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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 of a local field. The functional equation

L(ρ,s) = ε(ρ,s)L(ρ,1−s)

of an Artin L-function 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(ρv, s, ψv)

of local constants ε(ρv, s, ψv) associated to primes v.

Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis. Dwork (1956) proved the existence of the local constant ε(ρv, s, ψv) up to sign. The original proof of the existence of the local constants by Langlands (1970) used local methods and was rather long and complicated, and never published. Deligne (1973) later discovered a simpler proof using global methods.

Properties

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

• If ρ is 1-dimensional then ε(ρ, s, ψE) 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, ψE) = ε(ρ1, s, ψE)ε(ρ2, s, ψE). As a result ε(ρ, s, ψE) 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, ψE) = ε(IndE/Kρ, s, ψK)

Brauer's theorem on induced characters implies that these three properties characterize the local constants.

Deligne (1976) 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, ψE) = ε(ρ⊗||s, 0, ψE) 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

Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8; 978-3-540-31486-8, MR2234120
Deligne, Pierre (1973), "Les constantes des équations fonctionnelles des fonctions L", Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture notes in mathematics, 349, Berlin, New York: Springer-Verlag, pp. 501–597, doi:10.1007/978-3-540-37855-6_7, MR0349635
Deligne, Pierre (1976), "Les constantes locales de l'équation fonctionnelle de la fonction L d'Artin d'une représentation orthogonale", Inventiones Mathematicae 35: 299–316, doi:10.1007/BF01390143, ISSN 0020-9910, MR0506172
Dwork, Bernard (1956), "On the Artin root number", American Journal of Mathematics 78: 444–472, ISSN 0002-9327, JSTOR 2372524, MR0082476
Langlands, Robert (1970), On the functional equation of the Artin L-functions, Unpublished notes
Tate, John T. (1977), "Local constants", in Fröhlich, A., Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Boston, MA: Academic Press, pp. 89–131, ISBN 978-0-12-268960-4, MR0457408
Tate, J. (1979), "Number theoretic background", Automorphic forms, representations, and L-functions Part 2,, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4