The commutativity of Galois groups of the maximal unramified pro-$p$-extensions over the cyclotomic $\mathbb{Z}_{p}$-extensions II

Abstract

Let $p$ be an odd prime number and $K_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of a Galois $p$-extension $K$ over an imaginary quadratic field. We consider the Galois group $\tilde{X}(K_{\infty})$ of the maximal unramified pro-$p$-extension of $K_{\infty}$. In this paper, under certain assumptions, we give certain $K$ such that $\tilde{X}(K_{\infty})$ is abelian. Also, we give an example such that a special value of the characteristic polynomial of the Iwasawa module of $K_{\infty}$ determines whether $\tilde{X}(K_{\infty})$ is abelian or not.