Mackey functors and abelian class field theories

TL;DR

This paper introduces a unified, group-theoretical framework for abelian class field theories using cohomological Mackey functors, extending classical approaches to more general settings including higher local fields.

Contribution

It proposes a general definition of abelian class field theory based on Mackey functors, enabling modeling of abelian extensions beyond traditional invariants.

Findings

Models abelian extensions using cohomological Mackey functors.

Extends class field theory to higher local fields of positive characteristic.

Provides a unified perspective on classical and modern class field theories.

Abstract

Motivated by the work of J\"urgen Neukirch and Ivan Fesenko we propose a general definition of an abelian class field theory from a purely group-theoretical and functorial point of view. This definition allows a modeling of abelian extensions of a field inside more general objects than the invariants of a discrete module over the absolute Galois group of the field. The main objects serving as such models are cohomological Mackey functors as they have enough structure to make several reduction theorems of classical approaches work in this generalized setting and, as observed by Fesenko, they even have enough structure to make Neukirch's approach to class field theories via Frobenius lifts work. This approach is discussed in full detail and in its most general setting, including the pro-P setting proposed by Neukirch. As an application and justification of this generalization we describe Fesenko's approach to class field theory of higher local fields of positive characteristic, where the modeling of abelian extensions takes place inside the cohomological Mackey functor formed by the Milnor-Par\v{s}in K-groups. The motivation for this work (which is the author's Diplom thesis) was the attempt to understand what a class field theory is and to give a single-line definition which captures certain common aspects of several instances of class field theories. We do not claim to prove any new theorem here, but we think that our general and uniform approach offers a point of view not discussed in this form in the existing literature.