nLab geometric class field theory

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 $k$, not necessarily a finite field and possibly of characteristic zero. In particular $k$ 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

• Brian Conrad, Geometric global class field theory (pdf)

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

• David Ben-Zvi, Geometric class field theory, 2002 (MSRI notes)

See also

• MO discussion

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

• Moritz Kerz, Shuji Saito?, Chow group of 0-cycles with modulus and higher dimensional class field theory, arXiv:1304.4400.

which is briefly summarized in

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