On Kleinian mock modular forms

Abstract

We give an explicit and computationally efficient construction of harmonic weak Maass forms which map to weight 2 newforms under the $\xi$-operator. Our work uses a new non-analytic completion of the Kleinian $\zeta$-function from the theory of Abelian functions.

1 Introduction and statement of results

Harmonic weak Maass forms are real analytic generalizations of classical modular forms which were introduced by Bruinier and Funke in [14]. By now, harmonic weak Maass forms are ubiquitous in number theory and many other areas of mathematics and theoretical physics (see, for instance, [8, 32] and the references therein). A harmonic weak Maass form of weight $k\in \mathbb {Z}$ for a congruence subgroup $\Gamma _0(N)$ is a smooth function on the upper half-plane $\mathfrak {H}$ which transforms like a usual (holomorphic) modular form of weight k under $\Gamma _0(N)$. Rather than being holomorphic, it is annihilated by the weight k hyperbolic Laplace operator

$$\begin{aligned} \Delta _k= -y^2\left( \frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2}\right) +iky\left( \frac{\partial }{\partial x}+i\frac{\partial }{\partial y}\right) , \end{aligned}$$

where we write $\tau =x+iy$ for $z\in \mathfrak {H}$. In addition, they need to satisfy a certain growth condition at the cusps.

One of the central tools in the theory of harmonic weak Maass forms is the $\xi$-operator defined by $\xi _k:=- 2i y^k \overline{\frac{\partial }{\partial \overline{\tau }}}$. As Bruinier and Funke first showed in [14], this operator yields a surjective map from the space $H_k(N)$ of harmonic weak Maass forms of weight k for $\Gamma _0(N)$ to the space $S_{2-k}(\Gamma _0(N))$ of (holomorphic) cusp forms of dual weight $2-k$. The image of a harmonic weak Maass form under the $\xi$-operator is called the shadow of its canonical holomorphic part, which is itself referred to as a mock modular form.

As there are infinitely many preimages of a given cusp form under the $\xi$-operator, we would like to identify distinguished preimages. This can be achieved by employing Poincaré series [10, 12] or holomorphic projection [3, 20, 28]. In recent work, Ehlen, Li, and Schwagenscheidt were able to construct the so-called good preimages of CM forms using a certain theta lift. In particular, the holomorphic parts of such preimages have algebraic Fourier coefficients at $\infty$. Previous work of Bruinier, Ono, and Rhoades [13] guarantees the existence of such good preimages. Ehlen, Li, and Schwagenscheidt were able to determine the exact algebraic number field containing their Fourier coefficients.

For weight 2 newforms associated with rational elliptic curves, i.e., with rational Fourier coefficients, the first author together with Griffin, Ono, and Rolen [1] constructed distinguished preimages under the $\xi$-operator extending earlier work of Guerzhoy [24]. This construction uses a lattice-invariant completion of the Weierstrass $\zeta$-function from the classical theory of elliptic functions. When evaluated at the Eichler integral of the newform, this gives essentially a harmonic weak Maass form of weight 0.

These results were recently extended to newforms with rational Fourier coefficients of positive weight by the authors in joint work with Funke and Rosu [2].

In this paper, we extend the construction from [1] to newforms with non-rational coefficients. In order to achieve this, we require an analog of the Weierstrass $\zeta$-function in the context of Abelian functions, as we shall explain in the following paragraphs (see also Sect. 2 for further details).

Let $K/\mathbb {Q}$ be a number field and let $f\in S_2(N)$ be a newform of weight 2 for $\Gamma _0(N)$ with coefficients in K with Galois conjugates $f_1,=f,f_2,\ldots ,f_r$. We define $\Lambda _V=\omega \mathbb {Z}^r+\omega '\mathbb {Z}^r\subset \mathbb {C}^r$ to be the associated period lattice, where, slightly abusing notation, we denote by $V\subseteq X_0(N)$ the preimage of $\mathbb {C}/\Lambda _V\subseteq J(X_0(N))$ under the Abel–Jacobi map. Note that in general V is not necessarily an algebraic variety over $\mathbb {Q}$. As it turns out (see Sect. 4.1), we can choose $\omega ,\omega '$ such that $\Omega =\omega ^{-1}\omega '\in \mathfrak {H}_r$ is in the Siegel upper half-space of genus r (see (2.1)).

For $u\in \mathbb {C}^r$ and $\alpha , \beta \in \mathbb {R}^r$, we define—following e.g., [15, 22] (see also Sect. 2 for further references)—the Kleinian $\sigma$-function by

where is the Riemann theta function of characteristic (compare (2.2)) and $\eta ,\eta '$ denote the quasi-periods of V. For brevity, we sometimes write .

We then define $\zeta (u):=\nabla _u \log \sigma (u)$. This function is analytic but not Abelian, i.e., invariant under translations by lattice points $\ell \in \Lambda _V$. Following an idea of Rolen [33], we find a non-meromorphic completion $\widehat{\zeta }$ of the Kleinian $\zeta$-function in Proposition 2.5, which indeed satisfies

$$\begin{aligned} \widehat{\zeta }(u+\ell )=\widehat{\zeta }(u) \text { for all } \ell =\lambda \omega +\mu \omega '\in \Lambda _V \end{aligned}$$

wherever it is defined.

We define the vector of Eichler integrals associated with $f =(f_1,\ldots , f_r)$ as

$$\begin{aligned} \mathcal {E}(\tau ) = \left( -2\pi i \int _\tau ^{i\infty } f_1(z)dz,\ldots , -2\pi i \int _\tau ^{i\infty } f_r(z)dz\right) . \end{aligned}$$

The obstruction to modularity of $\mathcal {E}(\tau )$ is an element of $\Lambda _V$, that is $\mathcal {E}(\gamma .\tau ) -\mathcal {E}(\tau ) \in \Lambda _V$, for $\gamma \in \Gamma _0(N)$. Evaluating the Kleinian $\zeta$-function at $\mathcal {E}(\tau )$ gives a distinguished preimage of $f$.

