# On the Fourier coefficients of negative index meromorphic Jacobi forms

## Abstract

In this paper, we consider the Fourier coefficients of meromorphic Jacobi forms of negative index. This extends recent work of Creutzig and the first two authors for the special case of Kac–Wakimoto characters which occur naturally in Lie theory and yields, as easy corollaries, many important PDEs arising in combinatorics such as the famous rank–crank PDE of Atkin and Garvan. Moreover, we discuss the relation of our results to partial theta functions and quantum modular forms as introduced by Zagier, which together with previous work on positive index meromorphic Jacobi forms illuminates the general structure of the Fourier coefficients of meromorphic Jacobi forms.

## 1 Introduction and statement of results

The general framework of Jacobi forms was laid down by Eichler and Zagier in [14]. This theory has played an important role in many areas of number theory, including the theory of Siegel modular forms [24], the study of central *L*-values and derivatives of twisted elliptic curves [16], and in the theory of umbral moonshine [10], just to name a few. Roughly speaking, a *Jacobi form* is a function \(\phi :{\mathbb {C}}\times {\mathbb {H}}\rightarrow {\mathbb {C}}\), where \({\mathbb {H}}:=\{\tau \in {\mathbb {C}}:{\mathrm {Im}}(\tau )>0\}\), which satisfies two transformations similar to the transformations of elliptic functions and of modular forms (see Sect. 2.1). We refer to the variable in \({\mathbb {C}}\) (denoted by *z*) as the *elliptic variable* and to the variable in \({\mathbb {H}}\) (denoted by \(\tau \)) as the *modular variable*. As any Jacobi form \(\phi \) is one-periodic as a function of *z*, it is natural to consider its Fourier expansion in terms of \(\zeta :=e(z)\), where \(e(x):=e^{2\pi ix}\). In the classical case of holomorphic Jacobi forms, the Fourier coefficients give rise to a vector-valued modular form via the *theta decomposition* of the Jacobi form (see Sect. 2.1).

If \(\phi \) has poles in the elliptic variable, the story becomes much more interesting and difficult. In this case, the Fourier coefficients depend on the choice of range of *z* and are not modular. Such coefficients played a key role in the study of the mock theta functions of Ramanujan in [28], where they were studied in relation to mock modular forms and certain Appell–Lerch sums. Subsequent extensions and applications to quantum black holes were given in [12] (see also [19] for the appearance of mock modular forms in the context of quantum gravity partition functions and AdS3/CFT2, as well as [18] for a relation between multicentered black holes and mock Siegel–Narain theta functions). Meromorphic Jacobi forms also played a key role in the study of Kac and Wakimoto characters (see [17]), as studied in [5, 15, 23]. Collectively, these works completed the picture in the case when the meromorphic Jacobi form has positive index.

*M*,

*N*)th Kac–Wakimoto character is, for \(M,N\in {\mathbb {N}}_0\) and after a change of variables, the function

Furthermore, for various choices of *M*, *N*, the functions \(\phi _{M,N}\) are of combinatorial interest. In particular, the function \(\phi _{0,1}\) is essentially the famous Andrews–Dyson–Garvan crank generating function, which was used by Andrews and Garvan [1] to provide a combinatorial explanation for the Ramanujan congruences for the partition function, as postulated by Dyson [13]. Hence, an explicit understanding of the Fourier coefficients of \(\phi _{0,N}\) gives relations between powers of the crank generating function and certain Appell–Lerch series, giving a family of PDEs generalizing the “rank–crank PDE” of Atkin and Garvan [2] (see Corollary 1.3), and generalizing families of PDEs studied by Chan et al. in [9] and by the third author in [29]. The beautiful identity of Atkin and Garvan gives a surprising connection between the rank and crank generating functions which can be used to show various congruences relating ranks and cranks, as well as useful relations between the rank and crank moments [2].

