Frobenius and non logarithmic ramification

TL;DR

This paper introduces the logarithmic Frobenius in the context of logarithmic ramification, showing its relation to classical Frobenius and providing a new perspective on ramification theory in number fields.

Contribution

It constructs the logarithmic Frobenius and demonstrates its equivalence to the classical Frobenius when usual and logarithmic ramification coincide.

Findings

Logarithmic Frobenius generalizes classical Frobenius in ramification theory.

Logarithmic and classical Frobenius coincide under certain ramification conditions.

Provides a new framework for understanding ramification in number field extensions.

Abstract

A $\ell$-extension is said logarithmically unramified if it is locally cyclotomic. The purpose of this article is to explain the construction of the logarithmic Frobenius, which plays the role usually played by the classical Frobenius, but in the context of the logarithmic ramification.The interesting point is that usual and logarithmic Frobenius coincide when usual and logarithmic ramification are the same.