On cyclotomic norms and the conjectures of Leopoldt and Gross-Kuz'min

TL;DR

This paper employs $\

Contribution

It introduces a new perspective on cyclotomic norms and conjectures using $\

Findings

Provides numerical examples illustrating the logarithmic approach

Recalls and completes classical results in the area

Offers counter-examples to certain conjectures

Abstract

We use $\ell$-adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross generalized conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI.