On the structure of the Galois group of the maximal pro-$p$ extension with restricted ramification over the cyclotomic ${\mathbb{Z}}_p$-extension

Abstract

Let $k_{\infty }$ be the cyclotomic ${\mathbb{Z}}_p$-extension of an algebraic number field $k$. We denote by $S$ a finite set of prime numbers which does not contain $p$, and $S(k_{\infty })$ the set of primes of $k_{\infty }$ lying above $S$. In the present paper, we will study the structure of the Galois group ${\mathcal{X}}_S(k_{\infty })$ of the maximal pro-$p$ extension unramified outside $S(k_{\infty })$ over $k_{\infty }$. We mainly consider the question whether ${\mathcal{X}}_S(k_{\infty })$ is a non-abelian free pro-$p$ group or not. In the former part, we treat the case when $k$ is an imaginary quadratic field and $S=\varnothing$ (here $p$ is an odd prime number which does not split in $k$). In the latter part, we treat the case when $k$ is a totally real field and $S\ne \varnothing$.

20 pages, changed several places, added sentences and references