Theorem 1.1

Let the notation be as above. The function

$$\begin{aligned} \widehat{\mathfrak {Z}}_{V}:\mathfrak {H}\rightarrow \mathbb {C}^{1\times r},\quad \tau \mapsto \widehat{\zeta }(\mathcal {E}(\tau )) \end{aligned}$$

is defined for all $\tau \in \mathfrak {H}$ such that the Kleinian $\sigma$-function $\sigma (\mathcal {E}(\tau );\Lambda _V)$ does not vanish. There it is $\Gamma _0(N)$-invariant and is annihilated by the hyperbolic Laplacian $\Delta _0$. Moreover, we have

$$\begin{aligned} \xi _0\widehat{\mathfrak {Z}}_V(\tau )=4\pi ^2 {f}^{tr}P^{-1} \end{aligned}$$

where

$$\begin{aligned} P=\frac{1}{2i}\left( {\overline{\omega }}\omega '^{tr}-\overline{\omega '}\omega ^{tr}\right) \end{aligned}$$

is positive definite.

For later reference, we denote the meromorphic part of $\widehat{\mathfrak {Z}}_V$ by

$$\begin{aligned} \mathfrak {Z}_V(\tau )=\zeta (\mathcal {E}(\tau ))-\frac{1}{2} {\mathcal {E}(\tau )}^{tr}\left( \omega ^{-1}\eta +\eta ^{tr}\omega ^{-tr}\right) +\pi \mathcal {E}(\tau )^{tr}P^{-1} \end{aligned}$$

(1.1)

so that

$$\begin{aligned} \widehat{\mathfrak {Z}}_V(\tau )=\mathfrak {Z}_V(\tau )-\pi \overline{\mathcal {E}(\tau )}^{tr}P^{-1}. \end{aligned}$$

We call $\mathfrak {Z}_V(\tau )$ the (polar) Kleinian mock modular form associated with V.

Remark 1.2

For $r=1$, i.e., a newform with rational coefficients, Theorem 1.1 yields

$$\begin{aligned} \xi _0 \widehat{\mathfrak {Z}}_V(\tau )=\frac{4\pi ^2}{{\text {Im}}(\overline{\omega }\omega ')}f(\tau )=\frac{4\pi ^2}{{\text {vol}}(\Lambda _V)}f(\tau ), \end{aligned}$$

recovering the corresponding result in [1].

The Kleinian mock modular form $\mathfrak {Z}_V(\tau )$ may have poles. This phenomenon also occurs in the setup of [1], where the authors show that there is a modular function which cancels the poles. We obtain the following result in this direction.

Theorem 1.3

Assume the notation as above and choose the characteristic of the Riemann theta function as the Riemann characteristic of the base point $\infty$. Assume in addition that V is an (n, s)-curve, i.e., an algebraic curve with a model of the form

$$\begin{aligned} y^n=x^s+\sum _{\begin{array}{c} i,j\ge 0 \\ in+j s< ns \end{array}} c_{ij}x^iy^j,\quad c_{ij}\in \mathbb {Q}\text { and }s>n\text { are coprime} \end{aligned}$$

(1.2)

and that for $\tau \in \mathfrak {H}$ the vector $\mathcal {E}(\tau )$ is not contained in the theta divisor except for points in $\Lambda _V$ (see (2.7)).

Then the principal part at $\infty$ of the scalar-valued function ${\mathfrak {z}}_V(\tau ):=\mathfrak {Z}_V(\mathcal {E}(\tau ))\cdot (1,...,1)^{tr}$ has coefficients in K and there exists a modular function $F_V$ for $\Gamma _0(N)$ with algebraic Fourier coefficients, such that ${\mathfrak {z}}_V+F_V$ is a harmonic weak Maaß form.

Remark 1.4

• 1. We note that every elliptic or hyperelliptic curve is in particular an (n, s)-curve. Ogg [31] famously proved that the modular curve $X_0(N)$ is hyperelliptic if and only if

  $$\begin{aligned} N\in \{22,23,26,28,29,30,31,33,35,37,39,41,46,47,50,59,71\}, \end{aligned}$$

  so that Theorem 1.3 applies in those cases. Furthermore, it is a well-known fact that every curve of genus 2 is hyperelliptic, wherefore our result also applies whenever the length r of the Galois orbit of the considered newform is $\le 2$ and V happens to be an algebraic curve.

• 2. Our proof of Theorem 1.3 relies on the algebraicity of the coefficients of the expansion of the Kleinian $\sigma$-function around 0 ([30, Theorem 3]). To the authors’ knowledge, this is only known for (n, s)-curves explaining this technical assumption on V.

• 3. The additional assumption that $\mathcal {E}(\tau )$ lies in the theta divisor only if $\mathcal {E}(\tau )\in \Lambda _V$ is also a technical one. It allows us to only consider the expansion of $\sigma$ around 0 rather than other points. The algebraic properties of expansions of $\sigma$ around other points do not seem to have been investigated in the literature.

• 4. The so-called Millson theta lifts of the functions ${\mathfrak {z}}_V+F_V$ fall into the framework of results of Bruinier and Ono [11], i.e., it is possible to relate the algebraicity of the Fourier coefficients of these lifts to the vanishing of the twisted central L-derivatives of the associated newform of weight 2 (compare [1] for the case of newforms with rational Fourier coefficients).

The following theorem gives the expansion of the Kleinian mock modular form at other cusps than $\infty$.

