# A problem of Petersson about weight 0 meromorphic modular forms

- Kathrin Bringmann
^{1}Email author and - Ben Kane
^{2}Email authorView ORCID ID profile

**3**:24

https://doi.org/10.1186/s40687-016-0072-y

© The Author(s) 2016

**Received: **3 March 2016

**Accepted: **26 June 2016

**Published: **1 December 2016

## Abstract

In this paper, we provide an explicit construction of weight 0 meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincaré series diverges and one must analytically continue via Hecke’s trick. Hecke’s trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which are modular if a certain principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of nonmeromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.

### Keywords

Meromorphic modular forms Polar harmonic Maass forms Hecke’s trick### Mathematics Subject Classification

11F03 11F12 11F25## 1 Introduction and statement of results

A special case of the Riemann–Roch theorem gives a sufficient and necessary condition for the existence of meromorphic modular forms with prescribed principal parts. Although this implies the existence of meromorphic modular forms with certain prescribed principal parts, it unfortunately fails to explicitly produce them. Using Poincaré series, Petersson achieved the goal of an explicit construction for negative-weight forms in [21, 22], but did not cover the case of weight 0 considered in this paper. This paper deals with difficulties caused by divergence of the naive Poincaré series and also views the problem from a different perspective than Petersson’s. In particular, the question is placed into the context of a larger space of nonmeromorphic modular forms, allowing the usage of modern techniques to avoid some of the difficulties of Petersson’s method.

To give a flavor of the differences between these methods, we delve a little deeper into the history of Petersson’s related work in [21]. The relevant meromorphic modular forms are constructed via a family of two-variable meromorphic Poincaré series if the corresponding group only has one cusp and the weight is negative. The Poincaré series have positive weight in one variable by construction, but the other variable can be used as a gateway between the space of positive-weight forms and their dual negative-weight counterparts. In this way, the existence of meromorphic modular forms implied by the special case of the Riemann–Roch theorem considered in [21] may be viewed as a sufficient and necessary condition for certain linear combinations of Poincaré series to satisfy (negative weight) modularity in the second variable. Following this logic, Satz 3 of [21] provides an explicit version of the existence implied by Riemann–Roch.

Petersson then pointed out two remaining tasks: The first one pertains to the inclusion of general subgroups. He later achieved this in Satz 16 of [22], with an explicit representation of the forms given in (56) of [22]. He then asked whether there is a generalization to weight 0. In this paper, we settle Petersson’s second question by viewing it in a larger space of the so-called polar harmonic Maass forms, generalizations of Bruinier–Funke harmonic Maass forms [7]. These are modular objects which are no longer meromorphic but which are instead annihilated by the hyperbolic Laplacian. In this larger space, principal parts may essentially be chosen arbitrarily and the principal part condition of Riemann–Roch translates into a condition which determines whether a polar harmonic Maass form with a given principal part is meromorphic. The subspace of harmonic Maass forms has appeared in a number of recent applications. For example, Zwegers [27] recognized the mock theta functions, introduced by Ramanujan in his last letter to Hardy, as “holomorphic parts” of half-integral weight harmonic Maass forms. Generating functions for central values and derivatives of quadratic twists of weight 2 modular *L*-functions were later proven to be weight 1/2 harmonic Maass forms by Bruinier and Ono [8]. Duke, Imamo\(\overline{\text {g}}\)lu, and Tóth [10] used weight 2 harmonic Maass forms to evaluate the inner products between modular functions.

In this paper, we instead investigate Maass forms which are also allowed to grow at points in the upper half-plane. Such forms are of growing interest because they yield lifts of meromorphic modular forms, which occur throughout various applications. Just to mention a few examples, Duke and Jenkins [9] studied traces of meromorphic modular forms and such functions are also of importance for constructing canonical lifts [2, 12].

*z*approaches \(\varrho \in \mathcal {S}_N\) and no other singularities in \(\mathbb {H}\cup \mathcal {S}_N\). Thus, for an explicit construction of weight 0 forms, it only remains to build forms with singularities in the upper half-plane. In particular, the main step in this paper is to use Hecke’s trick to explicitly define a family of polar harmonic Maass forms \(\mathcal {Y}_{0,n,N}(\mathfrak {z},z)\) in (5.9) with principal parts \(X_{\mathfrak {z}}^{n}(z)\) at \(z=\mathfrak {z}\) and no other singularities in \(\Gamma _0(N)\backslash \mathbb {H}\cup \mathcal {S}_N\), where

### Theorem 1.1

*n*th Fourier coefficient of

*g*at the cusp \(\varrho \) and \(a_{g,\mathfrak {z}}(n)\) is the

*n*th coefficient in the elliptic expansion of

*g*around \(\mathfrak {z}\). Specifically, the weight \(2-2k\) polar harmonic Maass form

### Remarks

- (1)
Note that (1.1) only has to be checked for \(\dim _{\mathbb {C}} S_{2k} (N)\) many cusp forms.

- (2)
For genus 0 subgroups there is a simpler direct proof of Theorem 1.1, using explicit basis elements.

- (3)
An alternative approach for constructing basis elements is to average coefficients in the elliptic expansion of a Maass form. For a good description of such types of Poincaré series, see [14], while the case of forms with singularities at the cusps [13, 16].

By computing the Fourier coefficients of the basis elements, Theorem 1.1 yields the Fourier expansions of all meromorphic modular forms.

### Corollary 1.2

*f*on \(\Gamma _0(N)\) with principal parts at each cusp \(\varrho \) equal to \(\sum _{n<0} a_{\varrho }(n) e^{\frac{2\pi i nz}{\ell _{\varrho }}}\) and principal parts in \(\mathbb {H}\) given by

*f*has a Fourier expansion which is valid for

*y*sufficiently large (depending on \(v_1,\ldots ,v_d\)). For \(m\in \mathbb {N}\), the

*m*th Fourier coefficient of

*f*is given by

*c*(

*g*,

*m*) the

*m*th coefficient of

*g*. The coefficients of

*f*are explicitly given, independent of \(\mathcal {P}_{0,n,N}^{\varrho }\) and \(\mathcal {Y}_{0,n,N}\), in Theorem 6.2.

### Remarks

- (1)
While the functions \(\mathcal {Y}_{2-2k,n,N}\) are useful for Theorem 1.1, for \(k>1\) there are forms constructed from \(\mathcal {Y}_{2-2k,-1,N}\) by applying another natural differential operator, known as the raising operator, which yield Fourier expansions closely resembling the expansions given by Hardy and Ramanujan for the reciprocal of the weight 6 Eisenstein series. The authors [5] applied this method to obtain the Fourier expansions of negative-weight meromorphic modular forms.

