be studied in our forthcoming paper We will describe our main result Let k be a totally real number field and K be a CM field containing k such that K k is finite and abelian We consider an odd prime number p and the p primary component AK ClK Zp of the ideal class group of K Suppose that p does not divide K k Then AK L is decomposed into AK A K where A K is the component which is an O module where O Zp Image for the precise definition see 1 1 and ranges over Qp conjugacy classes of Qp valued characters of Gal K k see also 1 1 For k Q and K Q p the cyclotomic field of p th roots of unity Rubin in 17 described the detail of Kolyvagin s method 10 Theorem 7 and determined the structure of A Q p as a Zp module for an odd by using the Euler system of Gauss sums Rubin 17 Theorem 4 4 We generalize this result to arbitrary CM fields In our previous paper 11 we proposed a new definition of the Stickelberger ideal In this paper for certain CM fields we define higher Stickelberger ideals which correspond to higher Fitting ideals In 3 using the Stickelberger elements of fields containing K we define the higher Stickelberger ideals i K Zp Gal K k for i 0 cf 3 2 Our definition is different from Rubin s Rubin defined the higher Stickelberger ideal using the argument of Euler systems We do not use the argument of Euler systems to define our i K We remark that our i K is numerically computable since the Stickelberger elements are numerically computable We consider the component i K We study the structure of the component A K as an O module We note that p is a prime element of O because the order of Image is prime to p Theorem 0 1 We assume that the Iwasawa invariant of K is zero cf Proposition 2 1 and is an odd character of Gal K k such that 6 where is the Teichmu ller character giving the action on p and that p 6 1 for every prime p of k above p Suppose that A K O pn1 O pnr with 0 n1 nr Then for any i with 0 i r we have pn1 nr i i K and i K 1 for i r Namely M A K i K i 1 K i 0 In the case K Q p and k Q Theorem 0 1 is equivalent to Theorem 4 4 in Rubin 17 Documenta Mathematica Extra Volume Kato 2003 539 563 The Structure of Ideal Class Groups 541 This theorem says that the structure of A K as an O module is determined by the Stickelberger elements Since the Stickelberger elements are defined from the partial zeta functions we may view our theorem as a manifestation of a very general phenomena in number theory that zeta functions give us information on various important arithmetic objects In general for a commutative ring R and an R module M such that f Rm Rr M 0 is an exact sequence of R modules the i th Fitting ideal of M is defined to be the ideal of R generated by all r i r i minors of the matrix corresponding to f for i with 0 i r If i r it is defined to be R For more details see Northcott 13 Using this terminology Theorem 0 1 can be simply stated as Fitti O A K i K for all i 0 The proof of Theorem 0 1 is divided into two parts We first prove the inclusion Fitti O A K i K To do this we need to consider a general CM field which contains K Suppose that F is a CM field containing K such that F k is abelian and F K is a p extension Put RF Zp Gal F k For a character satisfying the conditions in Theorem 0 1 we consider RF O Gal F K and A F AF RF RF where Gal K k acts on O via For the component F RF of the Stickelberger element of F cf 1 2 we do not know whether F Fitt0 RF A F always holds or not cf Popescu 15 for function fields But we will show in Corollary 2 4 that the dual version of this statement holds namely F Fitt0 R 1 A F F where RF RF is the map induced by 7 1 for Gal F k and A F is the Pontrjagin dual of A F We can also determine the right hand side Fitt0 R 1 A F In …