Harmonic Maass form explained
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 (for the group) if it satisfies the following three conditions:
(1) For every matrix
\begin{pmatrix}a&b\ c&d\end{pmatrix}\inSL(2,Z)
the function
satisfies the modular transformation law
(2)
is an eigenfunction of the weight hyperbolic Laplacian
\Deltak=-y2\left(
+
\right)+iky\left(
+i
\right),
where
(3)
has at most linear exponential growth at the cusp, that is, there exists a constant such that as
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
f(z)=
c+(n)qn+
c-(n)\Gamma(1-k,-4\piny)qn,
where, and
are integers depending on
Moreover,
denotes the incomplete gamma function (which has to be interpreted appropriately when). 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
\xik(f)(z)=2iyk\overline{
f(z)}.
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 by Bruinier and Funke (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
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 (scalar and vector valued) multiplier systems.
Examples
- Every weakly holomorphic modular form is a harmonic Maass form.
- The non-holomorphic Eisenstein series
of weight 2 is a harmonic Maass form of weight 2.
- Zagier's Eisenstein series of weight 3/2 is a harmonic Maass form of weight 3/2 (for the group). 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's J-function (normalized such that the constant term vanishes) 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.
Works cited
- Weierstrass mock modular forms and elliptic curves . Alfes . Claudia . Griffin . Michael . Ono . Ken . Rolen . Larry . Research in Number Theory . 2015 . 1 . 24 . 1406.0443.
- On two geometric theta lifts . Bruinier . Jan Hendrik . Funke . Jens . . 2004 . 125 . 1 . 45–90 . math/0212286 . 10.1215/S0012-7094-04-12513-8 . 0012-7094 . 2097357 . 2078210.
- Harmonic weak Maass forms: a geometric approach . Candelori . Luca . . 2014 . 360 . 1–2 . 489–517 . 10.1007/s00208-014-1043-5 . 119474785.
- Cycle integrals of the j-function and mock modular forms . Duke . William . Imamoḡlu . Özlem . Tóth . Árpad . . 2011 . 173 . 2 . 947–981 . Second Series . 10.4007/annals.2011.173.2.8 . free.
- Fourier coefficients of the resolvent for a Fuchsian group . Fay . John . . 1977 . 294 . 143–203.
- Book: Hejhal, Dennis . The Selberg Trace Formula for PSL(2,R) . 1983 . Springer-Verlag . 1001 . Lecture Notes in Mathematics.
- On the derivative of an Eisenstein series of weight one . Kudla . Steve . Rapoport . Michael . Yang . Tonghai . . 1999 . 1999 . 7 . 347–385 . 10.1155/S1073792899000185. free .
- Nombres de classes et formes modulaires de poids 3/2 . Zagier . Don . Comptes Rendus de l'Académie des Sciences, Série A . 1975 . 281 . 883–886 . fr.
- PhD thesis. Mock Theta Functions . Zwegers . S.P. . University of Utrecht . 2002 . 978-903933155-2.
Further reading
- Book: Ono, Ken . Unearthing the visions of a master: harmonic Maass forms and number theory . 2009 . Current developments in mathematics . Jerison . David . Mazur . Barry . Mrowka . Tomasz . Schmid . Wilfried . Stanley . Richard P. . Yau . Shing-Tung . International Press of Boston . 2008 . 347–454 . 978-157146139-1 . none.