- (2)
The expansions of

*f*at the other cusps can easily be derived from Lemma 5.4 and the definition (5.9) of \(\mathcal {Y}_{0,n,N}\). - (3)
The result for \(k>1\) was proven by Petersson (see (69) of [22]). Furthermore, in this case, one can recognize the

*m*th coefficient of \(\mathcal {Y}_{2-2k,-1,N}\) as a constant multiple of the weight 2*k*Poincaré series with principal part \(e^{-2\pi i m \mathfrak {z}}\) at \(i\infty \). Using this, one can write the Fourier coefficients of a given weight \(2-2k\) meromorphic modular form as the image of an operator acting on weight 2*k*meromorphic modular forms. The explicit version of Corollary 1.2 given in Theorem 6.2 yields an analogous operator on weight 2 meromorphic modular forms, but we do not work out the details here.

*Q*of discriminant \(D<0\) yields a weight 2 version of the functions

*k*cusp forms which were investigated by Zagier [25] and played an important role in Kohnen and Zagier’s construction of a kernel function [15] for the Shimura [23] and Shintani [24] lifts. Kohnen and Zagier then used this kernel function to prove the nonnegativity of twisted central

*L*-values of cusp forms. As shown by Bengoechea [3], the \(f_{k,D}\) functions, with \(D<0\), are meromorphic modular forms of weight 2

*k*with poles of order

*k*at \(\mathfrak {z}=\tau _Q\) and which decay like cusp forms toward the cusps. The authors [4] proved that inner products of these meromorphic modular forms with other meromorphic modular forms lead to a new class of modular objects, the first case of which is a polar harmonic Maass form, and that they also appear as theta lifts, which was generalized to vector-valued forms by Zemel [26]. Noting the applications of the \(f_{k,D}\) functions for \(D>0\), it may be interesting to investigate the properties of \(\mathcal {Y}_{0,-1,N}\) if

*z*is a CM-point. In particular, the periods of these forms are of interest because they have geometric applications. This will be studied further in future research.

We do however investigate one aspect of the properties in the \(\mathfrak {z}\)-variable here. Recall that if *f* is a weight 2 meromorphic modular form, then \(f(\mathfrak {z})d\mathfrak {z}\) is a meromorphic differential, and, in the classical language, we say that the differential is of the *first kind* if *f* is holomorphic, it is of the *second kind* if *f* is not holomorphic but the residue vanishes at every pole, and of the *third kind* if all of the poles are simple (for further information about the connection between differentials and meromorphic modular forms, see page 182 of [20]). One can use \(\mathcal {Y}_{0,-1,N}\) to construct differentials of all three kinds.

### Theorem 1.3

- (1)
As a function of \(\mathfrak {z}\), \(\mathcal {Y}_{0,-1,N}(\mathfrak {z},z)\) corresponds to a differential of the third kind.

- (2)
The function \(z\mapsto \xi _{0,z}(\mathcal {Y}_{0,-1,N}(\mathfrak {z},z))\) corresponds to a differential of the first kind (as a function of

*z*). In other words, \(\xi _{0,z}(\mathcal {Y}_{0,-1,N}(\mathfrak {z},z))\) is a cusp form. - (3)
The function \(z\mapsto D_z\left( \mathcal {Y}_{0,-1,N}(\mathfrak {z},z)\right) \) corresponds to a differential of the second kind.

Although we only investigate the Fourier coefficients of \(\mathcal {Y}_{0,n,N}\) in the *z*-variable, the same techniques can be applied to compute the Fourier coefficients in the \(\mathfrak {z}\)-variable. Noting the connections to differentials given above, it might be interesting to explicitly determine the behavior toward each cusp in order to compute the differential of the third kind associated with \(\mathcal {Y}_{0,-1,N}\).

Another direction future research may take is the question of whether \(\mathcal {Y}_{0,n,N}\) may be constructed in a similar manner for more general subgroups. The methods in this paper can indeed be extended to obtain more general subgroups. The main difficulty lies in proving analytic continuation of the Kloosterman zeta functions for these subgroups. Finally, we want to mention that the properties as functions of \(\mathfrak {z}\) are of interest.

The paper is organized as follows. In Sect. 2, we give background on polar harmonic Maass forms and in particular harmonic Maass forms. We then determine the shape of the elliptic expansions of polar harmonic Maass forms. In Sect. 3 we use Hecke’s trick together with a splitting of [18] to analytically continue two-variable elliptic Poincaré series \( y^{2k-1} \Psi _{2k,N}(\mathfrak {z},z)\) to include \(k=1\). After that, we determine the properties of the analytic continuation as a function of *z* in Sect. 4. In particular, \(y\Psi _{2,N}(\mathfrak {z},z)\) yields \(\mathcal {Y}_{0,-1,N}(\mathfrak {z},z)\) up to a constant multiple of the nonholomorphic weight 2 Eisenstein series \(\widehat{E}_2(\mathfrak {z})\) and is invariant under the action of \(\Gamma _0(N)\) as functions of *z*. The Fourier expansions of \(y\Psi _{2,N}(\mathfrak {z},z)\) at each cusp are then computed in Sect. 5, and an explicit basis of all polar harmonic Maass forms is constructed. Finally, in Sect. 6, we extend a pairing of Bruinier and Funke [7] to obtain a pairing between weight 0 polar harmonic Maass forms and weight 2 cusp forms. For a fixed polar harmonic Maass form, this pairing is trivial if and only if the polar harmonic Maass form is a meromorphic modular form. We conclude the paper by computing the pairing between (1.2) and every cusp form in \(S_2(N)\), yielding Theorems 1.1 and 6.2.

## 2 Preliminaries

In this section, we define the space of polar harmonic Maass forms and some of its distinguished subspaces and then determine the shape of the elliptic expansions of such forms. For background on the well-studied subspace of harmonic Maass forms, which were introduced by Bruinier and Funke, we refer the reader for example to [7, 11, 13, 16].

### 2.1 Polar harmonic Maass forms

We are now ready to define the modular objects which are central for this paper.

