Abelian $p$-extensions with restricted $p$-ramification and the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q} (\sqrt{q})$

Abstract

We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted $p$-ramification. Let $K$ be an algebraic number field, and $p$ a prime number which splits into two distinct primes $\mathfrak{p}$, $\mathfrak{p}'$ in $K$. Assume that $K$ and $p$ satisfies several (somewhat strict) conditions. Let $K_\infty /K$ be the cyclotomic $\mathbb{Z}_p$-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module $X (K_\infty)$ by using abelian $p$-extensions unramified outside $\mathfrak{p}$. We give a sufficient condition for $|X (K_\infty)|$ to be finite in terms of such extensions. We also give a similar criterion for $X (K_\infty)$ to be finitely generated over $\mathbb{Z}_p$. In the latter part of the present paper, we consider the case where $k = \mathbb{Q} (\sqrt{q})$ with an odd prime number $q$ and apply our results to the cyclotomic $\mathbb{Z}_2$-extension $k_\infty /k$. We give a necessary and sufficient condition for $X (k_\infty)$ to be cyclic over $\mathbb{Z}_2$, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.