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

TL;DR

This paper investigates the structure of Galois groups of maximal pro-p extensions with restricted ramification over cyclotomic Z_p-extensions, focusing on whether these groups are non-abelian free pro-p groups in specific number field cases.

Contribution

It provides new insights into the non-abelian free pro-p nature of Galois groups over cyclotomic extensions for imaginary quadratic and totally real fields.

Findings

Galois group is non-abelian free pro-p for imaginary quadratic fields with S empty.

Galois group structure analyzed for totally real fields with non-empty S.

Results contribute to understanding ramification and Galois group structure in Iwasawa theory.

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 $S = \emptyset$ (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 $S \neq \emptyset$.