Explicit Class Field Theory for global function fields

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.