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 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 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 ${\displaystyle p^{7/64}}$, due to Kim and Sarnak.

See also

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

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
• 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.