Stickelberger ideals and Fitting ideals of class groups for abelian number fields
Let \(K\) be an imaginary, abelian number field, denote its Galois group by \(G\) and let \(R_K^-\) denote the minus part of \(\mathbb Z [\frac 12][G]\). To investigate the Galois structure of the minus part \((Cl_K')^-\) of the class group of \(K\) (and neglecting the \(2\)-part), one is interested in its structure as an \(R_K^-\)-module and especially in its annihilator, which contains the Fitting ideal as well as some Stickelberger ideal \(\Theta_K^-\) (which need not coincide with the Stickelberger ideal as defined by Iwasawa and Sinnott, if \(K\) is not a cyclotomic field). The main result of this paper shows that these two ideals coincide, i.e. \[ \text{Fitt}_{R_K^-} \bigl( (Cl_K')^- \bigr) = \Theta_K^-, \] as conjectured by the first author [J. Reine Angew. Math. 561, 39--86 (2003; Zbl 1056.11063)] and already proved in special cases. The method of proof also applies to show that the Stickelberger element of relative abelian extensions \(K/k\), where \(K\) is a CM-field, belongs to the Fitting ideal (Theorem 0.3). For the proof, one localizes at an odd prime \(p\) and considers the components of tame characters. Depending on whether \(p\) is tamely or wildly ramified, resp., the proofs are given in sections 3 and 4.
scientific article; zbMATH DE number 5939566
scientific article