mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an automorphic ''L''-function is a function ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an

automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

π of a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

''G'' over a

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...

and a finite-dimensional complex representation ''r'' of the

Langlands dual group In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a fie ...

^''L''''G'' of ''G'', generalizing the

Dirichlet L-series In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...

of a

Dirichlet character In analytic number theory and related branches of mathematics, a complex-valued arithmetic function \chi:\mathbb\rightarrow\mathbb is a Dirichlet character of modulus m (where m is a positive integer) if for all integers a and b: :1) \chi ...

and the

Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used i ...

of a

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...

. 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, \pi, r)

should be a product over the places

v

F

of local

L

functions.

L(s, \pi, r) = \prod_v L(s, \pi_v, r_v)

Here the

\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.

General linear groups

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 In number theory, Tate's thesis is the 1950 PhD thesis of completed under the supervision of Emil Artin at Princeton University. In it, Tate used a translation invariant integration on the locally compact group of ideles to lift the zeta function ...

. 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 mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic rep ...

. In general, the

Langlands functoriality In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic ...

conjectures imply that automorphic L-functions of a connected

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.

References

* * * * * * * * * * {{L-functions-footer Automorphic forms Zeta and L-functions Langlands program