In the mathematical theory of

automorphic form 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 o ...

s, a converse theorem gives sufficient conditions for a

Dirichlet series In mathematics, a Dirichlet series is any series of the form \sum_^\infty \frac, where ''s'' is complex, and a_n is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in anal ...

to be 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 ...

. More generally a converse theorem states that a representation of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

over the adeles is automorphic whenever the L-functions of various twists of it are well-behaved.

Weil's converse theorem

The first converse theorems were proved by who characterized the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

by its functional equation, and by who showed that if a Dirichlet series satisfied a certain

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

and some growth conditions then it was the

of a modular form of level 1. found an extension to modular forms of higher level, which was described by . Weil's extension states that if not only the Dirichlet series :

L(s)=\sum\frac

but also its twists :

L_\chi(s)=\sum\frac

by some

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 ...

s χ, satisfy suitable functional equations relating values at ''s'' and 1−''s'', then the Dirichlet series is essentially the Mellin transform of a modular form of some level.

Higher dimensions

J. W. Cogdell, H. Jacquet, I. I. Piatetski-Shapiro and J. Shalika have extended the converse theorem to automorphic forms on some higher-dimensional groups, in particular GL_''n'' and GL_''m''×GL_''n'', in a long series of papers.

References

* * * * * * * *{{Citation , last1=Weil , first1=André , author1-link=André Weil , title=Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen , doi=10.1007/BF01361551 , mr=0207658 , year=1967 , journal=

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

, issn=0025-5831 , volume=168 , pages=149–156

External links

Cogdell's papers on converse theorems
Automorphic forms Theorems in number theory