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. If the eigenvalue of under the Laplacian is zero, then 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.
Definition
A complex-valued smooth function on the upper half-plane is called a weak Maass form of integral weight if it satisfies the following three conditions: If is a weak Maass form with eigenvalue 0 under, that is, if, then is called a harmonic weak Maass form, or briefly a harmonic Maass form.
Basic properties
Every harmonic Maass form of weight has a Fourier expansion of the form where, and are integers depending on Moreover, denotes the incomplete gamma function. The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of There is a complex anti-linear differential operator defined by Since, the image of a harmonic Maass form is weakly holomorphic. Hence, defines a map from the vector space of harmonic Maass forms of weight to the space of weakly holomorphic modular forms of weight It was proved in 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 is the space 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 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 multiplier systems.
Zagier's Eisenstein series of weight 3/2 is a harmonic Maass form of weight 3/2. Its image under is a non-zero multiple of the Jacobi theta function
The derivative of the incoherent Eisenstein series of weight 1 associated to an imaginary quadratic order is a harmonic Maass forms of weight 1.
A mock modular form is the holomorphic part of a harmonic Maass form.
Poincaré series built with the M-Whittaker function are weak Maass forms,. When the spectral parameter is specialized to the harmonic point they lead to harmonic Maass forms.
The evaluation of the Weierstrass zeta function at the Eichler integral of the weight 2 new form corresponding to a rational elliptic curve can be used to associate a weight 0 harmonic Maass form to .
The simultaneous generating series for the values on Heegner divisors and integrals along geodesic cycles of Klein'sJ-function is a harmonic Maass form of weight 1/2.
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. 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. An algebraic theory of integral weight harmonic Maass forms in the style of Katz was developed by Candelori.
