Weak harmonic Maass forms of weight and a mock modular form for the partition function
We construct a natural basis for the space of weak harmonic Maass forms of weight on the full modular group. The nonholomorphic part of the first element of this basis encodes the values of the ordinary partition function . We obtain a formula for the coefficients of the mock modular forms of weight in terms of regularized inner products of weakly holomorphic modular forms of weight , and we obtain Hecke-type relations among these mock modular forms.
2010 Mathematics Subject Classification:Primary 11F37; Secondary 11P82
A number of recent works have considered bases for spaces of weak harmonic Maass forms of small weight. Borcherds  and Zagier  (in their study of infinite product expansions of modular forms, among many other topics) made use of the basis defined by for the space of weakly holomorphic modular forms of weight in the Kohnen plus space of level . Duke, Imamoḡlu and Tóth  extended this to a basis for the space of weak harmonic Maass forms of the same weight and level and interpreted the coefficients in terms of cycle integrals of the modular -function. In subsequent work,  they constructed a similar basis in the case of weight for the full modular group, and related the coefficients of these forms to regularized inner products of an infinite family of modular functions. To construct these bases requires various types of Maass-Poincaré series. These have played a fundamental role in the theory of weak harmonic Maass forms (see for example ,  among many other works).
Here, we will construct a natural basis for the space of weak harmonic Maass forms of weight on with a certain multiplier. To develop the necessary notation, let the Dedekind eta-function be defined by
We have the generating function
where is the ordinary partition function. The transformation property
defines a multiplier system of weight on which takes values in the -th roots of unity. Let denote the space of weakly holomorphic modular forms of weight and multiplier on . Then is the first element of a natural basis for this space. To construct the basis, we set , and for each we define , where is a suitable monic polynomial in the Hauptmodul such that . We list the first few examples here:
Denote by the space of weak harmonic Maass forms of weight and multiplier on . These are real analytic functions which transform as
which are annihilated by the weight hyperbolic Laplacian
and which have at most linear exponential growth at (see the next section for details). If denotes the space of weakly holomorphic modular forms of weight and multiplier , then we have .
Here we construct a basis for the space . For negative we have , while for positive we have , where is the -operator defined in (2.2) below. In particular, the nonholomorphic part of encodes the values of the partition function.
To state the first result it will be convenient to introduce the notation
where the incomplete gamma function is given by .
For each , there is a unique with Fourier expansion of the form
The set forms a basis for . For we have , while for we have
Furthermore, for , the coefficients are real.
When , we have
Using Lemma 7 below we obtain the following corollary.
The function defined by
is a weight weak harmonic Maass form on with multiplier .
The coefficients can be computed using the formula in Proposition 11 below. We have
We will construct the functions in Section 3 using Maass-Poincaré series. Unfortunately, the standard construction does not produce any nonholomorphic forms in , and we must therefore consider derivatives of these series with respect to an auxiliary parameter. This method was recently used by Duke, Imamoḡlu and Tóth  in the case of weight (see also  and ). The construction provides an exact formula for the coefficients . In particular, when we obtain the famous exact formula of Rademacher  for as a corollary (see Section 4 for details).
Work of Bruinier and Ono  provides an algebraic formula for the coefficients (and in particular the values of ) as the trace of certain weak Maass forms over CM points. Forthcoming work of the second author  investigates the analogous arithmetic and geometric nature of the coefficients . In analogy with  and , the coefficients are interpreted as the real quadratic traces (i.e. sums of cycle integrals) of weak Maass forms.
For , the formula for which results from the construction is an infinite series whose terms are Kloosterman sums multiplied by a derivative of the -Bessel function in its index. Here we give an alternate interpretation of these coefficients involving the regularized Petersson inner product, in analogy with  and .
For modular forms and of weight , define
provided that this limit exists. Here denotes the usual fundamental domain for truncated at height .
For positive with we have
We note that when the integral defining this inner product does not converge.
The following is an immediate corollary of Theorem 3.
For positive we have
There are also Hecke relations among the forms . Let denote the Hecke operator of index on (see Section 6 for definitions).
For any and for any prime we have
Using Theorem 5 it is possible to deduce many relations among the coefficients . We record a typical example as a corollary.
