# Topological behaviour of logarithmic invariants

## TL;DR

This paper investigates the algebraic structure of logarithmic modules associated with certain number field extensions, establishing their noetherian and torsion properties under specific conjectures, and deriving implications for logarithmic invariants.

## Contribution

It proves that the logarithmic module is noetherian and torsion under the Gross-Kuz'min conjecture, providing new insights into the structure of logarithmic invariants in number theory.

## Key findings

- Logarithmic module is noetherian over the Iwasawa algebra.
- Under the Gross-Kuz'min conjecture, the module is torsion.
- Derived local and global properties of logarithmic invariants.

## Abstract

Let $\ell$ be a rational prime number and $K$ a number field. We prove that the logarithmic module $X_{d}$ attached to a $\mathbb{Z}_{\ell}^{d}$-extension $K_{d}$ of $K$ is a noetherian $\Lambda_{d}$-module. Moreover, under the Gross-Kuz'min conjecture we prove that it is also torsion. We exploit this fact to deduce local and global information of the logarithmic invariants $\tilde{\mu}$ and $\tilde{\lambda}$ of $\mathbb{Z}_{\ell}$-extensions.