### Definition

*polar harmonic Maass form*of weight \(\kappa \) on \(\Gamma _0(N)\) is a function \(F:\mathbb {H}\rightarrow \overline{\mathbb {C}} := \mathbb {C}\cup \{\infty \}\) which is real analytic outside of a discrete set and which satisfies the following conditions:

- (1)For every \(M=\left( {\begin{matrix}a&{}b\\ c&{}d\end{matrix}}\right) \in \Gamma _0(N)\), we have \(F|_{\kappa }M=F\), wherewith \(j(M,z):=cz+d\).$$\begin{aligned} F(z)|_{\kappa }M :=j(M,z)^{-\kappa }F(Mz) \end{aligned}$$
- (2)The function
*F*is annihilated by the*weight*\(\kappa \)*hyperbolic Laplacian*$$\begin{aligned} \Delta _{\kappa }:=-y^2\left( \frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}\right) +i\kappa y\left( \frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right) . \end{aligned}$$ - (3)
For all \(\mathfrak {z}\in \mathbb {H}\), there exists \(n\in \mathbb {N}_0\) such that \((z-\mathfrak {z})^n F(z)\) is bounded in a neighborhood of \(\mathfrak {z}\).

- (4)
The function

*F*grows at most linear exponentially toward cusps of \(\Gamma _0(N)\).

*polar Maass form*. Moreover,

*weak Maass forms*are polar Maass forms which do not have any singularities in \(\mathbb {H}\).

*F*satisfies weight \(\kappa \) modularity, then \(\xi _{\kappa }(F)\) is modular of weight \(2-\kappa \). The kernel of \(\xi _{\kappa }\) is the subspace \(\mathcal {M}_{\kappa }(N)\) of meromorphic modular forms, while one sees from the decomposition (2.1) that if \(F\in \mathcal {H}_{\kappa }(N)\), then \(\xi _{\kappa }(F)\in \mathcal {M}_{2-\kappa }(N)\). It is thus natural to consider the subspace \(\mathcal {H}_{\kappa }^{{\text {cusp}}}(N)\subseteq \mathcal {H}_{\kappa }(N)\) consisting of those

*F*for which \(\xi _{\kappa }(F)\) is a cusp form. The space \(\mathcal {H}_{\kappa }^{{\text {cusp}}}(N)\) decomposes into the direct sum of the subspace \(H_{\kappa }^{{\text {cusp}}}(N)\) of harmonic Maass forms which map to cusp forms under \(\xi _{\kappa }\) and the subspace \(\mathbb {H}_{\kappa }^{{\text {cusp}}}(N)\) of polar harmonic Maass forms whose singularities in \(\mathbb {H}\) are all poles and which are bounded toward all cusps. In addition to \(\xi _{\kappa }\), further operators on polar Maass forms appear in another natural splitting \(\Delta _{\kappa }= -R_{\kappa -2}\circ L_{\kappa }\). Here \(R_{\kappa }:=2i \frac{\partial }{\partial z} + \kappa y^{-1}\) is the Maass

*raising operator*and \(L_{\kappa }:=-2i y^2\frac{\partial }{\partial \overline{z}}\) is the Maass

*lowering operator.*The raising (resp. lowering) operator sends weight \(\kappa \) polar Maass forms to weight \(\kappa +2\) (resp. \(\kappa -2\)) polar Maass forms with different eigenvalues.

*incomplete gamma function*. The first sum is the

*meromorphic part*of \(F_{\varrho }\), and the second sum is the

*nonmeromorphic part*of \(F_{\varrho }\). We call \(\sum _{n< 0} a_{F,\varrho }(n)e^{\frac{2\pi i nz}{ \ell _{\varrho }}}\) the

*principal part of*

*F*

*at*\(\varrho \). Furthermore, for each \(\mathfrak {z}\in \mathbb {H}\), there exist finitely many \(c_n\in \mathbb {C}\) such that, in a neighborhood around \(\mathfrak {z}\),

*principal part of*

*F*

*at*\(\mathfrak {z}\).

### 2.2 Construction of weak Maass forms

*M*-Whittaker function. Then let

*Maass–Poincaré series*associated with the cusp \(\varrho \)

*I*-Bessel function and define for \(\varrho =\alpha /\gamma \in \mathcal {S}_N\) and \(n,j\in \mathbb {Z}\), the Kloosterman sum

### Theorem 2.1

### Remark

The Fourier expansion of \(\mathcal {P}_{\kappa ,n,N}^{\varrho }\) was explicitly computed in Theorem 1.1 of [6]. Note that there are two small typos in the formula in [6]; the condition \((ad,c)=1\) in (1.11) of [6] should be replaced by \(ad\equiv 1\pmod {c}\) and the power of the cusp width \(t_{\mu }\) (denoted \(\ell _{\varrho }\) in this paper) in (1.15) should be \(1/2-k/2\) instead of \(-1/2-k/2\) (with *k* written as \(\kappa \) here).

### 2.3 Elliptic expansions of Maass forms

In this section, we determine the elliptic expansions of polar harmonic Maass forms. We assume throughout that \( k\in \mathbb {N}\).

### Proposition 2.2

- (1)Let \(\mathfrak {z}\in \mathbb {H}\) and assume that
*F*satisfies \(\Delta _{2-2k} (F) = 0\) and that there exists \(n_0\in \mathbb {N}\) such that \(r_{\mathfrak {z}}^{n_0}(z) F(z)\) is bounded in some neighborhood \(\mathcal {N}\) around \(\mathfrak {z}\). Then there exist \(a_n, b_n\in \mathbb {C}\), such that, for \(z\in \mathcal {N}\),where \(r_{\mathfrak {z}}(z):=|X_{\mathfrak {z}}(z)|\) and \(\beta (y;a,b):=\int _{0}^{y} t^{a-1}(1-t)^{b-1}\mathrm{d}t\) is the incomplete beta function.$$\begin{aligned} F(z)= & {} (z-\overline{\mathfrak {z}})^{2k-2} \sum _{n\ge -n_0} a_n X_{\mathfrak {z}}^n(z) - (z-\overline{\mathfrak {z}})^{2k-2}\nonumber \\&\times \sum _{0\le n\le n_0} b_n\beta \left( 1-r_{\mathfrak {z}}^2(z);2k-1,-n\right) X_{\mathfrak {z}}^n(z)\nonumber \\&+ (z-\overline{\mathfrak {z}})^{2k-2}\sum _{n\le -1} b_n\beta \left( r_{\mathfrak {z}}^2(z);-n,2k-1\right) X_{\mathfrak {z}}^n(z), \end{aligned}$$(2.3) - (2)
If \(F\in \mathcal {H}_{2-2k}(N)\), then, for every \(\mathfrak {z}\in \mathbb {H}\), the sums in (2.3) only run over those

*n*which satisfy \(n\equiv k-1\pmod {\omega _{\mathfrak {z}}}\). Furthermore, if \(F\in \mathcal {H}_{2-2k}^{{\text {cusp}}}(N)\), then the second sum in (2.3) vanishes.

