On the mu and lambda invariants of the logarithmic class group

TL;DR

This paper investigates the behavior of the logarithmic class group in $ ext{Z}_ ext{l}$-extensions of number fields, establishing a formula for its size growth and exploring relations with classical invariants, supported by numerical examples.

Contribution

It introduces a formula for the logarithmic class group's exponent in $ ext{Z}_ ext{l}$-extensions assuming the Gross-Kuz'min conjecture and relates logarithmic invariants to classical ones.

Findings

The logarithmic class group's order follows a linear formula in the extension level.

Relations between classical and logarithmic invariants are established.

Numerical examples illustrate the theoretical results.

Abstract

Let $\ell$ be a rational prime number. Assuming the Gross-Kuz'min conjecture along a $\Zl$-extension $K\_{\infty}$ of a number field $K$, we show that there exist integers $\mut$, $\lat$ and $\widetilde{\nu}$ such that the exponent $\tilde{e}\_{n}$ of the order $\ell^{\tilde{e}\_{n}}$ of the logarithmic class group $\Clog{n}$ for the $n$-th layer $K\_{n}$ of $K\_{\infty}$ is given by $\tilde{e}\_{n}=\widetilde{\mu}\ell^{n}+\widetilde{\lambda} n + \widetilde{\nu}$, for $n$ big enough. We show some relations between the classical invariants $\mu$ and $\lambda$, and their logarithmic counterparts $\mut$ and $\lat$ for some class of $\Zl$-extensions. Additionally, we provide numerical examples for the cyclotomic and the non-cyclotomic case.