Theorem 1.5

In the situation of Theorem 1.1, let Q be an exact divisor of N (i.e., $\gcd (Q,N/Q))=1$) and $W_Q$ the corresponding Atkin–Lehner involution. Denote by $\lambda _Q\in \{\pm 1\}$ the eigenvalue of the newform f under $W_Q$, i.e., $f|W_Q=\lambda _Qf$. Further let

$$\begin{aligned} L_Q(f):=-2\pi i \int _{W_Q^{-1}.\infty }^\infty f(t)\partial t. \end{aligned}$$

Then we have

$$\begin{aligned} \widehat{\mathfrak {Z}}_V|W_Q(\tau )=\widehat{\zeta }(\lambda _Q(\mathcal {E}(\tau )-L_Q(f))). \end{aligned}$$

We remark that if the level N is square-free, this allows the computation of a Fourier expansion of $\mathfrak {Z}_V$ at all cusps since in this case it is well known that the Atkin–Lehner operators act transitively on the set of cusps of $\Gamma _0(N)$.

The paper is organized as follows. In Sect. 2, we introduce the Riemann theta function, define the Kleinian $\sigma$ and $\zeta$-function, and construct the completed Kleinian $\zeta$-function. We prove Theorem 1.1, 1.3, and 1.5 in Sect. 3 3. In Sect. 4, we present some computational examples.

2 Construction of Kleinian Abelian functions

In this section, we construct the analog of the lattice-invariant Weierstrass $\zeta$-function in the Abelian case. We employ the approach using Riemann theta functions to construct the $\sigma$-function.

2.1 The Riemann $\theta$-function

We review the definition and some basic properties of the Riemann $\theta$-function.

We denote by

$$\begin{aligned} \mathfrak {H}_g:=\{\Omega \in \mathbb {C}^{g\times g}\,:\, \Omega ^{tr}=\Omega ,\ {\text {Im}}(\Omega )>0\} \end{aligned}$$

(2.1)

the Siegel upper half-space of genus g. Here, we write $A>0$ to indicate that a real symmetric matrix A is positive definite.

Let $\alpha ,\beta \in \mathbb {R}^g$. For $u\in \mathbb {C}^g$ and $\Omega \in \mathfrak {H}_g$, we define the Riemann theta function of characteristic $\left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right]$ by

$$\begin{aligned} \theta \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] ( u;\Omega ):=\sum _{ m\in \mathbb {Z}^g} e\left( ( m+\alpha )^{tr}(u+\beta )+\frac{1}{2} (m+\alpha )^{tr}\Omega (m+\alpha )\right) , \end{aligned}$$

(2.2)

where $e(x)=\exp (2\pi i x)$. We write $\theta (u;\Omega )=\theta [0](u;\Omega )$.

This function is well known to satisfy the following transformation properties (see, e.g., [29, pp. 123, 194, 195] or [6, Theta Transformation Formula 8.6.1])

$$\begin{aligned} \theta \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] (u+\lambda \Omega +\mu ;\Omega )&=e\left( -\frac{1}{2} \lambda ^{tr} \Omega \lambda - u^{tr}\lambda -\lambda ^{tr}\beta +\mu ^{tr}\alpha \right) \theta \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] (u;\Omega ),\quad \lambda ,\mu \in \mathbb {Z}^g, \nonumber \\ \end{aligned}$$

(2.3)

$$\begin{aligned} \theta (u;\Omega +S)&=\theta (u;\Omega ),\quad \text {if } S\in \mathbb {Z}^{g\times g}_{sym}\text { even},\end{aligned}$$

(2.4)

$$\begin{aligned} \theta (\Omega ^{-1}u;-\Omega ^{-1})&=(\det -i\Omega )^{1/2}e\left( \frac{1}{2} u^{tr}\Omega ^{-1} u\right) \theta (u;\Omega ). \end{aligned}$$

(2.5)

These transformations imply that $\theta (u;\Omega )$ is a Siegel–Jacobi form of weight and index 1/2 (see [39]).

2.2 Kleinian $\zeta$-functions and their completions

Following the classical treatments by Klein [26] and Baker [4] and the more modern discussions, e.g., in [15, 22], we construct Abelian functions via the Riemann theta function.

Let V be a non-singular algebraic curve of genus g with period matrices $\omega ,\omega '\in \mathbb {C}^{g\times g}$, so that the Jacobian of V is isomorphic to the quotient $\mathbb {C}^g/\Lambda _V$ where

$$\begin{aligned} \Lambda _V=\{\omega m+\omega ' n\,:\, m,n\in \mathbb {Z}^g\}. \end{aligned}$$

Here and throughout, we choose the period matrices $\omega ,\omega '$ such that $\Omega =\omega ^{-1}\omega '\in \mathfrak {H}_g$ lies in the Siegel upper half-space. Further let $\eta ,\eta '\in \mathbb {C}^{g\times g}$ denote the quasi-period matrices of V. For an arbitrary characteristic $\left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] \in \mathbb {R}^{2\,g}$ and $u=(u_1,...,u_g)^{tr}\in \mathbb {C}^g$, we define the Kleinian $\sigma$-function of characteristic $\left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right]$ by

$$\begin{aligned} \sigma \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] (u)=\sigma \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] (u;\Lambda _V)=\exp \left( \frac{1}{2} u^{tr}\omega ^{-1}\eta u\right) \theta \left[ {\begin{matrix}\alpha \\ \beta \end{matrix}}\right] \left( \omega ^{-1}u;\omega ^{-1}\omega '\right) . \end{aligned}$$

(2.6)

Remark 2.1

• (1) The definition of the Kleinian $\sigma$-function above differs from that in [15, 22] by a constant depending on the curve V.

• (2) In [15, 22], the characteristic is fixed as the Riemann characteristic of the base point $\infty$. In particular, this implies that $\alpha ,\beta$ are half-integral. We usually choose the characteristic with half-integer entries, but not necessarily corresponding to the base point $\infty$.

• (3) If $V=E$ is an elliptic curve, then the Kleinian $\sigma$-function of characteristic $\left[ {\begin{matrix} 1/2 \\ 1/2\end{matrix}}\right]$ coincides, up to a constant factor, with the classical Weierstrass $\sigma$-function.

