Local class field theory

In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; 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 (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the p-adic numbers Q_p (where p is any prime number), or a finite extension of the field of formal Laurent series F_q((T)) over a finite field F_q.

It is the analogue for local fields of global class field theory.

Connection to Galois groups

Local class field theory gives a description of the Galois group G of the maximal abelian extension of a local field K via the reciprocity map which acts from the multiplicative group K^×=K\{0}. 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 absolute Galois group G of K is compact and the group K^× is not compact. Taking the case where K is a finite extension of the p-adic numbers Q_p or formal power series over a finite field, the group K^× is the product of a compact group with an infinite cyclic group Z. The main topological operation is to replace K^× by its profinite completion, which is roughly the same as replacing the factor Z by its profinite completion Z^{^}. The profinite completion of K^× is the group isomorphic with G via the local reciprocity map.

The actual isomorphism used and the existence theorem is described in the theory of the norm residue symbol. There are several different approaches to the theory, using central division algebras or Tate cohomology or an explicit description of the reciprocity map. There are also two different normalizations of the reciprocity map: in the case of an unramified extension, one of them asks that the (arithmetic) Frobenius element corresponds to the elements of "K" of valuation 1; the other one is the opposite.

Lubin–Tate theory

Main article: Lubin–Tate theory

Lubin–Tate theory is important in explicit local class field theory. The unramified part of any abelian extension is easily constructed, Lubin–Tate finds its value in producing the ramified part. This works by defining a family of modules (indexed by the natural numbers) over the ring of integers consisting of what can be considered as roots of the power series repeatedly composed with itself. The compositum of all fields formed by adjoining such modules to the original field gives the ramified part.

A Lubin–Tate extension of a local field K is an abelian extension of K obtained by considering the p-division points of a Lubin–Tate group. If g is an Eisenstein polynomial, f(t) = t g(t) and F the Lubin–Tate formal group, let θ_n denote a root of gf^n-1(t)=g(f(f(⋯(f(t))⋯))). Then K(θ_n) is an abelian extension of K with Galois group isomorphic to U/1+pⁿ where U is the unit group of the ring of integers of K and p is the maximal ideal.

Higher local class field theory

For a higher-dimensional local field $K$ 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 $K$ is an $n$-dimensional local field then one uses $\mathrm{K}^{\mathrm{M}}_n(K)$ or its separated quotient endowed with a suitable topology. When $n=1$ the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if $n>1$. Higher-dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh. Saito in the general case.

See also

• Quasi-finite field
• Local Langlands conjectures
• Norm group

Further reading

• Milne, James, Class Field Theory.