Converse theorem explained

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 who characterized the Riemann zeta function by its functional equation, and by 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. 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 (a _{n})/(n ^{s}) but also its twists L _{\chi(s) = \sum (\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.

External links

• Cogdell's papers on converse theorems