On generators of unramified Iwasawa module over an imaginary quadratic field

Let $p$ be an odd prime number and $k_p$ be the maximal pro-$p$ extension unramified outside $p$ of an imaginary quadratic field $k$. Let $\tilde{k}$ be the maximal multiple ${\mathbb{Z}}_p$ extension over $k$ and $M_{\tilde{k}}$ be the maximal abelian pro-$p$ extension unramified outside $p$ over $\tilde{k}$. In this paper, we give an explicit description of a generating system of $\mathrm{G}\mathrm{a}\mathrm{l}(M_{\tilde{k}}\mathrm{/}\tilde{k})$ and study its structure. As an application, we apply it to describe the unramified Iwasawa module $X_k$ for $k$.