In mathematics, a weak Maass form is a smooth function
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.
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.
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
is a weak Maass form with eigenvalue 0 under
is called a harmonic weak Maass form, or briefly a harmonic Maass form.
Moreover, denotes the incomplete gamma function (which has to be interpreted appropriately when n=0 ).
, the image of a harmonic Maass form is weakly holomorphic.
of harmonic Maass forms of weight
of weakly holomorphic modular forms of weight
It was proved by Bruinier and Funke[1] (for arbitrary weights, multiplier systems, and congruence subgroups) that this map is surjective.
Consequently, there is an exact sequence providing a link to the algebraic theory of modular forms.
If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight
equipped with the Petersson metric over the modular curve, then this differential operator can be viewed as a composition of the Hodge star operator and the antiholomorphic differential.
The notion of harmonic Maass forms naturally generalizes to arbitrary congruence subgroups and (scalar and vector valued) multiplier systems.
The above abstract definition of harmonic Maass forms together with a systematic investigation of their basic properties was first given by Bruinier and Funke.
Independently, Zwegers developed a theory of mock modular forms which also connects to harmonic Maass forms.
[4] An algebraic theory of integral weight harmonic Maass forms in the style of Katz was developed by Candelori.