Explicit Class Field Theory for global function fields

TL;DR

This paper constructs an explicit isomorphism between the Galois group of the maximal abelian extension of a global function field and its idele class group, providing a concrete description of class field theory in this setting.

Contribution

It explicitly constructs the isomorphism between the Galois group and the idele class group for global function fields using Drinfeld modules and dic representations.

Findings

Established an explicit isomorphism  the Galois group and the idele class group.

Provided an explicit description of the maximal abelian extension of a global function field.

Connected Galois actions on Drinfeld modules with class field theory.

Abstract

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism \rho: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class field theory, we shall show that our \rho is an isomorphism of topological groups whose inverse is the Artin map of F. As a consequence of the construction of \rho, we obtain an explicit description of F^ab. Fix a place \infty of F, and let A be the subring of F consisting of those elements which are regular away from \infty. We construct \rho by combining the Galois action on the torsion points of a suitable Drinfeld A-module with an associated \infty-adic representation studied by J.-K. Yu.