Wulf-Dieter Geyer (Erlangen): Class Field Theory

Abstract:
Class field theory describes the abelian etale coverings of certain schemes X
by data from X.
It starts in the 19th century for Riemann surfaces X with the study of the
class group of their function fields (the torsion of their Jacobians) and with
Kronecker's discoveries about the abelian extensions of the rational numbers
and imaginary quadratic number fields.
In the first half of the 20th century class field theory of number fields
developped into a powerful theory, starting with Hilbert's class field,
culminating in Artin's reciprocity isomorphism.
In the second half of the 20th century several attempts were made to
generalize this theory to higher dimensional arithmetic schemes.
In the 1980's Kato and Saito gave a description for the abelian fundamental
group of regular arithmetic schemes by higher dimensional Milnor K-theory.
In our century A. Schmidt (Regensburg) uses this theory to present a
representation of the tame abelian fundamental group without K-theory.
Goetz Wiesend (Erlangen) recently gave an independent approach to the
description of the full abelian fundamental group of regular arithmetic schemes
without K-theory.
Walter Hofmann (Erlangen) generalized this approach to singular arithmetic
schemes.
The talk will concentrate on Wiesend's approach.