nLab
geometric class field theory

Contents

Idea

In the spirit of the function field analogy, geometric class field theory is analogous to class field theory, but where the latter studies finite abelian extensions of global fields, geometric class field theory studies finite covering spaces of suitable algebraic curves over any constant perfect field kk, not necessarily a finite field and possibly of characteristic zero. In particular kk may be the complex numbers \mathbb{C}, in which case the theory is about covering of curves.

As such, geometric class field theory has become part of the geometric Langlands program and of higher dimensional class field theory.

References

A brief survey of classical results is in

A fairly comprehensive review of the theory is in the thesis

  • Peter Toth, Geometric abelian class field theory, 2011 (web)

Discussion in the context of the geometric Langlands correspondence includes

See also

Higher dimensional class field theory, using the Chow group with modulus, is developed in

which is briefly summarized in

  • Chow group of 0-cycles with modulus and higher dimensional class field theory, pdf.

Last revised on January 21, 2015 at 11:22:43. See the history of this page for a list of all contributions to it.