Maass wave form

In mathematics, a Maass wave 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 a half-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 $\gamma = \left(\begin{smallmatrix} a & b \\ c & d\end{smallmatrix}\right) \in \Gamma$ and all $\tau \in \mathbb{H}$, we have $f\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^k f(\tau)$.
• We have $\Delta_{k} f = s f$, where $\Delta_{k}$ is the weight k hyperbolic laplacian defined as $\Delta_{k} = -y^{2} \left(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\right)+ i k y \frac{\partial}{\partial x}$.
• The function f is of at most polynomial growth at cusps.

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

Major Results

Let $f$ be a weight 0 Maass cusp form. Its normalized Fourier coefficient at a prime $p$ is bounded by $p^{7/64}$, due to Kim and Sarnak.

See also

• Harmonic Maass form
• Mock modular form
• Real analytic Eisenstein series

References

• K. Bringmann, A. Folsom, Almost harmonic Maass forms and Kac–Wakimoto characters, Crelle's Journal, Volume 2014, Issue 694, Pages 179–202 (2013). DOI: 10.1515/crelle-2012-0102
• W. Duke, J. B. Friedlander and H. Iwaniec, The subconvexity problem for Artin L-Functions’', Inventiones Mathematicae, 149, pp. 489–577 (2002). Section 4. DOI: 10.1007/BF01329622.