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).
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 and all , we have .
- We have , where is the weight k hyperbolic Laplacian defined as
- 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.
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.
- 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.