Converse theorem

Template:Short description Template:For In the mathematical theory of automorphic forms, a converse theorem gives sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. More generally a converse theorem states that a representation of an algebraic group 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 Template:Harvs who characterized the Riemann zeta function by its functional equation, and by Template:Harvtxt who showed that if a Dirichlet series satisfied a certain functional equation and some growth conditions then it was the Mellin transform of a modular form of level 1. Template:Harvtxt found an extension to modular forms of higher level, which was described by Template:Harvtxt. Weil's extension states that if not only the Dirichlet series

$L(s)=\sum \frac{a_n}{n^s}$

but also its twists

$L_{\chi }(s)=\sum \frac{\chi (n)a_n}{n^s}$

by some Dirichlet characters 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

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External links

• Cogdell's papers on converse theorems