In mathematics, the Langlands–Deligne local constant (or local Artin root number un to an elementary function of s) is an elementary function associated with a representation of the Weil group

Weil group

In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is a profinite group generally denoted WF...

 of a local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

. The functional equation

L(ρ,s) = ε(ρ,s)L(ρ^∨,1−s)

of an Artin L-function

Artin L-function

In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...

 has an elementary function ε(ρ,s) appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε(ρ,s) can be written in a canonical way as a product

ε(ρ,s) = Π ε(ρ_v, s, ψ_v)

of local constants ε(ρ_v, s, ψ_v) associated to primes v.

Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis

Tate's thesis

In number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....

.
proved the existence of the local constant ε(ρ_v, s, ψ_v) up to sign.
The original proof of the existence of the local constants by used local methods and was rather long and complicated, and never published. later discovered a simpler proof using global methods.

Properties

The local constants ε(ρ, s, ψ_E) depend on a representation ρ of the Weil group and a choice of character ψ_E of the additive group of E. They satisfy the following conditions:

• If ρ is 1-dimensional then ε(ρ, s, ψ_E) is the constant associated to it by Tate's thesis as the constant in the functional equation of the local L-function.
• ε(ρ₁⊕ρ₂, s, ψ_E) = ε(ρ₁, s, ψ_E)ε(ρ₂, s, ψ_E). As a result ε(ρ, s, ψ_E) can also be defined for virtual representations ρ.
• If ρ is a virtual representation of dimension 0 and E contains K then ε(ρ, s, ψ_E) = ε(Ind_E/Kρ, s, ψ_K)

Brauer's theorem on induced characters

Brauer's theorem on induced characters

Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group...

implies that these three properties characterize the local constants.

showed that the local constants are trivial for real (orthogonal) representations of the Weil group.

Notational conventions

There are several different conventions for denoting the local constants.

• The parameter s is redundant and can be combined with the representation ρ, because ε(ρ, s, ψ_E) = ε(ρ⊗||^s, 0, ψ_E) for a suitable character ||.
• Deligne includes an extra parameter dx consisting of a choice of Haar measure on the local field. Other conventions omit this parameter by fixing a choice of Haar measure: either the Haar measure that is self dual with respect to ψ (used by Langlands), or the Haar measure that gives the integers of E measure 1. These different conventions differ by elementary terms that are positive real numbers.