### Proof

- (1)
The claim follows precisely as in work of Hejhal, who computed the parabolic expansions of eigenfunctions under a differential operator closely related to the hyperbolic Laplacian in Proposition 4.3 of [13].

- (2)The stabilizer group \(\Gamma _\mathfrak {z}\subseteq \Gamma _0(N)\) of \(\mathfrak {z}\) is cyclic, and we denote by
*E*one of its generators. In (2a.15) of [21], Petersson showed thatIn particular, \(r_{\mathfrak {z}}(z) \) is invariant under \(\Gamma _{\mathfrak {z}}\) and modularity of$$\begin{aligned} X_{\mathfrak {z}}(Ez)= e^{\frac{2\pi i}{\omega _{\mathfrak {z}}}} X_{\mathfrak {z}}(z). \end{aligned}$$*g*together with uniqueness of expansions in \(e^{in\theta }\) implies that, for \(c_n=a_n\) or \(c_n=b_n\),Moreover, using \(E\mathfrak {z}=\mathfrak {z}\), we have$$\begin{aligned} c_n(z-\overline{\mathfrak {z}})^{2k-2}=c_n e^{\frac{2\pi i n}{\omega _{\mathfrak {z}}}} j(E,z)^{2k-2}(Ez-\overline{\mathfrak {z}})^{2k-2}. \end{aligned}$$(2.4)Then, by (26) of [22], we have \(j\left( M,\mathfrak {z}\right) =e^{-\frac{\pi i}{\omega _{\mathfrak {z}}}}\), and thus$$\begin{aligned} j(E,z)^{2k-2}(Ez-\overline{\mathfrak {z}})^{2k-2}= j\left( E,\mathfrak {z}\right) ^{2k-2}\left( z-\overline{\mathfrak {z}}\right) ^{2k-2}. \end{aligned}$$Hence (2.4) holds if and only if \(c_n=0\) or \(n\equiv k-1 \pmod {\omega _{\mathfrak {z}}}\), yielding the first statement in (2).$$\begin{aligned} c_n=c_n e^{\frac{2\pi i \left( n+1-k\right) }{\omega _{\mathfrak {z}}}}. \end{aligned}$$To conclude the second statement in (2), we apply \(\xi _{2-2k}\) to (2.3). Sinceis a cusp form, we require that \(b_n=0\) for \(n\ge 0\). \(\square \)$$\begin{aligned} \xi _{2-2k}(F(z))=- \left( 4\mathfrak {z}_2\right) ^{2k-1}(z-\overline{\mathfrak {z}})^{-2k} \sum _{n\le n_0}\overline{b_n} X_{\mathfrak {z}}^{-n-1}(z) \end{aligned}$$

## 3 Weight zero polar harmonic Maass–Poincaré series

In this section, we define a family of Poincaré series \(\mathcal {P}_{N,s}\) via the Hecke trick and analytically continue them to \(s=0\). We follow an argument of Petersson [18], who analytically continued certain cuspidal elliptic Poincaré series. However, technical difficulties arise because the Poincaré series \(\mathcal {P}_{N,s}\) have poles in \(\mathbb {H}\). We show in Lemma 4.4 that the analytic continuations \(z\mapsto y\Psi _{2,N}(\mathfrak {z},z)\) of \(\mathcal {P}_{N,s}\) to \(s=0\) are elements of \(\mathbb {H}_{0}^{{\text {cusp}}}(N)\). Applying certain differential operators in the \(\mathfrak {z}\) variable, we construct a family of functions \( z\mapsto \mathcal {Y}_{0,m,N}(\mathfrak {z},z) \in \mathbb {H}_{0}^{{\text {cusp}}}(N)\) with principal parts \(X_{\mathfrak {z}}^m(z)\) for \(m\in -\mathbb {N}\). In Proposition 5.6 we then prove that these functions, together with constant functions and the harmonic Maass form Poincaré series \(\mathcal {P}_{0,n,N}^{\varrho }\) (with \(n<0\)), generate \(\mathcal {H}_{0}^{{\text {cusp}}}(N)\).

### 3.1 Construction of the Poincaré series and their analytic continuations

*z*. Note that the analytic continuation \(\mathcal {P}_{N,0}\), if it exists, is the weight 2 analogue (as a function of \(\mathfrak {z}\)) of Petersson’s elliptic Poincaré series

*k*modularity. This function converges absolutely uniformly for \(k>1\) with \(\mathfrak {z},z\) in compact sets in which \(M\mathfrak {z}\ne z\) for any \(M\in \Gamma _0(N)\) (see Sections 1 and 2 of [19]). In particular, the absolute convergence of \(\mathcal {P}_{N,s}\) follows from Petersson’s work by majorizing by the absolute values in \(\Psi _{2+\sigma , N,\nu _{2+\sigma }}\) for \(\sigma =\mathrm {Re}(s)>0\).

*Kloosterman sums*

### Theorem 3.1

*s*if \(\sigma >-1/2 \) for any \(\mathfrak {z},z\) for which \(M\mathfrak {z}=z\) has no solution \(M\in \Gamma _0(N)\), which we assume throughout. Furthermore, we claim that \(\sum _3\) converges absolutely locally uniformly for \(\sigma >0\) and has an analytic continuation via its Fourier expansion to

*s*with \(\sigma >-1/4\). To validate reordering, we note that since the overall expression is absolutely locally uniformly convergent for \(\sigma >0\), \(\sum _3\) is absolutely locally uniformly convergent if both \(\sum _1\) and \(\sum _2\) converge absolutely locally uniformly. We prove this convergence for \(\sum _2\) in Lemma 3.3 and for \(\sum _1\) in Lemma 3.5.

### 3.2 Analytic continuation of \(\sum _3\)

We begin by analytically continuing \(\sum _3\) in (3.1).

### Lemma 3.2

### Proof

*t*becomes

*w*equals

*c*against

*n*and

*m*converges absolutely. Since the integrands in (3.4) and (3.5) are analytic in the integration variable for \(|{\text {Im}}(t)|<y\) and \(|{\text {Im}}(w)|<\mathfrak {z}_2\), respectively, we may shift the path of integration to \({\text {Im}}(t)=-\mathrm {sgn}(n)\alpha \) and \({\text {Im}}(w)=-\mathrm {sgn}(m)\beta \), respectively, for any \(0<\alpha <y\) and \(0<\beta <\mathfrak {z}_2\). A straightforward change of variables then shows that for any \(-1/2<\sigma _0<\sigma \), the absolute value of (3.4) may be bounded against

*s*for \(-1/4<\sigma _0<\sigma \). For this we require a well-known result of Weil, which implies that for, any \(\varepsilon >0\),

*c*. Combining (3.9) with (3.7) and (3.8), the terms in (3.6) with \((m,n)\ne (0,0)\) may be bounded against

*s*for \(-1/4<\sigma _0<\sigma \).

