# Maass wave form

In mathematics, a Maass wave form, Maass cusp form or Maass 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 Hans Maass in Maass (1949).

## Definition

Let k be an integer, s be a complex number, and Γ be a discrete subgroup of SL2(R). A Maass form of weight k for Γ with Laplace eigenvalue s is a smooth function from the upper half-plane to the complex numbers satisfying the following conditions:

• For all ${\displaystyle \gamma =\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma }$ and all ${\displaystyle z\in \mathbb {H} }$, we have ${\displaystyle f\left({\frac {az+b}{cz+d}}\right)=\left({\frac {cz+d}{|cz+d|}}\right)^{k}f(z)}$.
• We have ${\displaystyle \Delta _{k}f=sf}$, where ${\displaystyle \Delta _{k}}$ is the weight k hyperbolic Laplacian defined as
${\displaystyle \Delta _{k}=-y^{2}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)+iky{\frac {\partial }{\partial x}}.}$
• The function ƒ is of at most polynomial growth at cusps.

A weak Maass form is defined similarly but with the third condition replaced by "The function ƒ has at most linear exponential growth at cusps". Moreover, ƒ is said to be harmonic if it is annihilated by the Laplacian operator.

## Major results

Let ƒ be a weight 0 Maass cusp form. Its normalized Fourier coefficient at a prime p is bounded by p7/64+p-7/64. This theorem is due to Kim and Sarnak.