Neukirch's Approach to the Reciprocity Map and Global Class Field theory in Positive Characteristic

TL;DR

This paper aims to extend Neukirch's axiomatic approach to class field theory from local and number fields to global function fields, bridging a gap in the existing mathematical framework.

Contribution

The paper proposes to verify Neukirch's axioms for global function fields, enabling the application of his approach to positive characteristic cases.

Findings

Verification of Neukirch's axioms for global function fields

Potential extension of class field theory methods to positive characteristic

Framework for future proofs in global function field theory

Abstract

Neukirch developed an axiomatic and explicit approach to class field theory. This was applied to local fields and number fields but was never done for global function fields since he believed that geometric approach is more suitable. Nonetheless all the axioms can be verified and we plan to prove that global function fields satisfy the axioms Neukirch developed.