Maass wave form

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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.

Major results[edit]

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.

See also[edit]