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 over $\tilde{k}$. In the present article, we discus classifying CM-fields $k$ which are decomposed completely at $p$ such that $X(\tilde{k})=0$.