Fesenko reciprocity map

TL;DR

This paper provides a detailed, proof-inclusive exposition of Fesenko's non-abelian local reciprocity map for totally-ramified APF Galois extensions, extending prior non-abelian class field theories.

Contribution

It offers a comprehensive, accessible account of Fesenko's non-abelian reciprocity map, including all previously omitted proofs, advancing understanding in local class field theory.

Findings

Detailed exposition of Fesenko's reciprocity map

Includes all skipped proofs from original work

Extends non-abelian local class field theory

Abstract

In recent papers, Fesenko has defined the non-abelian local reciprocity map for every totally-ramified arithmetically profinite ($APF$) Galois extension of a given local field $K$ by extending the works of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work including all the skipped proofs.