Some remarks on finite submodules of the unramified Iwasawa module of totally real fields

Abstract

Let $k$ be a number field and $p$ a prime number. It is conjectured by Greenberg that the Iwasawa $\lambda$- and $\mu$-invariants of the cyclotomic $\mathbf{Z}_{p}$-extension of $k$ always vanish if $k$ is totally real. In this article, we will discuss a weak version of Greenberg’s conjecture, and give results analogous to Greenberg’s and Ozaki’s results.