Maass cusp form

In mathematics, a Maass cusp form, Maass wave form, Maaß 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 $\gamma =\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma$ and all $z\in \mathbb {H}$ , we have $f\left({\frac {az+b}{cz+d}}\right)=\left({\frac {cz+d}{|cz+d|}}\right)^{k}f(z)$ .
We have $\Delta _{k}f=sf$ , 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)+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 p^7/64 + p^−7/64. This theorem is due to Henry Kim and Peter Sarnak. It is an approximation toward Ramanujan-Petersson conjecture.

Higher dimensions

Maass cusp forms can be regarded as automorphic forms on GL(2). It is natural to define Maass cusp forms on GL(n) as spherical automorphic forms on GL(n) over the rational number field. Their existence is proved by Miller, Mueller, etc.

References

Bump, Daniel (1997), Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, doi:10.1017/CBO9780511609572, 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.

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Maass cusp form

Definition

Major results

Higher dimensions

See also

References