We require the following result on the zeros of the Kleinian $\sigma$-function (see, e.g., [29, Corollary II.3.6] and [22, p.1661]).

Lemma 2.2

Choosing $\left[ {\begin{matrix}\alpha \\ \beta \end{matrix}}\right] \in \frac{1}{2}\mathbb {Z}^{2\,g}$ as the Riemann characteristic of the base point $\infty$, we have that $\sigma (u)=0$ if and only if

$$\begin{aligned} u=\pm \left( \int _\infty ^{P_1}du+...+\int _\infty ^{P_{g-1}}du\right) +\ell , \end{aligned}$$

(2.7)

where $(P_1,...,P_{g-1})\in {\text {Sym}}^{g-1}(V)$, where ${\text {Sym}}^k(V)$ denotes the kth symmetric power of V, and $\ell \in \Lambda$.

From now on, we suppress the characteristic from the notation when it can be chosen arbitrarily.

We define the ith Kleinian $\zeta$-function by

$$\begin{aligned} \zeta _i(u)=\partial _{u_i}\log \sigma (u) \end{aligned}$$

and the Kleinian $\zeta$-function by

$$\begin{aligned} \zeta (u):=(\zeta _1(u),...,\zeta _g(u))=\nabla _u \log \sigma (u). \end{aligned}$$

The Kleinian $\wp$-functions $\wp _{ij}(u):=-\partial _{u_j}\zeta _i(u)$ are then Abelian functions with respect to the lattice $\Lambda _V$, i.e. they satisfy the transformation law

$$\begin{aligned} \wp _{ij}(u+\ell )=\wp (u),\quad \ell \in \Lambda _V, \end{aligned}$$

wherever they are defined. The $\zeta$-functions, however, are not Abelian, but rather satisfy

$$\begin{aligned} \zeta (u+\omega m+\omega ' n)=\zeta (u)+(m^{tr}\eta +n^{tr}\eta '). \end{aligned}$$

(2.8)

Again, for $V=E$ an elliptic curve this reduces to the well-known transformation law of the Weierstrass $\zeta$-function.

It is a classical fact going back to Eisenstein that the Weierstrass $\zeta$-function admits a non-analytic completion which is invariant under translation by lattice points [38]. We adapt Rolen’s proof of this property [33] to the setting of Kleinian $\zeta$-functions.

The following lemma follows by a straightforward computation.

Lemma 2.3

Let $F:\mathbb {C}^g\times \mathfrak {H}_g\rightarrow \mathbb {C}$ be a smooth function which satisfies the following functional equation

$$\begin{aligned} F(u+\Omega \lambda +\mu ;\Omega )=e\left( -2m\lambda ^{tr} u+f(\Omega ,\lambda ,\mu )\right) F(u;\Omega ),\qquad \lambda ,\mu \in \mathbb {Z}^g, \end{aligned}$$

for some real number m and some fixed function $f:\mathfrak {H}_g\times \mathbb {R}^g\times \mathbb {R}^g\rightarrow \mathbb {C}$.

Then the function defined by

$$\begin{aligned} \widetilde{F}(u;\Omega ):=\frac{1}{F(u;\Omega )}\left( i\nabla _u-4\pi m {\text {Im}}(u)^{tr}{\text {Im}}(\Omega )^{-1}\right) F(u;\Omega ) \end{aligned}$$

satisfies

$$\begin{aligned} {\widetilde{F}}(u+\Omega \lambda +\mu ;\Omega )=\widetilde{F}(u;\Omega ),\quad \text {for all }\lambda ,\mu \in \mathbb {Z}^g, \end{aligned}$$

wherever $F(u,\Omega )\ne 0$.

Remark 2.4

The differential operator

$$\begin{aligned} {\mathcal {Y}}_+:=i\nabla _u-4\pi m {\text {Im}}(u)^{tr}{\text {Im}}(\Omega )^{-1} \end{aligned}$$

(2.9)

in Lemma 2.3 looks similar to the raising operator

$$\begin{aligned} Y_+=i\partial _z-4\pi m\frac{{\text {Im}}z}{{\text {Im}}\tau } \end{aligned}$$

in the theory of Jacobi forms (see, e.g., [5]). This operator maps Jacobi forms of weight k and index m to non-holomorphic Jacobi forms of weight $k+1$ and index m. An extension of this operator in the context of Siegel–Jacobi forms is given in [39]. However, the operator ${\mathcal {Y}}_+$ in (2.9) does not respect the action of the symplectic group in the same way as (2.5). For the purpose of this paper, this is not required and the operators in [39] are not suitable for our setup.

We obtain the following result which is analogous to that in [33].

Proposition 2.5

The function

$$\begin{aligned} \widehat{\zeta }(u)=\zeta (u)-\frac{1}{2} u^{tr}\left( \omega ^{-1}\eta +\eta ^{tr}\omega ^{-tr}\right) +2\pi i{\text {Im}}(\omega ^{-1}u)^{tr}{\text {Im}}(\omega ^{-1}\omega ')^{-1}\omega ^{-1} \end{aligned}$$

is a non-meromorphic Abelian function for the lattice $\Lambda _V$, i.e. for any $\ell =\lambda \omega +\mu \omega '\in \Lambda _V$ we have

$$\begin{aligned} \widehat{\zeta }(u+\ell )=\widehat{\zeta }(u) \end{aligned}$$

wherever both sides are defined.

