In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor
or local Artin root number (up to an elementary real function of ''s''), is an
elementary function
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
associated with a
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of the
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 generally denoted ''WF''. There also exists "finite leve ...
of a
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 t ...
. The
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
:L(ρ,''s'') = ε(ρ,''s'')L(ρ
∨,1−''s'')
of an
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 1923 by Emil Artin, in connection with his research into class field theory. T ...
has an elementary function ε(ρ,''s'') appearing in it, equal to a constant called the
Artin root number 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 1923 by Emil Artin, in connection with his research into class field theory. T ...
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
In number theory, Tate's thesis is the 1950 thesis, PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta f ...
.
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(ρ
1⊕ρ
2, ''s'', ψ
''E'') = ε(ρ
1, ''s'', ψ
''E'')ε(ρ
2, ''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''/''K''ρ, ''s'', ψ
''K'')
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, within representation theory of a finite group.
Backgrou ...
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.
References
*
*
*
*
*
*
*
External links
*
{{DEFAULTSORT:Langlands-Deligne local constant
Representation theory
Zeta and L-functions
Class field theory