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 Template:Harvs.

Template:Harvtxt and Template:Harvtxt 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 $L(s,\pi ,r)$ should be a product over the places $v$ of $F$ of local $L$ functions.

$L(s,\pi ,r)=\prod _{v}L(s,{\pi }_v,r_v)$

Here the automorphic representation $\pi =\otimes {\pi }_v$ is a tensor product of the representations ${\pi }_v$ 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

$L(s,\pi ,r)=ϵ(s,\pi ,r)L(1-s,\pi ,r^{\vee })$

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

$ϵ(s,\pi ,r)=\prod _vϵ(s,{\pi }_v,r_v,{\psi }_v)$

almost all of which are 1.

General linear groups

Template:Harvtxt 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.

See also

• Grand Riemann hypothesis

References

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