Proof

For simplicity, we let

$$\begin{aligned} \vartheta (u):=\theta \left[ {\begin{matrix}\alpha \\ \beta \end{matrix}}\right] \left( \omega ^{-1}u;\omega ^{-1}\omega '\right) . \end{aligned}$$

It follows immediately from Lemma 2.3 that the function

$$\begin{aligned} \frac{1}{\vartheta (u)}\nabla _u\vartheta (u)+2\pi i {\text {Im}}(\omega ^{-1}u)^{tr}{\text {Im}}(\omega ^{-1}\omega ')^{-1}\omega ^{-1} \end{aligned}$$

(2.10)

is a non-meromorphic Abelian function where it is defined. From (2.6), we obtain that

$$\begin{aligned} \zeta (u)=\nabla _u\left( \frac{1}{2} u^{tr}\omega ^{-1}\eta u +\log \vartheta (u)\right) =\frac{1}{2} u^{tr}(\omega ^{-1}\eta +\eta ^{tr}\omega ^{-tr})+\frac{1}{\vartheta (u)}\nabla _u(\vartheta (u)), \end{aligned}$$

so the claim follows. $\square$

3 Proofs of the main results

In this section, we prove the main results of the paper.

Proof of Theorem 1.1

We first note that $\zeta (u)$ is defined whenever u does not lie in the divisor of the Kleinian $\sigma$-function.

We now prove the $\Gamma _0(N)$-invariance. Let $\gamma \in \Gamma _0(N)$ and $\tau \in \mathfrak {H}$. We then have

$$\begin{aligned} \mathcal {E}(\gamma .\tau )=-2\pi i\int _{\gamma .\tau }^\infty f(t) \partial t=-2\pi i\left( \int _\tau ^\infty -\int _{\tau }^{\gamma .\tau }\right) f(t) \partial t. \end{aligned}$$

Since all components of $f$ are holomorphic cusp forms, we have

$$\begin{aligned} \int _\tau ^{\gamma .\tau } f(t)\partial t=\int _{{\mathfrak {a}}}^{\gamma .{\mathfrak {a}}} f(t)\partial t \end{aligned}$$

for an arbitrary cusp ${\mathfrak {a}}$ of $\Gamma _0(N)$ (see, e.g., [34, Proposition 10.5]). Since a path from ${\mathfrak {a}}$ to $\gamma .{\mathfrak {a}}$ lies in the cuspidal homology of $X_0(N)$, we find by definition of the period lattice $\Lambda _V$ that $\ell =-2\pi i\int _\tau ^{\gamma .\tau }f(t)dt\in \Lambda _V$. The completed Kleinian $\zeta$-function $\widehat{\zeta }(u;\Lambda _V)$ is invariant under translations by points in $\Lambda _V$ by Proposition 2.5. Therefore, it follows that the function

$$\begin{aligned} \widehat{\mathfrak {Z}}_V(\gamma .\tau )=\widehat{\zeta }(\mathcal {E}(\gamma .\tau ))=\widehat{\zeta }(\mathcal {E}(\tau )-\ell )=\widehat{\zeta }(\mathcal {E}(\tau ))=\widehat{\mathfrak {Z}}_V(\tau ) \end{aligned}$$

is indeed $\Gamma _0(N)$-invariant wherever it is defined.

We now proceed to show the properties under the action of the differential operators $\Delta _0$ and $\xi _0$. Using Proposition 2.5 and (2.10), we can write

$$\begin{aligned} \widehat{\zeta }(u)&=\frac{1}{\theta (\omega ^{-1}; \Omega )}\left( \nabla _u\theta \right) (\omega ^{-1}u; \Omega )\omega ^{-1}+2\pi i {\text {Im}}(\omega ^{-1}u)^{tr}{\text {Im}}(\Omega )^{-1}\omega ^{-1}\nonumber \\&=\frac{1}{\theta (\omega ^{-1}u;\Omega )}\left( \nabla _u\theta \right) (\omega ^{-1}; \Omega )\omega ^{-1}+\pi u^{tr}\omega ^{-tr}{\text {Im}}(\Omega )^{-1}\omega ^{-1}\nonumber \\&\qquad -\pi \overline{u}^{tr}{\overline{\omega }}^{-tr}{\text {Im}}(\Omega )^{-1}\omega ^{-1}. \end{aligned}$$

(3.1)

This immediately implies that

$$\begin{aligned} \xi _0\left( \widehat{\mathfrak {Z}}_V(\tau )\right) =4\pi ^2f^{tr}\omega ^{-tr}{\text {Im}}(\Omega )^{-1}\omega ^{-1}. \end{aligned}$$

A straightforward computation gives

$$\begin{aligned} \omega {\text {Im}}(\Omega )\omega ^{tr}=\frac{1}{2i}(\overline{\omega }\omega '^{tr}-{\overline{\omega }}'\omega ^{tr})=P. \end{aligned}$$

Since $\Omega =\omega ^{-1}\omega '\in \mathfrak {H}_r$, we have that

$$\begin{aligned} 2{\text {Im}}(\Omega )=i(\overline{\Omega }^{tr}-\Omega )=i(\overline{\omega '}^{tr}\overline{\omega }^{-tr}-\omega ^{-1}\omega ') \end{aligned}$$

is positive definite. Therefore, the same is true for

$$\begin{aligned} i\omega (\overline{\omega '}^{tr}\overline{\omega }^{-tr}-\omega ^{-1}\omega '){\overline{\omega }}^{tr}=2\overline{P} \end{aligned}$$

and hence P is positive definite. $\square$

The proof of Theorem 1.3 is in part analogous to that of the corresponding result in [1]. Since the proof given there is rather short, we give a more detailed version here.

Proof of Theorem 1.3

By our assumption on $\mathcal {E}$ and Lemma 2.2, we see that by construction $\mathfrak {Z}_V$ has a pole in $\tau$ if and only if $\mathcal {E}(\tau )\in \Lambda _V$. Since $\widehat{\zeta }(u)$ is lattice-invariant, it is therefore enough to consider the expansion of $\zeta (u)$ around $u=0$. By [30, Theorem 3], we have

$$\begin{aligned} \sigma (u)=S_{\lambda (n,s)}(u)+\text {higher-order terms}, \end{aligned}$$

where $S_{\lambda (n,s)}$ is the so-called Schur function associated with the curve V (for a precise definition see p. 192 of loc.cit.). This polynomial has rational coefficients. Since each newform $f_1,...,f_r$ has coefficients in K, so do the functions $\mathcal {E}_1,...,\mathcal {E}_r$. Therefore, we see that upon plugging $\mathcal {E}(\tau )$ into $\partial _{u_j}\sigma (u)/\sigma (u)$, the principal part of $\zeta (\mathcal {E}(\tau ))$ at $\infty$ has coefficients in K as well.

