In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, local class field theory, introduced by
Helmut Hasse
Helmut Hasse (; 25 August 1898 – 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of ''p''-adic numbers to local class field theory and ...
, is the study of
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s of
local field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
s; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
(as a topological field) to the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every re ...
R, the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
C, a
finite extension of the
''p''-adic numbers Q
''p'' (where ''p'' is any
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
), or a finite extension of the field of
formal Laurent series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...
F
''q''((''T'')) over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F
''q''.
Approaches to local class field theory
Local class field theory gives a description of the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
''G'' of the maximal abelian extension of a local field ''K'' via the reciprocity map which acts from the multiplicative group ''K''
×=''K''\. For a finite abelian extension ''L'' of ''K'' the reciprocity map induces an isomorphism of the quotient group ''K''
×/''N''(''L''
×) of ''K''
× by the norm group ''N''(''L''
×) of the extension ''L''
× to the Galois group Gal(''L''/''K'')
of the extension.
The existence theorem in local class field theory establishes a one-to-one correspondence between open subgroups of finite index in the multiplicative group ''K''
× and finite abelian extensions of the field ''K''. For a finite abelian extension ''L'' of ''K'' the corresponding open subgroup of finite index is the norm group ''N''(''L''
×). The reciprocity map sends higher groups of units to higher ramification subgroups, see e.g. Ch. IV of.
Using the local reciprocity map, one defines the Hilbert symbol and its generalizations. Finding explicit formulas for it is one of subdirections of the theory of local fields, it has a long and rich history, see e.g.
Sergei Vostokov
Sergei Vladimirovich Vostokov (russian: Сергей Владимирович Востоков; born April 13, 1945) is a Russian mathematician. He made major contributions to local number theory. He is a professor at St. Petersburg State Univer ...
's review.
There are cohomological approaches and non-cohomological approaches to local class field theory. Cohomological approaches tend to be non-explicit, since they use the cup-product of the first Galois cohomology groups.
For various approaches to local class field theory see Ch. IV and sect. 7 Ch. IV of They include the Hasse approach of using the Brauer group,
cohomological
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
approaches, the explicit methods of
Jürgen Neukirch,
Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and a ...
, the
Lubin-Tate theory and others.
Generalizations of local class field theory
Generalizations of local class field theory to local fields with quasi-finite residue field were easy extensions of the theory, obtained by G. Whaples in the 1950s, see chapter V of.
Explicit p-class field theory for local fields with perfect and imperfect residue fields which are not finite has to deal with the new issue of norm groups of infinite index. Appropriate theories were constructed by
Ivan Fesenko
Ivan Fesenko is a mathematician working in number theory and its interaction with other areas of modern mathematics.
Education
Fesenko was educated at St. Petersburg State University where he was awarded a PhD in 1987.
Career and research
Fes ...
.
Fesenko's noncommutative local class field theory for arithmetically profinite Galois extensions of local fields studies appropriate local reciprocity cocycle map and its properties.
This arithmetic theory can be viewed as an alternative to the representation theoretical local Langlands correspondence.
Higher local class field theory
For a
higher-dimensional local field there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the
Milnor K-group of the field. Namely, if
is an
-dimensional local field then one uses
or its separated quotient endowed with a suitable topology. When
the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if
. General higher-dimensional local class field theory was developed by
K. Kato and
I. Fesenko.
Higher local class field theory is part of
higher class field theory which studies abelian extensions (resp. abelian covers) of rational function fields of proper regular schemes flat over integers.
See also
*
Higher local field
*
Local Langlands conjectures
*
Norm group
References
Further reading
*
*
*
*
*
*{{Citation , last1=Serre , first1=Jean-Pierre , author1-link=Jean-Pierre Serre , title=Corps Locaux (English translation: Local Fields) , orig-year=1962 , url=https://books.google.com/books?id=DAxlMdw_QloC , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Graduate Texts in Mathematics , isbn=978-0-387-90424-5 , mr=0150130 , year=1979 , volume=67
Class field theory