Langlands–Deligne local constant

Template:Short description 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 Template:Harvtxt 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 $E$Template:Clarification needed. 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

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External links

• Template:Springer