User:Stephen Miller Math/Automorphic L-functions

In mathematics, Automorphic L-functions are a particular type of L-function that are attached (still partially conjecturally) to automorphic forms or representations. They conjecturally cover all known constructions of L-functions.

Definition
Like the Riemann Zeta Function, Automorphic L-functions are defined as Euler products over the primes. The local L-factor for the prime p is always the evaluation of a polynomial at 1/p^s, and the factor at infinity is a product of gamma factors, more precisely shifts of Gamma(s) or Gamma(s/2), times a power of pi. The defintion may be recast in terms of the places of a number field, though this description is always valid.

Robert Langlands defined various automorphic L-functions for general automorphic forms in terms of finite dimensional representations of his L-groups.

Primitive L-functions and cusp forms
Often L-functions factor in terms of smaller L-functions. One which does not is called primitive. It was conjectured by Ilya Piatetski-Shapiro and others that the primitive L-functions are precisely the L-functions of cuspidal automorphic representations on GL(n) over the rationals Q. More precisely, all L-functions come from cusp forms on Gamma\GL(n,R), where Gamma is a congruence subgroup of GL(n,Z). For this reason, GL(n,R) has been termed the "mother of all L-functions".

Analytic properties
Like other L-functions, automorphic L-functions are conjectured to have meromorphic continuations to the complex plane, with poles at well understood places. The only primitive L-function expected to have a pole is the Riemann Zeta function itself, so poles should only arise when Zeta is a factor of a larger product of L-functions. Knowing this full analytic continuation in general would give a tremendous number of examples of Langlands' functoriality conjectures, via the converse theorem.

There are currently three methods to study the analytic properties directly: the Rankin-Selberg method of integral representations; the Langlands-Shahidi method of Fourier expansions of Eisenstein series; and a newer method of automorphic distributions.