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.
Categories: math.NT