On class groups and Iwasawa modules of CM-fields

Abstract

Our aim in this paper is to give a clear view of what can be proved on the Fitting ideal of the Pontryagin dual of the minus class group of a CM-field as a Galois module. The class group we study is the classical full class group, and not the ray class group mod $T$ (we are particularly interested in the Teichmüller character component of the class group). We show that several theorems in the existing literature hold unconditionally, using recent groundbreaking results by Dasgupta, Kakde, Silliman, and Wang. In particular, we give a simple proof of the equivariant Iwasawa main conjecture including the case $p=2$, using their keystone theorems due to Dasgupta and Kakde in [Ann. of Math. (2) 197 (2023), 289–388] and due to Dasgupta, Kakde, Silliman and Wang in [arXiv:2310.16399]. We unconditionally compute the Fitting ideal of the Pontryagin dual of the minus class group of the cyclotomic ${\mathbb{Z}}_p$-extension of a CM-field, and also the Fitting ideal of $S$-ramified Iwasawa modules for totally real number fields. The numbers of minimal generators of these Iwasawa modules are also studied.