Class field theory

In mathematics, class field theory is the branch of algebraic number theory concerned with describing the Galois extensions of local and global fields.[1] Hilbert is often credited for the notion of class field. But it was already familiar for Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out.[2] This theory has its origins in the proof of quadratic reciprocity by Gauss at the end of the 18th century. These ideas were developed over the next century, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin. These conjectures and their proofs constitute the main body of class field theory.

One major result states that, given a number field F, and writing K for the maximal abelian unramified extension of F, the Galois group of K over F is canonically isomorphic to the ideal class group of F. This statement can be generalized to the Artin reciprocity law; writing CF for the idele class group of F, and taking L to be any finite abelian extension of F, this law gives a canonical isomorphism

A standard method for developing global class field theory since the 1930s is to develop local class field theory, which describes abelian extensions of local fields, and then use it to construct global class field theory. This was first done by Artin and Tate using the theory of group cohomology, and in particular by developing the notion of class formations. Later, Neukirch found a proof of the main statements of global class field theory without using cohomological ideas.

The Langlands program gives one approach for generalizing class field theory to non-abelian extensions. This generalization is mostly still conjectural. For number fields, class field theory and the results related to the modularity theorem are the only cases known.

In modern mathematical language class field theory can be formulated as follows. Consider the maximal abelian extension A of a local or global field K. It is of infinite degree over K; the Galois group G of A over K is an infinite profinite group, so a compact topological group, and it is abelian. The central aims of class field theory are: to describe G in terms of certain appropriate topological objects associated to K, to describe finite abelian extensions of K in terms of open subgroups of finite index in the topological object associated to K. In particular, one wishes to establish a one-to-one correspondence between finite abelian extensions of K and their norm groups in this topological object for K. This topological object is the multiplicative group in the case of local fields with finite residue field and the idele class group in the case of global fields. The finite abelian extension corresponding to an open subgroup of finite index is called the class field for that subgroup, which gave the name to the theory.

The fundamental result of general class field theory states that the group G is naturally isomorphic to the profinite completion of CK, the multiplicative group of a local field or the idele class group of the global field, with respect to the natural topology on CK related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism (the Artin reciprocity map)

of the abelianization of the Galois group of the extension with the quotient of the idele class group of K by the image of the norm of the idele class group of L.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension (this is done inside local class field theory) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field. The latter property is called the global reciprocity law and is a far reaching generalization of the Gauss quadratic reciprocity law.

One of the methods to construct the reciprocity homomorphism uses class formation which derives class field theory from axioms of class field theory. This derivation is purely topological group theoretical, while to establish the axioms one has to use the ring structure of the ground field.[3]

There are methods which use cohomology groups, in particular the Brauer group, and there are methods which do not use cohomology groups and are very explicit and fruitful for applications.

The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions.

However, these very explicit theories could not be extended to more general number fields. General class field theory used different concepts and constructions which work over every global field.

The famous problems of David Hilbert stimulated further development, which led to the reciprocity laws, and proofs by Teiji Takagi, Phillip Furtwängler, Emil Artin, Helmut Hasse and many others. The crucial Takagi existence theorem was known by 1920 and all the main results by about 1930. One of the last classical conjectures to be proved was the principalisation property. The first proofs of class field theory used substantial analytic methods. In the 1930s and subsequently saw the increasing use of infinite extensions and Wolfgang Krull's theory of their Galois groups. This combined with Pontryagin duality to give a clearer if more abstract formulation of the central result, the Artin reciprocity law. An important step was the introduction of ideles by Claude Chevalley in the 1930s to replace ideal classes, essentially clarifying and simplifying the description of abelian extensions of global fields. Most of the central results were proved by 1940.

Later the results were reformulated in terms of group cohomology, which became a standard way to learn class field theory for several generations of number theorists. One drawback of the cohomological method is its relative inexplicitness. As the result of local contributions by Bernard Dwork, John Tate, Michiel Hazewinkel and a local and global reinterpretation by Jürgen Neukirch and also in relation to the work on explicit reciprocity formulas by many mathematicians, a very explicit and cohomology-free presentation of class field theory was established in the 1990s. (See, for example, Class Field Theory by Neukirch.)

Class field theory is used to prove Artin-Verdier duality.[4] Very explicit class field theory is used in many subareas of algebraic number theory such as Iwasawa theory and Galois modules theory.

Most main achievements toward the Langlands correspondence for number fields, the BSD conjecture for number fields, and Iwasawa theory for number fields use very explicit but narrow class field theory methods or their generalizations. The open question is therefore to use generalizations of general class field theory in these three directions.

There are three main generalizations, each of great interest. They are: the Langlands program, anabelian geometry, and higher class field theory.

Often, the Langlands correspondence is viewed as a nonabelian class field theory. If and when it is fully established, it would contain a certain 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: the concept of class fields is absent in the Langlands correspondence. There are several other nonabelian theories, local and global, which provide alternatives to the Langlands correspondence point of view.

Another generalization of class field theory is anabelian geometry, which studies algorithms to restore the original object (e.g. a number field or a hyperbolic curve over it) from the knowledge of its full absolute Galois group or algebraic fundamental group.[5]