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.
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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.
Definition
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:
(1) For every matrixIf f is a weak Maass form with eigenvalue 0 under
Basic properties
Every harmonic Maass form f of weight k has a Fourier expansion of the form
where q=e2πiz, and n+, n− are integers depending on f . Moreover,
There is a complex anti-linear differential operator
Since
2-k of weakly holomorphic modular forms of weight 2-k . It was proved in (Bruinier & Funke 2004) (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. An important subspace of Hk is the space H+
k of those harmonic Maass forms which are mapped to cusp forms under
If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight k 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.
Examples
History
The above abstract definition of harmonic Maass forms together with a systematic investigation of their basic properties was first given by Bruinier and Funke (Bruinier & Funke 2004). However, many examples, such as Eisenstein series and Poincaré series, had already been known earlier. Independently, Zwegers developed a theory of mock modular forms which also connects to harmonic Maass forms (Zwegers 2002).
An algebraic theory of integral weight harmonic Maass forms in the style of Katz was developed by Candelori (Candelori 2014).