*c*equals in this case \(F(N, 2+2s)\), where

*s*for \(\sigma >-1/2\). Since \(\zeta (2s+2)\) does not vanish for \(\sigma >-1/2\) and \(\zeta (2s+1)\) only has a simple pole at \(s=0\), the function \(s\frac{\zeta (2s+1)}{\zeta (2s+2)}\) is analytic for \(\sigma >-1/2\). The finite factor in (3.13) is clearly analytic away from a pole at \(s=-1\) and hence in particular analytic for \(\sigma >-1/2\). Similarly, the ratio of the gamma factors in (3.10) is analytic for

*s*if \(\sigma >-1/2\) and \(\mathfrak {z}_2^{-1-2s}\) is analytic for \(s\in \mathbb {C}\). It thus remains to show that \(\int _{\mathbb {R}}\varphi _s (t,z) \hbox {d}t\) is analytic in

*s*. Since \(s\mapsto \varphi _{s}(t,z)\) is clearly analytic, it suffices to bound the integrand locally uniformly. For this, we shift \(t\mapsto ty+\mathrm {Re}(z)\), to obtain

### 3.3 Analytic continuation of \(\sum _2\)

In this subsection we show that \(\sum _2\) converges absolutely uniformly inside the rectangle \(\mathcal {R}\) defined by \(|{\text {Im}}(s)|\le R\) and \(-1/2<\sigma _0\le \sigma \le \sigma _1\).

### Lemma 3.3

If \(M\mathfrak {z}=z\) has no solution \(M\in \Gamma _0(N)\), then the series \(\sum _2\) converges absolutely uniformly in \(\mathcal {R}\).

Before proving Lemma 3.3, we show a technical lemma which proves useful later.

### Lemma 3.4

### Proof

*s*| and \(\sigma \) are bounded from above, we can conclude (3.14)

We next prove Lemma 3.3.

### Proof of Lemma 3.3

*M*as \(T^nM\) with \(n\in \mathbb {Z}\) and \(M=\left( {\begin{matrix} a&{}b\\ c&{}d\end{matrix}}\right) \in \Gamma _{\infty }\backslash \Gamma _0(N)\) such that \(|\frac{a}{c}|\le \frac{1}{2}\). Abbreviating \(w:=\frac{a}{c}+n-\overline{z}\) and \(M^*\mathfrak {z}:=M\mathfrak {z}-\frac{a}{c}=-\frac{1}{c(c\mathfrak {z}+d)}\), the terms in the series \(\sum _2\) equal

*w*| and \(|M^*\mathfrak {z}|\), with constants only depending on \(\mathcal {R}\). For this, we rewrite the first term in (3.16) as

*n*and apply Lemma 3.4 for all

*n*with |

*n*| sufficiently large. In particular, one can show that if \(|n|\ge |z|+1/2+2/\mathfrak {z}_2\), then (3.14) implies that (3.19) can be bounded against

*n*, (3.18) can be bounded against \(3/2\cdot |w|\). We then use (3.15), once with

*s*and twice with \(s=1/2\), estimating (3.17) against

*M*is half of the termwise absolute value of the weight \(3+2\sigma \) Eisenstein series without its constant term and hence converges absolutely uniformly in \(\mathfrak {z}\) and \(\sigma>\sigma _0>-1/2\).

*M*is a majorant for the weight \(3+2\sigma \) Eisenstein series minus its constant term and hence uniformly converges for \(\mathfrak {z}\in \mathbb {H}\) and \(\sigma>\sigma _0>-1/2\).

*M*which does not satisfy (3.22), denoting the set of such (

*M*,

*n*) by \(\mathcal {T}\). We naively bound the contributions of \(\mathcal {T}\) to (3.1), using the original splitting from the definition of \(\sum _2\) instead of the splitting from (3.16). If

*M*does not satisfy (3.22), then \(c^2\mathfrak {z}_2\le |c|\cdot |c\mathfrak {z}+d|< 2y^{-1}\) and hence

*c*satisfying (3.24), if (3.22) is not satisfied, then \( |c\mathfrak {z}_1+d|<|c\mathfrak {z}+d| <2(yc)^{-1}\), and hence

*z*and \(\mathfrak {z}\). We moreover bound

### 3.4 Analytic continuation of \(\sum _1\)

We finally consider \(\sum _1\).

### Lemma 3.5

If \(M\mathfrak {z}=z\) has no solution \(M\in \Gamma _0(N)\), then the series \(\sum _1\) converges absolutely uniformly in \(\mathcal {R}\).

### Proof

*n*, we use \(|T^n\mathfrak {z}-\overline{z}|\ge {\text {Im}}\left( T^n\mathfrak {z}-\overline{z}\right) \ge y\) to estimate

Theorem 3.1 is now a direct consequence of (3.1) and Lemmas 3.2, 3.3, and 3.5.

## 4 Properties of \(y\Psi _{2,N}\)

In this section, we explicitly compute the analytic continuation \(y\Psi _{2,N}(\mathfrak {z},z):=\mathcal {P}_{N,0}(\mathfrak {z},z)\) and investigate its properties. In particular, we show that it is modular and harmonic in both variables.

### 4.1 The term \(\sum _3\)

In this section, we evaluate the analytic continuation of \(\sum _3\). To state the result, we let \(c_N:=-6 N^{-1}\prod _{p|N} (1+p^{-1})^{-1}=-6/[{\text {SL}}_2(\mathbb {Z}):\Gamma _0(N)]\).

### Proposition 4.1

### Proof of Proposition 4.1

*m*and

*n*, we rewrite

*y*in the sums over

*m*and

*n*, respectively. Hence the function is harmonic in both \(\mathfrak {z}\) and

*z*because it is termwise. \(\square \)

### 4.2 The term \(\sum _2\)

We next consider \(\sum _2\) in (3.1).

### Proposition 4.2

The series \(\sum _2\) with \(s=0\) converges absolutely locally uniformly in \(\mathfrak {z}\) and *z* if \(M\mathfrak {z}=z\) is not solvable for \(M\in \Gamma _0(N)\) and is meromorphic as a function of \(\mathfrak {z}\) and harmonic as a function of *z*.

### Proof

*z*. It hence suffices to prove locally uniform convergence in \(\mathfrak {z}\) and

*z*to show that \(\sum _2\) has the desired properties. Since the argument for

*z*is similar, we only prove the statement for \(\mathfrak {z}\).