Next we show the existence of the meromorphic modular function $F_V$ which cancels all the poles of ${\mathfrak {z}}_V$ within the upper half-plane: It is well known (see, e.g., Sect. 2.2) that any partial derivative of the Kleinian $\zeta$-function yields (up to sign) a Kleinian $\wp$-function, thus a meromorphic Abelian function. Therefore, the function

$$\begin{aligned} {\mathfrak {p}}_V(\tau )=\sum _{j=1}^r\partial _{u_j}^2\log \sigma (\mathcal {E}(\tau )) \end{aligned}$$

is a meromorphic modular function with respect to $\Gamma _0(N)$ by the same argument employed in the proof of Theorem 1.1. The poles of this function within the upper half-plane are clearly at the same points as those of ${\mathfrak {z}}_V$, but strictly with higher order. As indicated in [1], we follow the proof of [19, Theorem 11.9], which states that every modular function for $\Gamma _0(N)$ is a rational function in $j(\tau )$ and $j(N\tau )$. Let $\gamma _1,...,\gamma _{\iota (N)}$, $\iota (N)=[\Gamma _0(1):\Gamma _0(N)]$, be a fixed set of representatives of $\Gamma _0(N)\backslash {\text {SL}}_2(\mathbb {Z})$ and assume $\gamma _1=\left( {\begin{matrix} 1 &{} 0 \\ 0 &{} 1 \end{matrix}}\right)$. We consider the function

$$\begin{aligned} G(X,\tau )=\sum _{i=1}^{\iota (N)} {\mathfrak {p}}_V(\gamma _i\tau )\prod _{j\ne i}(X-j(\gamma _j\tau )). \end{aligned}$$

This is clearly a polynomial in X whose coefficients are meromorphic functions in $\tau$. In fact, it is not hard to show that these coefficients are modular functions for ${\text {SL}}_2(\mathbb {Z})$, whence they are all rational functions in $j(\tau )$. We may therefore write

$$\begin{aligned} G(X,j(\tau ))=\sum _{k=0}^{\iota (N)-1} \frac{p_k\left( j(\tau )\right) }{q_k\left( j(\tau )\right) }X^k \end{aligned}$$

for certain polynomials $p_k,q_k\in \mathbb {C}[Y]$.

In fact, we can choose $p_k,q_k$ with algebraic coefficients. By assumption, V is an (n, s)-curve, so it follows from [30, Theorem 3] that the coefficients of the Taylor expansion of $\sigma$, and therefore, those of the Laurent expansions of both ${\mathfrak {z}}_V$ and ${\mathfrak {p}}_V$ are rational polynomials in the curve coefficients $c_{ij}$ in (1.2), and hence algebraic. Since the newform f has algebraic Fourier coefficients at all cusps, it also follows that the modular function ${\mathfrak {p}}_V$ has algebraic Fourier coefficients at all cusps.

Let

$$\begin{aligned} Q={\text {lcm}}(q_1,...,q_{\iota (N)})=\prod _{\ell =1}^M(Y-\alpha _\ell ) \end{aligned}$$

for some $\alpha _\ell \in {\overline{\mathbb {Q}}}$ and $M\in \mathbb {N}_0$. Now arguing exactly as in the aforementioned proof of [19, Theorem 11.9], we find that we can write

$$\begin{aligned} {\mathfrak {p}}_V(\tau )=\frac{\sum _{k=0}^{\iota (M)-1}\tilde{p}_k(j(\tau ))j(N\tau )}{\prod _{\ell =1}^M(j(\tau )-\alpha _\ell )\cdot \prod _{m\ne 1}(j(N\tau )-j(N\gamma _m\tau ))}. \end{aligned}$$

(3.2)

Note that the numerator in (3.2) is holomorphic in $\mathfrak {H}$ and each factor in the denominator yields a simple pole of ${\mathfrak {p}}_V$ in $\mathfrak {H}$. (We ignore the slight technical complication of elliptic fixed points for the sake of simplicity.) By multiplying through by all but one of the factors in the denominator (after canceling against potential zeros in the numerator), we obtain a modular function with algebraic Fourier coefficients with a simple pole precisely where ${\mathfrak {z}}_V$ has a pole. Thus, we can cancel all the poles using only modular functions with algebraic coefficients. $\square$

As the proof of Theorem 1.5 is almost literally the same as that of the analogous result in [1] (Theorem 1.2), we omit it here.

4 Examples

4.1 Computational aspects

We briefly outline how to compute the quantities required for the construction of the Kleinian mock modular forms.

Most of the facts in this section are by now fairly standard and more or less implemented in computer algebra systems like Sage [35], Magma [7], or Pari/Gp [36]. We loosely follow the accounts in [16, 37] and Kapitel VI of [23].

Let $f\in S_2(N)$ be a newform whose coefficients lie in a number field $K/\mathbb {Q}$, and let $f_1=f,f_2,...,f_r$ denote its Galois conjugates. The vector of all these conjugates is denoted by $f=(f_1,...f_r)^{tr}$. Suppose we have Fourier expansions

$$\begin{aligned} f_j(\tau )=\sum _{n=1}^\infty a_j(n)q^n,\quad q=e^{2\pi i\tau },\quad a_j(n)\in K. \end{aligned}$$

Then there is a component over $\mathbb {Q}$ of the modular curve $X_0(N)$ associated with the Galois orbit of f. Its Jacobian is given by $\mathbb {C}^r/\Lambda _f$ for the period lattice $\Lambda _f.$ We can find a basis for this lattice by computing the integrals

$$\begin{aligned} -2\pi i \int _\gamma {f}(z) \partial z, \end{aligned}$$

where $\gamma$ runs through a basis of the integral homology $H^1(X_0(N),\mathbb {Z})$, which can in turn be determined using the available functions in Sage or Magma.

