Class Field Theory - Generalizations of Class Field Theory

Generalizations of Class Field Theory

One natural development in number theory is to understand and construct nonabelian class field theories which provide information about general Galois extensions of global fields. Often, the Langlands correspondence is viewed as a nonabelian class field theory and indeed when fully established it will contain a very rich theory of nonabelian Galois extensions of global fields. However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case. It also does not include an analog of the existence theorem in class field theory, i.e. the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view.

Another natural development in arithmetic geometry is to understand and construct class field theory which describes abelian extensions of higher local and global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localization and completions. Higher local and global class field theory uses algebraic K-theory and appropriate Milnor K-groups replace which is in use in one-dimensional class field theory. Higher local and global class field theory was developed by A. Parshin, Kazuya Kato, Ivan Fesenko, Spencer Bloch, Shuji Saito and other mathematicians. There are attempts to develop higher global class field theory without using algebraic K-theory (G. Wiesend), but his approach does not involve higher local class field theory and a compatibility between the local and global theories.