Langlands–Deligne local constant

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 }

.