Harmonic Maass form

In mathematics, a weak Maass form is a smooth function  f  on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of  f  under the Laplacian is zero, then  f  is called a harmonic weak Maass form, or briefly a harmonic Maass form.

A weak Maass form which has actually moderate growth at the cusps is a classical Maass wave form.

The Fourier expansions of harmonic Maass forms often encode interesting combinatorial, arithmetic, or geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties.

A complex-valued smooth function  f  on the upper half-plane  H = {z ∈ C:  Im(z) > 0}  is called a weak Maass form of integral weight k (for the group SL(2, Z)) if it satisfies the following three conditions:

If  f  is a weak Maass form with eigenvalue 0 under ${\displaystyle \Delta _{k}}$, that is, if ${\displaystyle \Delta _{k}f=0}$, then  f  is called a harmonic weak Maass form, or briefly a harmonic Maass form.

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