This may be achieved very efficiently by evaluating holomorphic Eichler integrals

$$\begin{aligned} \mathcal {E}_j(\tau ):=-2\pi i\int _\tau ^\infty f_j(z)\partial z =\sum _{n=1}^\infty \frac{a_j(n)}{n}q^n \end{aligned}$$

at suitable points $\tau$ in the upper half-plane.

It follows from work of Hida [25] together with standard linear algebra that we can choose a basis $(a_1,...,a_r,b_1,...,b_r)$ of $H^1(X_0(N),\mathbb {Z})$ with the property that the cycles follow the intersection pattern

$$\begin{aligned} a_i\circ a_j =0,\quad b_i\circ b_j=0,\quad a_i\circ b_j=e_i\delta _{ij}, \end{aligned}$$

where $\delta _{ij}$ denotes the usual Kronecker delta and $e_1\mid e_2\mid ...\mid e_r$ are positive integers. With respect to this basis, we obtain matrices

$$\begin{aligned} \omega =-2\pi i\left( \int _{a_i}f_j(z) \partial z\right) _{i,j=1,...r},\quad \omega '=-2\pi i\left( \int _{b_i}f_j(z)\partial z\right) _{i,j=1,...,r} \end{aligned}$$

with the property that $\Omega :=\omega ^{-1}\omega '\in \mathfrak {H}_r$ lies in the Siegel upper half-space. An algorithm to compute this basis was found by Merel [27] and is implemented, e.g., in Magma.

Note that since we have

$$\begin{aligned} \widehat{\zeta }(u)=\frac{1}{\vartheta (u)}\nabla _u\vartheta (u)+\pi u^{tr}P^{-1}-\pi \overline{u}^{tr}P^{-1} \end{aligned}$$

with $\vartheta (u)=\theta \left[ {\begin{matrix} \alpha \\ \beta \end{matrix}}\right] (\omega ^{-1}u,\omega ^{-1}\omega ')$ and P as in Theorem 1.1 by (3.1), we do not require the quasi-periods $\eta ,\eta '$ to compute the Kleinian mock modular form.

4.2 Level 27

We consider the unique newform $f=\eta (3\tau )^2\eta (9\tau )^2\in S_2(27)$ associated with the elliptic curve

$$\begin{aligned} y^2+y=x^3-7 \quad \text {(LMFDB label 27.a3)}. \end{aligned}$$

It has rational coefficients and complex multiplication by $\mathbb {Q}(\sqrt{-3})$.

Since f has rational coefficients, the results in [1] apply and we can compute a mock modular form whose shadow is f (up to a constant multiple). Alternatively, we can apply the strategy of this paper and find that the period lattice of f is generated by

$$\begin{aligned} \omega =-0.294439 - 0.509984i\quad \text {and}\quad \omega '=-1.01996i. \end{aligned}$$

Using the Kleinian zeta function with characteristic $\alpha =\beta =1/2$, we employ Theorem 1.1 to construct the function

$$\begin{aligned} \mathfrak {Z}_V(\tau )=q^{-1}+\frac{1}{2} q^2-\frac{701}{5}q^5+\frac{1407}{4}q^8-\frac{40776}{11}q^{11}+\frac{37961}{2}q^{14}-\frac{2125098}{17}q^{17}+O(q^{20}), \end{aligned}$$

whose shadow is $4\pi ^2 f$. Note that the Fourier coefficients above are indeed rational numbers, which can be shown using work of Bruinier–Ono–Rhoades [13, Theorem 1.3] or Ehlen–Li–Schwagenscheidt [21, Corollary 1.2].

4.3 Level 23

The modular curve $X_0(23)$ has genus 2 and there is one Galois orbit of newforms, generated by the form with Fourier expansion

$$\begin{aligned} f(\tau )=q-\phi q^2+(2\phi -1)q^3+(\phi -1)q^4-2\phi q^5+O(q^6),\quad \phi =\frac{1-\sqrt{5}}{2}. \end{aligned}$$

We denote the Galois conjugate of f by $f^\sigma$.

The four elements $\{-1/19,0\},\{-1/17,0\},\{-1/15,0\},\{-1/11,0\}$ form a basis of $H_1(X_0$$(23),\mathbb {Z})$. Consequently we find the following basis for the period lattice

$$\begin{aligned} c_1&=\begin{pmatrix} -1.062972 + 2.060558i \\ 0.642714 + 0.710672i \end{pmatrix},\ c_2=\begin{pmatrix} 1.062972 + 2.060558i \\ -0.642714 + 0.710672i \end{pmatrix},\\ c_3&=\begin{pmatrix} 1.719925 + 0.787063i\\ 0.397219 + 1.860563i \end{pmatrix}, \ c_4=\begin{pmatrix} 1.313906\\ 2.079867 \end{pmatrix}. \end{aligned}$$

Computing the intersection pairing with the help of Magma, we compute the period matrices

$$\begin{aligned} \omega&=(c_1-c_2+c_3,c_2)=\begin{pmatrix} -0.406019 + 0.787063i &{} 1.062972 + 2.060558i\\ 1.682647 + 1.860563i &{} -0.642714 + 0.710672i \end{pmatrix},\\ \omega '&=(-c_2+c_3-c_4,c_2-c_3)=\begin{pmatrix} -0.656953 - 1.273495i &{} -0.656953 + 1.273495i \\ -1.039933 + 1.149891i &{} -1.039933 - 1.149891i \end{pmatrix}, \end{aligned}$$

so that we have

$$\begin{aligned} \Omega =\omega ^{-1}\omega '=\begin{pmatrix} 0.01074169 &{} -0.3817894 \\ -0.3817894 &{} 0.3888885 \end{pmatrix}+i\begin{pmatrix} 0.7666448 &{}-0.1730782 \\ -0.1730782&{} 0.6607763 \end{pmatrix} \in \mathfrak {H}_2. \end{aligned}$$

