Maass wave form

In mathematics, a Maass wave form is a function on the upper half plane that transforms like a modular form but need not be holomorphic. They were first studied by Maass (1949).

Definition

A Maass wave form is defined to be a continuous complex-valued function f of τ = x + iy in the upper half plane satisfying the following conditions:

f is invariant under the action of the group SL₂(Z) on the upper half plane.
f is an eigenvector of the Laplacian operator $-y^2\left(\frac{\partial^2}{\partial x^2} %2B \frac{\partial^2}{\partial y^2}\right).$
f is of at most polynomial growth at cusps of SL₂(Z).

A weak Maass wave form is defined similarly but without the growth condition at cusps.

References

Bump, Daniel (1997), Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, ISBN 978-0-521-55098-7, MR 1431508
Maass, Hans (1949), "Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen 121: 141–183, doi:10.1007/BF01329622, MR 0031519

Maass wave form

Definition

See also

References