Automorphic L-function

In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by Langlands (1967, 1970, 1971).

Borel (1979) and Arthur & Gelbart (1991) gave surveys of automorphic L-functions.

Properties

Automorphic

L

{\displaystyle L}

-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function

L

(

s

,

π

,

r

)

{\displaystyle L(s,\pi ,r)}

should be a product over the places

v

{\displaystyle v}

of

F

{\displaystyle F}

of local

L

{\displaystyle L}

functions.

L

(

s

,

π

,

r

)

=

∏

v

L

(

s

,

π

v

,

r

v

)

{\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}

Here the automorphic representation

π

=

⊗

π

v

{\displaystyle \pi =\otimes \pi _{v}}

is a tensor product of the representations

π

v

{\displaystyle \pi _{v}}

of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex

s

{\displaystyle s}

, and satisfy a functional equation

L

(

s

,

π

,

r

)

=

ϵ

(

s

,

π

,

r

)

L

(

1

−

s

,

π

,

r

∨

)

{\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}

where the factor

ϵ

(

s

,

π

,

r

)

{\displaystyle \epsilon (s,\pi ,r)}

is a product of "local constants"

ϵ

(

s

,

π

,

r

)

=

∏

v

ϵ

(

s

,

π

v

,

r

v

,

ψ

v

)

{\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}

almost all of which are 1.

General linear groups

Godement & Jacquet (1972) constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.