If and is a prime with , then
In the next section we provide some background material. In Section 3 we adapt the method of  to construct the basis described in Theorem 1. The last three sections contain the proofs of the remaining results.
We thank Kathrin Bringmann, Jan Bruinier, and Karl Mahlburg for their comments.
2. Weak harmonic Maass forms
for every . Here denotes the slash operator defined for with by
We choose the argument of each nonzero in , and we define using the principal branch of the logarithm. For any , let denote the Maass-type differential operator which acts on differentiable functions on by
This operator satisfies
for any . So if is modular of weight , then is modular of weight . Furthermore, if and only if is holomorphic. We define the weight hyperbolic Laplacian by
In this paper, we are interested in the multiplier system which is attached to the Dedekind eta function. An explicit description of is given, for instance, in [20, Section 2.8]. Defining , we have and . For with , we have
where is the Dedekind sum
If satisfies (2.1) with then we must have , since
In what follows, we assume this consistency condition so that the forms in question are not identically zero.
Suppose that is real analytic and satisfies
for all . Then has a Fourier expansion at which is supported on exponents of the form with . If, in addition, satisfies
then by the discussion in Section 3.2 below we have the Fourier expansion
Let denote the space of functions satisfying (2.7) and (2.8) with the additional property that only finitely many of the with in (2.9) are nonzero. We call elements of weak harmonic Maass forms of weight and multiplier . Note that these forms are allowed to have poles in the nonholomorphic part. Let denote the subspaces of cusp forms, modular forms, and weakly holomorphic modular forms, respectively.
The next lemma follows from a computation. Care must be taken with the two cases and .
For any and any constant , we have
3. Construction of harmonic Maass forms and proof of Theorem 1
In order to construct the basis described in Theorem 1, we first construct Poincaré series attached to the usual Whittaker functions (similar constructions can be found in [16, 15, 8, 5, 21, 7] among others). It turns out that for positive these series are identically zero. So we must differentiate with respect to an auxiliary parameter in order to obtain nontrivial forms in this case. The construction is carried out in several subsections and is summarized in Proposition 11 below.
3.1. Poincaré series
In this section, fix with . Suppose that is a smooth function satisfying . Since , the expression
only depends on the coset , where . So the series
is well-defined if is chosen so that it converges absolutely. Each coset of corresponds to a pair with and . Because of (2.6) the terms corresponding to and are equal.
Let and define
with as for some . By comparison with the Eisenstein series of weight on , we find that
converges absolutely for .
The function has polynomial growth as , and is periodic with period 1. So we have the Fourier expansion
Using a standard argument (see, for example, [8, Proposition 3.1]) we compute the Fourier coefficients as
Here indicates that the sum is restricted to residue classes coprime to , and denotes the inverse of modulo . The second equality comes from writing and from making the change of variable . The last equality comes from writing with and gluing together the integrals for each .
If we write , then by (2.4) we obtain
where denotes the Kloosterman sum
Therefore we have
3.2. Whittaker functions and nonholomorphic Maass-Poincaré series
The Poincaré series clearly has the desired transformation behavior; in order to construct harmonic forms, we specialize to be a function which has the desired behavior under . The Whittaker functions and are linearly independent solutions of Whittaker’s differential equation
are linearly independent solutions of the differential equation
At the special value , we have (by [14, (13.18.2), (13.18.5)])
For and , define
Then as , so the series
converges for . Thus if one of the special values or is larger than , then is harmonic at this value. The following proposition describes the Fourier expansion of .
Let and be as above, and suppose that and . Then we have
Here and denote the usual Bessel functions and is defined in (3.2).
Then (3.3) becomes
Using (3.4) we find that
The integral in (3.10) can be written as
The integral is computed in [7, p. 33] in the case , and the case is similar. Write . Then
Now let and . By [19, p. 357] we have
Combining this with the expression for from [7, p. 33], we obtain
3.3. Derivatives of nonholomorphic Maass-Poincaré series in weight .
Since , we see that
In order to construct nontrivial forms when , we apply the method of  and consider the derivative
By (3.6) we have
By (3.13) we find that ; it follows that
The following proposition gives the Fourier expansion of , and is an analogue of Proposition 4 of .
For let be defined as in (3.14). Then we have