Explicit class field theory for global 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

Applications of the Class Field Theory of Global Fields

Class field theory of global fields provides a description of finite abelian extensions of number fields and of function fields of transcendence degree 1 over finite fields. After a brief review of the handling of both function and number fields in Magma, we give a introduction to computational class field theory focusing on applications: We show how to construct tables of small degree extensio...

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