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Automorphic L-function

Mathematical concept

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Source updated June 20, 2025

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.

01Properties

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

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

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

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

almost all of which are 1.

02General 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.

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Sources and credits

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Automorphic L-function

01Properties

02General linear groups

Sources and credits

Related topics

Grand Riemann hypothesis