*M*), we have \(\left| M\mathfrak {z}-z\right| >\delta \). For \(c\ne 0\) we rewrite

*M*), we conclude that \(|M\mathfrak {z}-z|\ge \delta \) and hence \(\mathfrak {z}\in \mathcal {N}_{z}(\delta ,v_0,|\tau _0|)\). But then \(\mathcal {N}_{z}(\delta ,v_0,|\tau _0|)\) contains the open ball around \(\tau _0\) of radius \(\varepsilon \) and is hence a neighborhood of \(\tau _0\).

We next claim that the series \(\sum _2\) converges uniformly in \(\mathcal {N}_{z}(\delta ,V,R)\). For this, we require a uniform bound for \(\sum _{\begin{array}{c} M\in \Gamma _\infty \backslash \Gamma _0 (N) \end{array}} |j(M,\mathfrak {z})|^{-3-2\sigma }\). Since this series is the termwise absolute value of the weight \(3+2\sigma \) Eisenstein series, it is well known to be smaller than a uniform constant times the value with \(\mathfrak {z}=i\). Thus (3.20) implies that the contribution to \(\sum _2\) from the terms with \(|n|>|z|+1/2+2/\mathfrak {z}_2\) may be bounded absolutely uniformly on any compact subset of \(\mathbb {H}\). Similarly, (3.23) implies a uniform estimate on compact subsets for the contribution of the terms with \(|n|\le |z|+1/2+2/\mathfrak {z}_2\) satisfying (3.22).

### 4.3 The term \(\sum _1\)

We next investigate the properties of the analytic continuation of \(\sum _1\) to \(s=0\).

### Proposition 4.3

The series \(\sum _1\) with \(s=0\) converges locally uniformly in \(\mathfrak {z}\) and *z* for which \(M\mathfrak {z}=z\) is not solvable with \(M\in \Gamma _0(N)\) and is meromorphic as a function of \(\mathfrak {z}\) and harmonic as a function of *z*.

### Proof

*z*. We again only prove locally uniform convergence in \(\mathfrak {z}\) and leave the analogous argument for

*z*to the reader. Noting that, for every \(\mathfrak {z}\in \mathcal {N}_z(\delta ,V,R)\), the inequality

### 4.4 Image under \(\xi _{0,z}\)

As mentioned in the introduction, we obtain the principal part condition from the Riemann–Roch theorem via a pairing of Bruinier and Funke [7]. This pairing is between weight 2*k* cusp forms and weight \(2-2k\) polar harmonic Maass forms which map to cusp forms under the operator \(\xi _{2k,z}\) in the splitting (2.1). For this reason, it is important to compute the image of \(y\Psi _{2,N}(\mathfrak {z},z)\) under \(\xi _{0,z}\) and prove that it is a cusp form, as we do in this section.

### Lemma 4.4

### Proof

*s*and then using (23) of [22] to switch the role of the variables, we conclude that

\(\square \)

## 5 Expansions of \(y\Psi _{2,N}(\mathfrak {z},z)\) in other cusps and limiting behavior toward the cusps and the proof of Theorem 1.3

In this section, we determine the principal parts of \(z\mapsto y\Psi _{2,N}(\mathfrak {z},z)\) and then construct a basis of \(\mathbb {H}_{0}^{{\text {cusp}}}(N)\) by applying differential operators to \(y\Psi _{2,N}(\mathfrak {z},z)\) in the \(\mathfrak {z}\) variable. For this, we show that \(y\Psi _{2,N}(\mathfrak {z},z)\) is the \(k=1\) analogue of \(y^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\) in the sense that their principal parts are as expected if \(k=1\). Before stating the proposition, we note that the principal parts coming from the meromorphic parts of \((2iy)^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\) were computed as a special case of (50) of [22]; we expound further upon this analogy in Lemma 5.5.

### Proposition 5.1

If \(\mathfrak {z}\) is an elliptic fixed point, then \(z\mapsto y\Psi _{2,N}(\mathfrak {z},z)\) vanishes identically. If \(\mathfrak {z}\in \mathbb {H}\) is not an elliptic fixed point, then, for every \(M\in \Gamma _0(N)\), \(y\Psi _{2,N}(\mathfrak {z},z)\) has principal part \(\frac{j(M,\overline{\mathfrak {z}})}{2\mathfrak {z}_2 j(M,\mathfrak {z})}X_{M\mathfrak {z}}^{-1}(z)\) around \(z=M\mathfrak {z}\) and is bounded toward all cusps.

*E*generating \(\Gamma _{\mathfrak {z}}\) and

*r*running \(\pmod {2\omega _{\mathfrak {z}}}\). The sum over

*r*then becomes

If \(\mathfrak {z}\) is not an elliptic fixed point, then the possible poles of \(y\Psi _{2,N}(\mathfrak {z},z)\) come from the terms in \(\sum _1\) and \(\sum _2\) for which \(M\mathfrak {z}=z\), since \(\sum _3\) converges for all \(\mathfrak {z},z\in \mathbb {H}\). Furthermore, by determining terms of (4.4) and (4.7) which contribute to the pole, a direct calculation yields that the residue of the principal part at \(z=M\mathfrak {z}\) must be \(i/j(M,\mathfrak {z})^2\). From this, one concludes that the principal part is \(\frac{j(M,\overline{\mathfrak {z}})}{2\mathfrak {z}_2 j(M,\mathfrak {z})} X_{M\mathfrak {z}}^{-1}(z)\).

In order to prove Proposition 5.1, it hence remains to determine the growth of \(y\Psi _{2,N}(\mathfrak {z},z)\) as *z* approaches a cusp. This is proven in a series of lemmas.

### 5.1 Cusp expansions

*z*is close to a cusp \(\alpha /\gamma \), with \(\gamma | N\), \(\gamma \ne N\), and \((\alpha ,N)=1\). Letting \(L:= \left( {\begin{matrix} \alpha &{} \beta \\ \gamma &{} \delta \end{matrix}}\right) \) with \(\alpha \delta \equiv 1\pmod {N}\), it is easy to see that

*z*approaches the cusp \(L(i\infty )\) may be determined by taking \(z\rightarrow i\infty \) on (the analytic continuation of) the right-hand side.

