Langlands–Deligne local constant


In mathematics, the Langlands–Deligne local constant is an elementary function associated with a representation of the Weil group of a local field. The functional equation
of an Artin L-function has an elementary function ε appearing in it, equal to a constant called the Artin root number times an elementary real function of s, and Langlands discovered that ε can be written in a canonical way as a product
of local constants ε associated to primes v.
Tate proved the existence of the local constants in the case that ρ is 1-dimensional in Tate's thesis.
proved the existence of the local constant ε 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 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:
Brauer's theorem on induced characters implies that these three properties characterize the local constants.
showed that the local constants are trivial for real representations of the Weil group.

Notational conventions

There are several different conventions for denoting the local constants.