Automorphic L-function
Encyclopedia
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 comlplex 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.
The L-function L(s,π,r) should be a product over the places v of F of local L functions.
Here the automorphic representation π=⊗πv is a tensor product of the representations πv 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
where the factor ε(s,π,r) is a product of "local constants"
almost all of which are 1.
.
The Langlands functoriality conjectures imply that all automorphic L-functions are equal to L-functions of general linear groups, so this would prove the analytic continuation and functional equation for them.
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
π of a reductive group G over a global field and a finite-dimensional comlplex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
and the Mellin transform of a modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
. They were introduced by .
and 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,πv,rv)
Here the automorphic representation π=⊗πv is a tensor product of the representations πv 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,πv,rv, ψv)
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 thesisTate's thesis
In number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....
.
The Langlands functoriality conjectures imply that all automorphic L-functions are equal to L-functions of general linear groups, so this would prove the analytic continuation and functional equation for them.