To determine this continuation, we decompose as in (3.1) and denote the corresponding sums by \(\sum _1(\alpha ,\gamma ),\sum _2(\alpha ,\gamma )\), and \(\sum _3(\alpha ,\gamma ).\) Note, that since \(\gamma \ne N\), \(\sum _1(\alpha ,\gamma )\) cannot occur. Moreover \(\sum _2(\alpha , \gamma )\) is treated analogously to \(\sum _2\), and in particular converges uniformly for \(-1/2<\sigma _0\le \sigma \le \sigma _1\) under the assumption that \(M\mathfrak {z}=z\) is not solvable in \(L^{-1}\Gamma _0 (N)\). Thus, if \(\sigma >-1/2\), we may directly compute the Fourier expansion of \(\sum _2(\alpha ,\gamma )\) for *y* sufficiently large to determine the growth toward the cusp \(\alpha /\gamma \). We do so in the proof of Lemma 5.4.

*cusp width*of \(\alpha /\gamma \), the minimal \(\ell \) such that \(\Gamma _{\infty }^{\ell }\) acts on \(L^{-1}\Gamma _0(N)\) from the left, is \(\ell =\ell _{\varrho }:= \frac{N/\gamma }{\left( N/\gamma , \gamma \right) }\). Moreover \(\Gamma _\infty \) acts on \(L^{-1}\Gamma _0(N)\) from the right. Thus we obtain, as in Sect. 3.2,

We next prove that the analogue of (3.9) holds. For this, we write \(K_{\alpha ,\gamma }\) in terms of the classical Kloosterman sums. To state the resulting identity, we require the natural splitting \(\ell = \ell _1 \ell _2\) with \(\ell _1\mid \gamma ^{\infty }\) and \((\ell _2, \gamma ) = 1\), where \(\ell _1\mid \gamma ^{\infty }\) means that there exists \(n\in \mathbb {N}_0\) such that \(\ell _1\mid \gamma ^n\). A straightforward calculation then shows the following.

### Lemma 5.2

From Lemma 5.2, the analogue of (3.9) then follows easily and we can argue as before for the terms \((m, n)\ne (0, 0)\).

We finally rewrite the contribution from \(m=n=0\) in a form which yields its analytic continuation to \(s=0\). In order to state the result, we let \(\ell _1 \gamma =A_1A_2\), with \(A_1\mid N_1^{\infty }\) and \(\left( A_2,N_1\right) =1\) and denote the Möbius function by \(\mu \).

### Lemma 5.3

### Proof

*c*as

*F*(

*N*,

*s*) is defined in (3.11). Using (3.12), (5.3) hence becomes (5.2), completing the proof. \(\square \)

### 5.2 Behavior toward the cusps

We are now ready to determine the growth as *z* approaches a cusp. For this we compute the Fourier expansion of \(z\mapsto y \Psi _{2,N} \left( \mathfrak {z}, z\right) \).

### Lemma 5.4

### Proof

One determines (5.5) by computing the Fourier expansion of the analytic continuation to \(s=0\) of each of the sums \(\sum _1(\alpha ,\gamma )\), \(\sum _2(\alpha ,\gamma )\), and \(\sum _3(\alpha ,\gamma )\) in the splitting analogous to (3.1) for the coset \(L^{-1}\Gamma _0(N)\). The behavior toward the cusp \(\alpha /\gamma \) then follows by taking the limit \(z\rightarrow i\infty \) termwise. The sum \(\sum _1(\alpha ,\gamma )\) vanishes unless \(L\in \Gamma _0(N)\). For \(\sum _3(\alpha ,\gamma )\) we plug \(s=0\) into the \((m,n)\ne (0,0)\) terms of (5.1) and note that by Lemma 5.3 the contribution from \(m=n=0\) is \(c_N(\gamma )/\mathfrak {z}_2\), where \(c_N(\gamma )\) is some constant. However, observing that, by Propositions 4.1, 4.2, and 4.3, \(\mathfrak {z} \mapsto y\Psi _{2,N}(\mathfrak {z},z)-c_N/\mathfrak {z}_2\) is meromorphic for every \(z\in \mathbb {H}\), we conclude that \(c_N(\gamma )=c_N\).

*y*and \(\mathfrak {z}_2\) implies that \(y>{\text {Im}}(M\mathfrak {z})\) for all \(M\in \Gamma _0(N)\). This follows because, for \(M\in \Gamma _{\infty }\), we have \({\text {Im}}(M\mathfrak {z})=\mathfrak {z}_2<y\) and for \(M=\left( {\begin{matrix}a&{}b\\ c&{}d\end{matrix}}\right) \) with \(c\ne 0\), we have

*y*and \(\mathfrak {z}_2\), one concludes that \(-\frac{1}{c^2w}-z\in -\mathbb {H}\) and hence (4.3) implies that the integral over

*t*equals

*I*-Bessel and

*J*-Bessel functions. Combining (5.8) with (4.1) and (5.6) yields the claimed expansion.

*y*sufficiently large. The Fourier expansion in Proposition 4.1 converges uniformly in \(y>y_0\) for any fixed \(y_0>0\), so the contribution to the limit from \(\sum _3\) equals

*z*. The sums \(\sum _1\) and \(\sum _2\) converge absolutely uniformly under the assumptions given in the lemma, so we may also take the limits \( z\rightarrow i\infty \) termwise; the contribution coming from \(\sum _1\) is \(2\pi \) and the limit of (5.8) vanishes, completing the proof. \(\square \)

### 5.3 A basis of polar Maass forms

### Remark

For \(k>1\), Petersson applied his differential operator \(\frac{\partial ^{-m-1}}{\partial X_{\tau _0}^{-m-1}(\mathfrak {z})}\) to the meromorphic part of \(y^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\) (see (49a) of [22]). He investigated this function and used the Residue Theorem to compute its principal part in (50) of [22].

The following lemma extends Lemma 4.4 of [5] to include \(k=1\) and level *N*.

### Lemma 5.5

### Proof

The necessary condition follows as in the proof of Lemma 4.4 of [5]. It hence suffices to show that \(z\mapsto \mathcal {Y}_{2-2k,n,N}(\mathfrak {z}, z)\) have prescribed principal parts and are indeed elements of \(\mathbb {H}_{2-2k}^{{\text {cusp}}}(N)\).

The principal parts for \(k>1\) are known by (50) of [22], but a little work is needed to translate the calculation into a statement useful for our purposes. Petersson technically only computed the principal part coming from acting with his differential operator on a function \(H_{2k,N}(\mathfrak {z},z)\). However, in Proposition 3.1 of [5] it was shown that \(H_{2k,N}\) is the meromorphic part of \((2iy)^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\). Furthermore, for \(k>1\), we also have \(y^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\in \mathbb {H}_{2-2k}^{{\text {cusp}}}(N)\) by Proposition 3.1 of [5]. Since acting by a differential operator in the independent variable \(\mathfrak {z}\) preserves both modularity and harmonicity in *z*, one easily concludes that \(\mathcal {Y}_{2-2k,n,N}\in \mathbb {H}_{2-2k}^{{\text {cusp}}}(N)\). Since the nonmeromorphic part of \(z\mapsto (2iy)^{2k-1}\Psi _{2k,N}(\mathfrak {z},z)\) is real analytic, its image under Petersson’s differential operator is also real analytic and hence does not contribute to the principal part. To conclude the claim for \(k>1\) we thus only need to plug in (50) of [22], which we next rewrite in our notation for the reader’s convenience.

