On the triviality of the unramified Iwasawa modules of the maximal multiple ${\mathbb{Z}}_p$-extensions

Abstract

For a number field $k$ and an odd prime number $p$, we consider the maximal multiple ${\mathbb{Z}}_p$-extension $\tilde{k}$ of $k$ and the unramified Iwasawa module $X(\tilde{k})$, which is the Galois group of the maximal unramified abelian $p$-extension of $\tilde{k}$. In this article, we classify the CM-fields $k$ in which $p$ splits completely and for which $X(\tilde{k})=0$. To this end, we provide a relation between the trivialities of $X(\tilde{k})$ and that of the $p$-split Iwasawa modules associated with certain anticyclotomic ${\mathbb{Z}}_p$-extensions. In addition, we provide an alternative proof of the sufficient condition for $X(\tilde{k})=0$, based on the study of the generalized Greenberg conjecture.