Automorphic L-function
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 ...
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 that has a finite kernel and is a ...
''G'' over a
global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...
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 f ...
''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 ana ...
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: # \chi(ab) = \ch ...
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 ...
of a
modular form In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modul ...
. 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 of F of local L functions. L(s, \pi, r) = \prod_v L(s, \pi_v, r_v) Here the
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 ...
\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-function In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands. Here, ''standard'' refers to the finite-dimensional representation r being the standard representation of the Langl ...
s) 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 functi ...
. 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 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 that has a finite kernel and is a ...
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 In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all Automorphic L-function, automorphic ''L''-functions lie on the critical line \f ...


References

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