A formal approach "\`a la Neukirch" of $\ell$-adic class field theory

TL;DR

This paper demonstrates how Neukirch's abstract class field theory framework can be used to derive Jaulent's e1-adic class field theory, involving the definition of degree maps, G-modules, and valuations.

Contribution

It provides a formal derivation of e1-adic class field theory from Neukirch's framework, unifying two approaches in algebraic number theory.

Findings

Successfully derives e1-adic class field theory from Neukirch's framework

Defines suitable degree maps, G-modules, and valuations for both local and global cases

Proves the class field axiom within this unified approach

Abstract

Neukirch developed abstract class field theory in his famous book "Class Field Theory". We show that it is possible to derive Jaulent's '-adic class field from Neukirch's framework. The proof requires in both cases (local case and global case) to define suitable degree maps, G-modules, valuations and to prove the class field axiom.