Automorphic L-function


In mathematics, an automorphic L-function is a function L 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.
and gave surveys of automorphic L-functions.

Properties

Automorphic L-functions should have the following properties.
The L-function should be a product over the places of of local functions.
Here the automorphic representation is a tensor product of the representations of local groups.
The L-function is expected to have an analytic continuation as a meromorphic function of all complex, and satisfy a functional equation
where the factor is a product of "local constants"
almost all of which are 1.

General linear groups

constructed the automorphic L-functions for general linear groups with r the standard representation 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 and GL. 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.