Harmonic Maass form

Template:Short description Template:Use shortened footnotes 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.

Definition

A complex-valued smooth function $f$ on the upper half-plane Template:Math is called a weak Maass form of integral weight Template:Mvar (for the group Template:Math) if it satisfies the following three conditions:

(1) For every matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \text{SL}(2,𝐙)$ the function $f$ satisfies the modular transformation law

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

(2) $f$ is an eigenfunction of the weight Template:Mvar hyperbolic Laplacian

${\mathrm{\Delta }}_k=-y^2(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2})+iky(\frac{\partial }{\partial x}+i\frac{\partial }{\partial y}),$

where $z=x+iy.$

(3) $f$ has at most linear exponential growth at the cusp, that is, there exists a constant Template:Math such that Template:Math as $y\to \mathrm{\infty }.$

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

Basic properties

Every harmonic Maass form $f$ of weight $k$ has a Fourier expansion of the form

$f(z)=\sum _{n\ge n^{+}}{c}^{+}(n)q^n+\sum _{n\le n^{-}}{c}^{-}(n)\mathrm{\Gamma }(1-k,-4\pi ny)q^n,$

where Template:Math, and $n^{+},n^{-}$ are integers depending on $f.$ Moreover,

$\mathrm{\Gamma }(s,y)=\underset{y}{\overset{\mathrm{\infty }}{\int }}{t}^{s-1}{e}^{-t}dt$

denotes the incomplete gamma function (which has to be interpreted appropriately when Template:Math). The first summand is called the holomorphic part, and the second summand is called the non-holomorphic part of $f.$

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

${\xi }_k(f)(z)=2i{y}^k\overline{\frac{\partial }{\partial \overline{z}}f(z)}.$

Since ${\mathrm{\Delta }}_k=-{\xi }_{2-k}{\xi }_k$, the image of a harmonic Maass form is weakly holomorphic. Hence, ${\xi }_k$ defines a map from the vector space $H_k$ of harmonic Maass forms of weight $k$ to the space $M_{2-k}^{!}$ of weakly holomorphic modular forms of weight $2-k.$ It was proved by Bruinier and FunkeTemplate:Sfn (for arbitrary weights, multiplier systems, and congruence subgroups) that this map is surjective. Consequently, there is an exact sequence

$0\to M_k^{!}\to H_k\to M_{2-k}^{!}\to 0,$

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

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

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

$E_2(z)=1-\frac{3}{\pi y}-24\sum _{n=1}^{\mathrm{\infty }}{\sigma }_1(n)q^n$

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

• Zagier's Eisenstein series Template:Math of weight 3/2Template:Sfn is a harmonic Maass form of weight 3/2 (for the group Template:Math). Its image under ${\xi }_{3/2}$ is a non-zero multiple of the Jacobi theta function

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

• The derivative of the incoherent Eisenstein series of weight 1 associated to an imaginary quadratic orderTemplate:Sfn is a harmonic Maass forms of weight 1.
• A mock modular formTemplate:Sfn is the holomorphic part of a harmonic Maass form.
• Poincaré series built with the M-Whittaker function are weak Maass forms.Template:SfnTemplate:Sfn 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 Template:Math can be used to associate a weight 0 harmonic Maass form to Template:Math.Template:Sfn
• 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.Template:Sfn

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.Template:Sfn 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.Template:Sfn

An algebraic theory of integral weight harmonic Maass forms in the style of Katz was developed by Candelori.Template:Sfn

Works cited

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Further reading

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