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 form 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-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) should be a product over the places v of F of local L functions.

L(s,π,r) = Π L(s,π_v,r_v)

Here the automorphic representation π=⊗π_v is a tensor product of the representations π_v of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex s, and satisfy a functional equation

L(s,π,r) = ε(s,π,r)L(1 – s,π,r^∨)

where the factor ε(s,π,r) is a product of "local constants"

ε(s,π,r) = Π ε(s,π_v,r_v, ψ_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.

In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.