Explicit class field theory for rational function fields

Explicit Class Field Theory for Global Function Fields

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 ρ : Gal(F /F ) → CF , where CF is the idele class group of F . Using class field theory, we shall show that our ρ is an isomorphism of topological groups whose inverse is the Artin map of F . As a consequence of the construction of ρ, we...

Explicit class field theory in function fields: Gross-Stark units and Drinfeld modules

Reduction theory for a rational function field

Let G be a split reductive group over a finite field Fq. Let F = Fq(t) and let A denote the adèles of F . We show that every double coset in G(F)\G(A)/K has a representative in a maximal split torus of G. Here K is the set of integral adèlic points of G. When G ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isom...

Explicit higher local class field theory

Consider now an approach based on a generalization [F2] of Neukirch’s approach [N]. Below is a modified system of axioms of class formations (when applied to topological K -groups) which imposes weaker restrictions than the classical axioms (cf. section 11). (A1). There is a Ẑ-extension of F . In the case of higher local fields let F pur/F be the extension which corresponds to K sep 0 /K0: F pu...

Complex Multiplication and Explicit Class Field Theory