Langlands–Deligne local constant

Short description: Elementary function in mathematics

In number theory, the Langlands–Deligne local constant, named after Robert Langlands and Pierre Deligne, also known as the local epsilon factor, is a function associated with a representation $\rho$ of the Weil group of a local field. The functional equation

$L(\rho ,s)=\epsilon (\rho ,s)L({\rho }^v,1-s)$

of the Artin L-function associated to $\rho$ has a function $\epsilon (\rho ,s)$ appearing in it, equal to a constant called the Artin root number times an elementary real function of $s$, and Langlands discovered that $\epsilon (\rho ,s)$ can be written in a canonical way as a product

$\epsilon (\rho ,s)=\prod \epsilon ({\rho }_v,s,{\psi }_v)$

of local constants $\epsilon ({\rho }_v,s,{\psi }_v)$ associated to primes $v$.

In his thesis, John Tate proved the existence of the local constants in the case that $\rho$ is one-dimensional. Bernard Dwork proved the existence of the local constant $\epsilon ({\rho }_v,s,{\psi }_v)$ 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 later discovered a simpler proof using global methods.

Properties

The local constants $\epsilon (\rho ,s,{\psi }_v)$ depend on a representation $\rho$ of the Weil group and a choice of character ${\psi }_E$ of the additive group of $E$. They satisfy the following conditions:

• If $\rho$ is one-dimensional then $\epsilon (\rho ,s,{\psi }_E)$ is the constant associated to it by Tate's thesis as the constant in the functional equation of the local L-function.
• $\epsilon ({\rho }_1\oplus {\rho }_2,s,{\psi }_E)=\epsilon ({\rho }_1,s,{\psi }_E)\epsilon ({\rho }_2,s,{\psi }_E).$As a result, $\epsilon (\rho ,s,{\psi }_E)$ can also be defined for virtual representations $\rho$.
• If $\rho$ is a virtual representation of dimension 0 and $E$ contains $K$ then $\epsilon (\rho ,s,{\psi }_E)=\epsilon ({\mathrm{Ind}}_{E/K}\rho ,s,{\psi }_K)$.

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

Deligne 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 $\rho$, 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 $\psi$ (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 
• 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, ISBN 978-3-540-06558-6 
• 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 
• Dwork, Bernard (1956), "On the Artin root number", American Journal of Mathematics 78 (2): 444–472, doi:10.2307/2372524, ISSN 0002-9327 
• Langlands, Robert (1970), On the functional equation of the Artin L-functions, Unpublished notes, http://publications.ias.edu/rpl/paper/61 
• 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, https://books.google.com/books?id=_QDvAAAAMAAJ 
• 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, https://www.ams.org/online_bks/pspum332/, retrieved 2021-04-28 

External links

• Hazewinkel, Michiel, ed. (2001), "Artin root numbers", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=a/a120270