Further examples of negative index Jacobi forms may also be found in the theory of vertex operator algebras. For example, they arise in the context of certain chiral two-point functions associated with lattice theories whose trace is restricted to a simple module of a Heisenberg vertex operator algebra. The interested reader is referred to [21] Corollary 3.15 for details, and more details can also be found in [20, 22].

In this paper, we generalize the work of [4], offering a completely general picture for negative index Jacobi forms. To describe our results, we let \(m\in -\frac{1}{2}{\mathbb {N}}\), \(\tau \in {\mathbb {H}}\), and \(\varepsilon \in \{0,1\}\) and consider meromorphic functions \(\phi : {\mathbb {C}}\rightarrow {\mathbb {C}}\) that satisfy the elliptic transformation law (2.1). For example, if \(\phi _{M,N}\) is a Kac–Wakimoto character, then it transforms according to (2.1) with \(\varepsilon =\varepsilon (N)\) and \(m=\frac{M-N}{2}\), where \(\varepsilon (N)\in \{0,1\}\) is such that \(\varepsilon (N)\equiv N\pmod 2\). Note that a Jacobi form also satisfies a modular transformation law (in the suppressed variable \(\tau \)), but for our main result, only assuming (2.1) suffices.

*x*and consider the level 2

*M*Appell–Lerch sum given for \(M\in \frac{1}{2}{\mathbb {N}}\) by

**Theorem 1.1**

*Remark*

As \(\phi \) is a meromorphic function, there are only finitely many nonzero terms in the sum over *n* in the right-hand side of (1.2).

*Remark*

*m*and \(\ell \), and \(\psi \) is a meromorphic Jacobi form of index

*m*(the interested reader is also referred to [25] for important extensions of this work). Hence, Theorem 1.1 allows one to further decompose the pieces \(\psi \) in these decompositions, which yields explicit decompositions of H-harmonic Maass–Jacobi forms.

As a corollary, applying this result to the Kac–Wakimoto characters \(\phi _{M,N}\) yields the following, which extends Theorem 1.3 in [4] to the case of general Kac–Wakimoto characters. Note that the only pole of these functions occurs at \( z = 0 \) (independent of \(\tau \)) and is of order precisely *N*.

**Corollary 1.2**

**Corollary 1.3**

*Remark*

Theorem 1.1 immediately implies other PDEs for combinatorial generating functions. For example, the results in Section 3.2 of [8] in relation to the overpartition generating function immediately follow from Theorem 1.1 as applied to \(\phi _{1,3}\).

**Theorem 1.4**

In particular, Theorem 1.4 directly implies the following result, which is analogous to Theorem 1.4 of [4] (where a different range for the Fourier coefficients is used).

**Corollary 1.5**

*Remark*

Following the proof of Theorem 1.5 of [4], one finds that the partial theta functions \(\vartheta ^+_{\ell ,\varepsilon -m}\) are all quantum modular forms, so that Theorem 1.4 implies that the Fourier coefficients of a general negative index Jacobi form are expressible as derivatives of quantum modular forms times quasimodular forms. This is further explained in Sect. 2.2 (see Theorem 2.2).

The paper is organized as follows. In Sect. 2.1, we review the basic theory of Jacobi forms, theta decompositions, and the definition of Fourier coefficients of Jacobi forms. In Sect. 2.2, we discuss the theory of quantum modular forms in the context of partial theta functions. We complete the proofs of the main results in Sect. 3.

## 2 Preliminaries

### 2.1 Jacobi forms and Fourier coefficients

*holomorphic Jacobi form*is a holomorphic function \(\phi \) on \({\mathbb {C}}\times {\mathbb {H}}\) which satisfies a

*modular transformation*, together with an

*elliptic transformation*. We are mainly interested in the explicit elliptic transformation property and consider functions which transform in the complex

*z*-variable as

*m*as the index of the Jacobi form. For example, in the motivating case of the Kac–Wakimoto characters, we require the Jacobi \(\vartheta \) function given by

*k*is the weight of the Jacobi form (see Chapter 2, Section 5 of [14]).

