Harmonic Maass form

A complex-valued smooth function ${\displaystyle f}$ 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 matrix ${\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in {\text{SL}}(2,\mathbf {Z} )}$ the function ${\displaystyle f}$ satisfies the modular transformation law

${\displaystyle f\left({\frac {az+b}{cz+d}}\right)=(cz+d)^{k}f(z).}$

(2) ${\displaystyle f}$ is an eigenfunction of the weight k hyperbolic Laplacian

${\displaystyle \Delta _{k}=-y^{2}\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)+iky\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right),}$

where ${\displaystyle z=x+iy.}$

(3) ${\displaystyle f}$ has at most linear exponential growth at the cusp, that is, there exists a constant C > 0 such that  f (z) = O(e^Cy) as ${\displaystyle y\to \infty .}$

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

Every harmonic Maass form ${\displaystyle f}$ of weight ${\displaystyle k}$ has a Fourier expansion of the form

${\displaystyle f(z)=\sum \nolimits _{n\geq n^{+}}c^{+}(n)q^{n}+\sum \nolimits _{n\leq n^{-}}c^{-}(n)\Gamma (1-k,-4\pi ny)q^{n},}$

where  q = e^2πiz, and ${\displaystyle n^{+},n^{-}}$ are integers depending on ${\displaystyle f.}$ Moreover,

${\displaystyle \Gamma (s,y)=\int _{y}^{\infty }t^{s-1}e^{-t}dt}$

denotes the incomplete gamma function (which has to be interpreted appropriately when  n=0 ). The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of ${\displaystyle f.}$

There is a complex anti-linear differential operator ${\displaystyle \xi _{k}}$ defined by

${\displaystyle \xi _{k}(f)(z)=2iy^{k}{\overline {{\frac {\partial }{\partial {\bar {z}}}}f(z)}}.}$

Since ${\displaystyle \Delta _{k}=-\xi _{2-k}\xi _{k}}$, the image of a harmonic Maass form is weakly holomorphic. Hence, ${\displaystyle \xi _{k}}$ defines a map from the vector space ${\displaystyle H_{k}}$ of harmonic Maass forms of weight ${\displaystyle k}$ to the space ${\displaystyle M_{2-k}^{!}}$ of weakly holomorphic modular forms of weight ${\displaystyle 2-k.}$ 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

${\displaystyle 0\to M_{k}^{!}\to H_{k}\to M_{2-k}^{!}\to 0,}$

providing a link to the algebraic theory of modular forms. An important subspace of ${\displaystyle H_{k}}$ is the space ${\displaystyle H_{k}^{+}}$ of those harmonic Maass forms which are mapped to cusp forms under ${\displaystyle \xi _{k}}$.

If harmonic Maass forms are interpreted as harmonic sections of the line bundle of modular forms of weight ${\displaystyle k}$ 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.

• Every weakly holomorphic modular form is a harmonic Maass form.
• The non-holomorphic Eisenstein series

${\displaystyle E_{2}(z)=1-{\frac {3}{\pi y}}-24\sum _{n=1}^{\infty }\sigma _{1}(n)q^{n}}$

of weight 2 is a harmonic Maass form of weight 2.

• Zagier's Eisenstein series  E_3/2  of weight 3/2[2] is a harmonic Maass form of weight 3/2 (for the group Γ₀(4)). Its image under ${\displaystyle \xi _{3/2}}$ is a non-zero multiple of the Jacobi theta function

${\displaystyle \theta (z)=\sum _{n\in \mathbb {Z} }q^{n^{2}}.}$

• The derivative of the incoherent Eisenstein series of weight 1 associated to an imaginary quadratic order[3] is a harmonic Maass forms of weight 1.
• A mock modular form[4] is the holomorphic part of a harmonic Maass form.
• Poincaré series built with the M-Whittaker function are weak Maass forms.[5][6] 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  E  can be used to associate a weight 0 harmonic Maass form to  E .[7]
• The simultaneous generating series for the values on Heegner divisors and integrals along geodesic cycles of Klein's J-function (normalized such that the constant term vanishes) is a harmonic Maass form of weight 1/2.[8]

The above abstract definition of harmonic Maass forms together with a systematic investigation of their basic properties was first given by Bruinier and Funke.[1] 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.[4]

An algebraic theory of integral weight harmonic Maass forms in the style of Katz was developed by Candelori.[9]