Class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.[1]
Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out.[2] The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem).
One of the major results is: given a number field F, and writing K for the
where denotes the idelic norm map from L to F. This isomorphism is named the reciprocity map.
The existence theorem states that the reciprocity map can be used to give a bijection between the set of abelian extensions of F and the set of closed subgroups of finite index of
A standard method for developing global class field theory since the 1930s was to construct 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 Emil 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. His method was explicit and algorithmic.
Inside class field theory one can distinguish[3] special class field theory and general class field theory.
Explicit class field theory provides an explicit construction of maximal abelian extensions of a number field in various situations. This portion of the theory consists of Kronecker–Weber theorem, which can be used to construct the abelian extensions of , and the theory of complex multiplication to construct abelian extensions of CM-fields.
There are three main generalizations of class field theory: higher class field theory, the Langlands program (or 'Langlands correspondences'), and anabelian geometry.
Formulation in contemporary language
In modern mathematical language, class field theory (CFT) can be formulated as follows. Consider the maximal abelian extension A of a local or
The fundamental result of general class field theory states that the group G is naturally isomorphic to the
of the
For some small fields, such as the field of rational numbers or its
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
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.[4]
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.
History
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
The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order. Much later, the theory of Shimura provided another very explicit class field theory for a class of algebraic number fields. In positive characteristic ,
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
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.)
Applications
Class field theory is used to prove
Most main achievements toward the
Generalizations of class field theory
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
Another natural generalization is higher class field theory, divided into higher local class field theory and higher global class field theory. It describes abelian extensions of higher local fields and higher global fields. The latter come as function fields of schemes of finite type over integers and their appropriate localizations and completions. It uses algebraic K-theory, and appropriate Milnor K-groups generalize the used in one-dimensional class field theory.
See also
- Non-abelian class field theory
- Anabelian geometry
- Frobenioid
- Langlands correspondences
Citations
- ^ Milne 2020, p. 1, Introduction.
- ^ Cassels & Fröhlich 1967, p. 266, Ch. XI by Helmut Hasse.
- S2CID 239667749.
- ^ Reciprocity and IUT, talk at RIMS workshop on IUT Summit, July 2016, Ivan Fesenko
- ^ Milne, J. S. Arithmetic duality theorems. Charleston, SC: BookSurge, LLC 2006
- ^ Fesenko, Ivan (2015), Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015 (PDF)
- ^ Fesenko, Ivan (2021), Class field theory, its three main generalisations, and applications, May 2021, EMS Surveys 8(2021) 107-133 (PDF)
References
- ISBN 978-0-201-51011-9
- Zbl 0153.07403
- Conrad, Keith, History of class field theory. (PDF)
- MR 1915966
- Gras, Georges (2003). Class Field Theory: from theory to practice. ISBN 978-3-540-44133-5.
- Zbl 0604.12014
- Kawada, Yukiyosi (1955), "Class formations", Duke Math. J., 22 (2): 165–177, Zbl 0067.01904
- Kawada, Yukiyosi; Zbl 0101.02902
- Milne, James S. (2020), Class Field Theory(4.03 ed.)
- ISBN 978-3-540-15251-4
- Zbl 0956.11021.