Langlands–Deligne local constant

In mathematics, the Langlands–Deligne local constant, also known as the local epsilon factor^[1] or local Artin root number (up to an elementary real 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(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. Dwork (1956) 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 Langlands (1970) used local methods and was rather long and complicated, and never published. Deligne (1973) later discovered a simpler proof using global methods.

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 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.

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.

• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
• 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, vol. 349, Berlin, New York: Springer-Verlag, pp. 501–597, doi:10.1007/978-3-540-37855-6_7, ISBN 978-3-540-06558-6, MR 0349635
• 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, MR 0506172, S2CID 119880957
• Dwork, Bernard (1956), "On the Artin root number", American Journal of Mathematics, 78 (2): 444–472, doi:10.2307/2372524, ISSN 0002-9327, JSTOR 2372524, MR 0082476
• Langlands, Robert (1970), On the functional equation of the Artin L-functions, Unpublished notes
• Tate, John T. (1977), "Local constants", in Fröhlich, A. (ed.), 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, MR 0457408
• Tate, J. (1979), "Number theoretic background", Automorphic forms, representations, and L-functions Part 2, Proc. Sympos. Pure Math., vol. XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 0-8218-1435-4

• Perlis, R. (2001) [1994], "Artin root numbers", Encyclopedia of Mathematics, EMS Press