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 ρ {\displaystyle \rho } of the Weil group of a local field. The functional equation
L ( ρ , s ) = ε ( ρ , s ) L ( ρ v , 1 − s ) {\displaystyle L(\rho ,s)=\varepsilon (\rho ,s)L(\rho ^{v},1-s)}
of the Artin L-function associated to ρ {\displaystyle \rho } has a function ε ( ρ , s ) {\displaystyle \varepsilon (\rho ,s)} appearing in it, equal to a constant called the Artin root number times an elementary real function of s {\displaystyle s} , and Langlands discovered that ε ( ρ , s ) {\displaystyle \varepsilon (\rho ,s)} can be written in a canonical way as a product
ε ( ρ , s ) = ∏ ε ( ρ v , s , ψ v ) {\displaystyle \varepsilon (\rho ,s)=\prod \varepsilon (\rho _{v},s,\psi _{v})}
of local constants ε ( ρ v , s , ψ v ) {\displaystyle \varepsilon (\rho _{v},s,\psi _{v})} associated to primes v {\displaystyle v} . In his thesis, John Tate proved the existence of the local constants in the case that ρ {\displaystyle \rho } is one-dimensional. Bernard Dwork proved the existence of the local constant ε ( ρ v , s , ψ v ) {\displaystyle \varepsilon (\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 ε ( ρ , s , ψ v ) {\displaystyle \varepsilon (\rho ,s,\psi _{v})} depend on a representation ρ {\displaystyle \rho } of the Weil group and a choice of character ψ E {\displaystyle \psi _{E}} of the additive group of E {\displaystyle E} . They satisfy the following conditions:
If ρ {\displaystyle \rho } is one-dimensional then ε ( ρ , s , ψ E ) {\displaystyle \varepsilon (\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.
ε ( ρ 1 ⊕ ρ 2 , s , ψ E ) = ε ( ρ 1 , s , ψ E ) ε ( ρ 2 , s , ψ E ) . {\displaystyle \varepsilon (\rho _{1}\oplus \rho _{2},s,\psi _{E})=\varepsilon (\rho _{1},s,\psi _{E})\varepsilon (\rho _{2},s,\psi _{E}).} As a result, ε ( ρ , s , ψ E ) {\displaystyle \varepsilon (\rho ,s,\psi _{E})} can also be defined for virtual representations ρ {\displaystyle \rho } . If ρ {\displaystyle \rho } is a virtual representation of dimension 0 and E {\displaystyle E} contains K {\displaystyle K} then ε ( ρ , s , ψ E ) = ε ( Ind E / K ρ , s , ψ K ) {\displaystyle \varepsilon (\rho ,s,\psi _{E})=\varepsilon (\operatorname {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.
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