Fitting Ideals of Iwasawa Modules and of the Dual of Class Groups

Abstract

In this paper we study some problems related to a refinement of Iwasawa theory, especially questions about the Fitting ideals of several natural Iwasawa modules and of the dual of the class groups, as a sequel to our previous papers [8], [3]. Among other things, we prove that the annihilator of $\mathbb{Z}_{p}(1)$ times the Stickelberger element is not in the Fitting ideal of the dualized Iwasawa module if the $p$-component of the bottom Galois group is elementary $p$-abelian with $p$-rank $\geq 4$. Our results can be applied to the case that the base field is $\mathbb{Q}$.