On the triviality of the unramified Iwasawa modules of the maximal multiple $\mathbb{Z}_p$-extensions

TL;DR

This paper classifies CM-fields where the unramified Iwasawa module vanishes in the maximal multiple $Z_p$-extension, providing new insights and an alternative proof related to Greenberg's conjecture.

Contribution

It offers a classification of CM-fields with trivial unramified Iwasawa modules and presents an alternative proof for the conditions under which this occurs.

Findings

Classified CM-fields with $X( ilde{k})=0$ where $p$ splits completely.

Provided an alternative proof for the sufficient condition of triviality of $X( ilde{k})$.

Connected results to the generalized Greenberg conjecture.

Abstract

For a number field $k$ and an odd prime number $p$, we consider the maximal multiple $\mathbb{Z}_p$-extension $\tilde{k}$ of $k$ and the unramified Iwasawa module $X(\tilde{k})$, which is the Galois group of the maximal unramified abelian $p$-extension of $\tilde{k}$. In this article, we classify the CM-fields $k$ in which $p$ splits completely and for which $X(\tilde{k}) = 0$. In addition, we provide an alternative proof of the sufficient condition for $X(\tilde{k})=0$, based on the ideas of Minardi, Itoh, and Fujii in the study of the generalized Greenberg conjecture.