Weak harmonic Maass forms of weight 5/2

Weak harmonic Maass forms of weight and a mock modular form for the partition function

Scott Ahlgren Department of Mathematics
University of Illinois
Urbana, IL 61801
sahlgren@illinois.edu
 and  Nickolas Andersen Department of Mathematics
University of Illinois
Urbana, IL 61801
nandrsn4@illinois.edu
July 27, 2019
Abstract.

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
The first author was supported by a grant from the Simons Foundation (#208525 to Scott Ahlgren).

1. Introduction

A number of recent works have considered bases for spaces of weak harmonic Maass forms of small weight. Borcherds [4] and Zagier [24] (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 [16] 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, [15] 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 [6], [8] 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

(1.1)

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:

(1.2)

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 .

Theorem 1.

For each , there is a unique with Fourier expansion of the form

and

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.

Corollary 2.

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 [15] in the case of weight (see also [5] and [21]). The construction provides an exact formula for the coefficients . In particular, when we obtain the famous exact formula of Rademacher [23] for as a corollary (see Section 4 for details).

Work of Bruinier and Ono [10] 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 [3] investigates the analogous arithmetic and geometric nature of the coefficients . In analogy with [16] and [11], the coefficients are interpreted as the real quadratic traces (i.e. sums of cycle integrals) of weak Maass forms.

Remark.

When , we can construct the forms directly as in (1). We list a few examples here. Let . Then

Together with the family (1), these form a “grid” in the sense of Guerzhoy [18] or Zagier [24] (note that the integers appearing as coefficients are the same up to sign as those in (1)).

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 [15] and [17].

For modular forms and of weight , define

(1.3)

provided that this limit exists. Here denotes the usual fundamental domain for truncated at height .

Theorem 3.

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.

Corollary 4.

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).

Theorem 5.

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.

Corollary 6.

If and is a prime with , then

Remark.

It is possible to derive results analogous to Theorem 5 and Corollary 6 involving the operators for any (see, for example, [1] or [2]). For brevity, we will not record these statements here.

In the next section we provide some background material. In Section 3 we adapt the method of [15] to construct the basis described in Theorem 1. The last three sections contain the proofs of the remaining results.

Acknowledgements

We thank Kathrin Bringmann, Jan Bruinier, and Karl Mahlburg for their comments.

2. Weak harmonic Maass forms

We require some preliminaries on weak harmonic Maass forms (see for example [12], [22], or [25] for further details). For convenience, we set . If then we say that has weight and multiplier for if

(2.1)

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

(2.2)

This operator satisfies

(2.3)

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

(2.4)

where is the Dedekind sum

(2.5)

If satisfies (2.1) with then we must have , since

(2.6)

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

(2.7)

for all . Then has a Fourier expansion at which is supported on exponents of the form with . If, in addition, satisfies

(2.8)

then by the discussion in Section 3.2 below we have the Fourier expansion

(2.9)

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 .

Lemma 7.

For any and any constant , we have

Suppose that has Fourier expansion (2.9). By Lemma 7 and (2.3) we find that

Since and , we see that if and only if for all . Because of this, each form in Theorem 1 is uniquely determined by a single term. If this term is and if this term is .

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

(3.1)

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

with

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

(3.2)

Therefore we have

(3.3)

with

(3.4)

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

(3.5)

and are defined in terms of confluent hypergeometric functions (see [14, §13.14] for definitions and properties). Using (3.5) we see after a computation that

are linearly independent solutions of the differential equation

(3.6)

At the special value , we have (by [14, (13.18.2), (13.18.5)])

(3.7)

and

(3.8)

For and , define

(3.9)

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 .

Proposition 8.

Let and be as above, and suppose that and . Then we have

where

and

Here and denote the usual Bessel functions and is defined in (3.2).

Proof.

In view of (3.9) and (3.1) we take

We write

Then (3.3) becomes

Using (3.4) we find that

(3.10)

The integral in (3.10) can be written as

(3.11)

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

Using this with (3.10) and (3.11) we find that

3.3. Derivatives of nonholomorphic Maass-Poincaré series in weight .

We specialize Proposition 8 to the situation and . Using (3.7) and (3.8) and noting that for , we obtain

(3.12)

Since , we see that

(3.13)

In order to construct nontrivial forms when , we apply the method of [15] and consider the derivative

(3.14)

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 [15].

Proposition 9.

For let be defined as in (3.14). Then we have

where

Proof.

We compute

By (3.12) and (3.13) we see that if , then

(3.15)

Therefore the second sum reduces to