*y*and \(\mathfrak {z}_2\), one easily determines that the resulting Fourier expansion converges for every \(\mathfrak {z}, z\in \mathbb {H}\) and hence these terms do not contribute to the principal parts. This completes the proof. \(\square \)

Lemma 5.5 then yields the following proposition.

### Proposition 5.6

*F*is unique up to addition by a constant (for \(k=1\)) and is explicitly given by

### 5.4 Differentials

In this section, we consider the properties of \(\mathcal {Y}_{0,-1,N}\). Using the connection between weight 2 forms and differentials, we prove Theorem 1.3.

### Proof of Theorem 1.3

(1) We have to show that \(y\Psi _{2,N}(\mathfrak {z},z)\) has precisely a simple pole at \(\mathfrak {z}=z\). As in the proof of Proposition 5.1 for nonelliptic fixed points, these correspond to the terms from \(\sum _1\) and \(\sum _2\) for which \(M\mathfrak {z}=z\), since \(\sum _3\) converges absolutely in \(\mathbb {H}\). One sees directly that the poles are at most simple and the only matrices contributing to the pole at \(\mathfrak {z}=z\) are \(M=\pm I\), yielding the principal part \(-i/(\mathfrak {z}-z)\). For elliptic fixed points, the argument is similar, except that we must make sure that the congruence conditions for the coefficients are satisfied. The congruence condition is \(-1\equiv -k\pmod {\omega _{z}}\), which in this case (\(k=1\)) is always satisfied.

## 6 Bruinier–Funke pairing and the proofs of Theorem 1.1 and Corollary 1.2

*pairing*

*Petersson inner-product*with \(\mu _N:=[{\text {SL}}_2(\mathbb {Z}):\Gamma _0(N)]\). The pairing \(\{g,F\}\) was computed for \(F\in H_{2-2k}^{{\text {cusp}}}(N)\) by Bruinier and Funke in Proposition 3.5 of [7]. The following proposition extends their evaluation of \(\{g,F\}\) to the entire space \(\mathcal {H}_{2-2k}^{{\text {cusp}}}(N)\).

### Proposition 6.1

*F*at the cusp \(\varrho \) as

### Proof

The proof closely follows the proof of Proposition 3.5 in [7]. By Proposition 5.6 and linearity, \(\mathcal {H}_{2-2k}^{{\text {cusp}}}(N)\) decomposes into the direct sum of \(H_{2-2k}^{{\text {cusp}}}(N)\) and \(\mathbb {H}_{2-2k}^{{\text {cusp}}}(N)\). Each of these subspaces then further splits into direct sums of subspaces consisting of forms with poles at precisely one cusp \(\varrho \) or precisely one point \(\mathfrak {z}\in \Gamma _0(N)\backslash \mathbb {H}\). Hence it suffices to assume that *F* has a pole at exactly one \(\mathfrak {z}\in \Gamma _0(N)\backslash \mathbb {H}\).

*F*and

*g*around \(z=\mathfrak {z}\). Since the nonmeromorphic part of

*F*is real analytic (and hence in particular has no poles) in \(\mathbb {H}\), we may directly plug in \(\varepsilon =0\) to see that the contribution from nonmeromorphic part to (6.2) vanishes. Therefore, the limit \(\varepsilon \rightarrow 0\) of (6.2) equals the limit \(\varepsilon \rightarrow 0\) of

We are finally ready to extend Satz 3 of [21] to include the case \(k=1\).

### Proof of Theorem 1.1

By Proposition 5.6, the linear combination (1.2) is a (unique, up to addition by a constant if \(k=1\)) weight \(2-2k\) polar harmonic Maass form \(F\in \mathbb {H}_{2-2k}^{{\text {cusp}}}(N)\) with the principal parts as in the theorem. Since \(F\in \mathcal {M}_{2-2k}(N)\) if and only if \(\xi _{2-2k}(F)=0\) and \(\xi _{2-2k}(F)\) is a cusp form, we conclude that \(F\in \mathcal {M}_{2-2k}(N)\) if and only if \(\xi _{2-2k}(F)\) is orthogonal to every cusp form \(g\in S_{2k}(N)\). However, by Proposition 6.1, we see that this occurs if and only if (6.1) holds for every \(g\in S_{2k}(N)\). This is the statement of the theorem.

We are now ready to give an explicit version of Corollary 1.2, which is an easy consequence of Theorem 1.1 together with Lemma 5.4. To state the theorem, we require the sum-of-divisors function \(\sigma (m):=\sum _{d\mid m} d\).

### Theorem 6.2

*y*sufficiently large (depending on \(v_1,\ldots ,v_d\)), we have the expansion

### Proof

The claim follows by computing the Fourier expansions of \(\mathcal {P}_{0,n,N}^{\varrho }\) and \(\mathcal {Y}_{0,n,N}\) for each \(n\in -\mathbb {N}\). However, since *F* is meromorphic by Theorem 1.1, we only need to compute the meromorphic parts of each Fourier expansion. We begin by plugging in the meromorphic parts of the Fourier expansions of \(\mathcal {P}_{0,n,N}^{\varrho }\) given in Theorem 2.1.

*y*is sufficiently large so that in particular there exists \(v_0>0\) satisfying \(2v_0<v_{d}<y-1/v_0\) for every \(d\in \{1,\ldots ,r\}\). By (5.10), to determine the expansions of \(\mathcal {Y}_{0,n,N}^{+}\), we only need to apply the differential operators in the definition (5.9) to the expansion of \(\mathcal {Y}_{0,-1,N}^+\). Furthermore, the expansion of \(\mathcal {Y}_{0,-1,N}^+\) may be directly obtained by taking the meromorphic part of the expansion given in Lemma 5.4 plus

### Acknowledgements

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 research of the second author was supported by Grant Project Numbers 27300314 and 17302515 of the Research Grants Council. The authors thank Paul Jenkins, Steffen Löbrich, Ken Ono, Martin Raum, Olav Richter, Larry Rolen, and Shaul Zemel for helpful comments on an earlier version of this paper.

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