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).
Let k be a half-integer and s be a complex number. 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 f is of at most polynomial growth at cusps.
A weak Maass wave 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.
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- 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
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- 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.