On tamely ramified Iwasawa modules for the cyclotomic $\mathbb {Z}_{p}$-extension of abelian fields

Abstract

Let $p$ be an odd prime, and $k_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of an abelian field $k$. For a finite set $S$ of rational primes which does not include $p$, we will consider the maximal $S$-ramified abelian pro-$p$ extension $M_{S} (k_{\infty})$ over $k_{\infty}$. We shall give a formula of the $\mathbb{Z}_{p}$-rank of $\mathrm{Gal}(M_{S} (k_{\infty})/k_{\infty})$. In the proof of this formula, we also show that $M_{\{q\}} (k_{\infty})/L(k_{\infty})$ is a finite extension for every real abelian field $k$ and every rational prime $q$ distinct from $p$, where $L(k_{\infty})$ is the maximal unramified abelian pro-$p$ extension over $k_{\infty}$.