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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).

Contents

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 γ = ( a b c d ) ∈ Γ and all z ∈ H , we have f ( a z + b c z + d ) = ( c z + d | c z + d | ) k f ( z ) .
  • We have Δ k f = s f , where Δ k 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

    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, due to Kim and Sarnak.

    References

    Maass wave form Wikipedia
    (Text) CC BY-SA