*z*and on the choice of path of integration. To this end, following [12], we define for \(z_0\in {\mathbb {C}}\) and \(\phi \) a function satisfying the transformation in (2.1) with \(m\in \frac{1}{2}{\mathbb {Z}}\) and \(\varepsilon \in \{0,1\}\), the (slightly modified) Fourier coefficients by

*u*by

*v*of an almost holomorphic modular form, where an almost holomorphic modular form is a function of \(\tau \in {\mathbb {H}}\) which transforms as a modular form and which is a polynomial in 1 /

*v*with holomorphic coefficients, where \(\tau =u+iv\) with \(u,v\in {\mathbb {R}}\).

### 2.2 Partial theta functions and quantum modular forms

In this section, we recall some basic facts concerning quantum modular forms. We begin with the following definition, where \(|_k\) is the usual Petersson slash operator (see [26] for more background on quantum modular forms).

**Definition 2.1**

*quantum modular form*of weight \(k\in \frac{1}{2}{\mathbb {Z}}\) on a congruence subgroup \(\Gamma \) if for all \(\gamma \in \Gamma \), the cocycle

*F*(

*q*), as studied by Zagier in [27], which is given by

*q*any root of unity. Zagier’s study of

*F*depends on the “sum of tails” identity

In the decomposition of Jacobi forms of negative index, we encounter the more general partial theta functions \(\vartheta ^+_{\ell ,M,\varepsilon }(z;\tau )\) defined in (1.4). These functions turn out to yield quantum modular forms.

**Theorem 2.2**

For any \(m\in -\frac{1}{2}{\mathbb {N}}\), \(\ell \in m+{\mathbb {Z}}\), \(\varepsilon \in \{0,1\}\), and \(z\in {\mathbb {Q}}\tau +{\mathbb {Q}}\), the partial theta function \(\vartheta ^+_{\ell ,\varepsilon ,-m}(z;\tau )\) is (up to multiplication by a rational power of *q*) a quantum modular form of weight 1 / 2 whose cocycles are real-analytic except at one point.

*Proof*

*q*, and after rescaling \(\tau \mapsto \tau /m\), we are led to study a partial theta series of the form

## 3 Proofs of the main results

We begin by giving the key properties of \(F_{M,\varepsilon }\) needed for the proof of Theorem 1.1, both of which follow from direct calculations.

**Lemma 3.1**

*u*, we have the elliptic transformation property

*u*, \(F_{M,\varepsilon }(z,u)\) is a meromorphic function having only simple poles in \(z+{\mathbb {Z}}\tau +{\mathbb {Z}}\) and residue \(\frac{1}{2\pi i}\) in \(u=z\).

We are now in a position to prove our main result, Theorem 1.1.

*Proof of Theorem 1.1*

*z*. Further let \(z_0\in {\mathbb {C}}\) be such that \(z\in P_{z_0}\) and that \(\phi \) has no poles on the boundary of \(P_{z_0}\). We consider the integral

*z*. In \(v=z\), the function has residue \(\frac{1}{2\pi i} \phi (z)\), so we have

Before giving the proof of Theorem 1.4, we require the following properties of the partial theta functions under consideration, which follow from a direct calculation.

**Lemma 3.2**

*Proof of Theorem 1.4*

*Proof of Corollaries 1.2 and 1.5*

By (2.3), we find that \(\phi _{M,N}\) transforms according to (2.1) with \(\varepsilon =\varepsilon (N)\) and \(m=\frac{M-N}{2}\). Further note that \(\phi _{M,N}\) is a function whose only poles are poles of order *N* in \({\mathbb {Z}}+{\mathbb {Z}}\tau \). Corollary 1.2 then follows directly by applying (1.2) with \(z_0=-\frac{1}{2}-\frac{\tau }{2}\). Similarly, Corollary 1.5 follows directly by plugging into Theorem 1.4. \(\square \)

## Notes

### Acknowledgements

The authors thank the referee for many comments which improved the exposition of this paper.

The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation, and the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 335220-AQSER. The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011.

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