On finiteness properties of the unramified Iwasawa module of a $\mathbb{Z}_p$-extension with restricted ramification

Abstract

In the present article, we consider $\mathbb{Z}_p$-extensions defined by restrictions on ramifications at $p$-adic primes. We study the finiteness, the existence of non-trivial finite submodules, and the triviality of the unramified Iwasawa modules of these $\mathbb{Z}_p$-extensions. In addition, we give a new sufficient condition for Greenberg's Generalized Conjecture with some non-trivial concrete examples.