Choosing the characteristic $\alpha =(1/2,0)^{tr}$, $\beta =(1/2,1/2)^{tr}$, we find that $\theta (0,\Omega )=0$, so we directly obtain from Theorem 1.1 that the components of the function

$$\begin{aligned} \frac{1}{4\pi ^2}\mathfrak {Z}_{X_0(23)}\omega {\text {Im}}(\Omega ){\overline{\omega }}^{tr} \end{aligned}$$

yield preimages of the newforms f and $f^\sigma$ resp. under $\xi _0$, up to the addition of a meromorphic modular form. Their meromorphic parts are given by

$$\begin{aligned} 0.259008q^{-1} + 1.000942 + 4.868978q + 18.294037q^2 + 68.247223q^3 + 252.912538q^4 \\ + 938.377980q^5 + 3477.898343q^6 + 12892.503560q^7 + 47787.961740q^8+O(q^9) \end{aligned}$$

and

$$\begin{aligned} -0.505669q^{-1} - 1.954167 - 6.9786217q - 26.191387q^2 - 97.573609q^3\\ - 361.535343q^4 - 1341.254086q^5 - 4971.053026q^6 - 18427.581035q^7\\ - 68304.578170q^8+O(q^9). \end{aligned}$$

We consider the sum of the components of the vector-valued function. This will be a scalar-valued polar mock modular form ${\mathfrak {z}}_V(\tau )$ whose shadow is some linear combination of the newforms f and $f^{\sigma }$. Note that the coefficient of $q^{43}$ and that of $q^{109}$ in f both vanish, so it follows from the work of Bruinier–Ono–Rhoades [13], that the coefficients of $q^{43}$ and $q^{109}$ of a good preimage of f under $\xi _0$ should be algebraic numbers. Even though ${\mathfrak {z}}_V$ is not guaranteed to be a good preimage in the sense of [13], we still find that

$$\begin{aligned} {\mathfrak {z}}_V(\tau )=q^{-1}+3.864515+0.142266q+0.319448q^2+0.193313q^3+0.304709q^4\\ +0.055558q^5+0.059060q^6+0.080332q^7+0.572492q^8-0.190607 q^9+O(q^{10}) \end{aligned}$$

and the coefficient of $q^{43}$ is (within computational precision) 27/43 and that of $q^{109}$ is 942/109.

We conclude this example by mentioning a few numerical observations.

• (1) In this particular case, we see that the matrix $P=\frac{1}{2i}\left( {\overline{\omega }}\omega '^{tr}-\overline{\omega '}\omega ^{tr}\right)$ from Theorem 1.1 is diagonal; in fact, we have up to computational precision

  $$\begin{aligned} P=\begin{pmatrix} 3.741508 &{} 0 \\ 0 &{} 5.347829 \end{pmatrix}=4\pi ^2\begin{pmatrix} \langle f,f\rangle &{} 0 \\ 0 &{} \langle f^\sigma ,f^\sigma \rangle \end{pmatrix}, \end{aligned}$$

  where $\langle \cdot ,\cdot \rangle$ denotes the Petersson inner product. Possibly, this is a consequence of Haberland’s formula for subgroups (see, e.g., [17, Theorem 5.2]).

• (2) By the Petersson coefficient formula, we can write the (conditionally convergent) cuspidal Poincaré series $\mathcal P(2,1,23;\tau )=\sum _{\gamma \in \Gamma _\infty \backslash \Gamma _0(23)} e^{2\pi i\tau }|_2 \gamma$ as

  $$\begin{aligned} {\mathcal {P}}(2,1,23;\tau )=\frac{1}{4\pi \langle f,f\rangle } f(\tau )+\frac{1}{4\pi \langle f^{\sigma },f^{\sigma }\rangle } f^\sigma (\tau ). \end{aligned}$$

  It is well known that the preimage of a cuspidal Poincaré series under the $\xi$-operator is given by the so-called Maass–Poincaré series of dual weight, denoted by $\mathcal {Q}(2,-1,23;\tau )$ (see, e.g., [8, Theorem 6.10]). Computing the Fourier expansion of this Poincaré series numerically (see [8, Theorem 6.10] for a description of the coefficients) strongly suggests that indeed

  $$\begin{aligned} \widehat{{\mathfrak {z}}}_V(\tau )=\mathcal {Q}(2,-1,23;\tau )+C \end{aligned}$$

  for some constant C, which would imply that $\widehat{{\mathfrak {z}}}_V$ indeed has no poles within the upper half-plane. Since their shadows are equal, we know that the difference $\widehat{{\mathfrak {z}}}_V(\tau )-\mathcal {Q}(2,-1,23;\tau )$ is a meromorphic modular function. By analyzing the behavior at cusps (see Theorem 1.5), we see that this function must have all its poles in the upper half-plane.

• (3) Since $f|W_{23}=-f$, it follows by construction that $\mathfrak {Z}_V(\tau )+\mathfrak {Z}_V|W_{23}(\tau )$ should be a meromorphic modular function for $\Gamma _0(23)$ or rather the group $\Gamma _0(23)^+$. Indeed we find, within computational accuracy, that

  $$\begin{aligned}&\mathfrak {Z}_V(\tau )+\mathfrak {Z}_V|W_{23}(\tau )\\ \quad&=\begin{pmatrix} C(q^{-1}+\alpha +4q+7q^2+13q^3+19q^4+33q^5+47q^6+O(q^7))\\ C'(q^{-1}+\alpha '+4q+7q^2+13q^3+19q^4+33q^5+47q^6+O(q^7)) \end{pmatrix} \end{aligned}$$

  for constants $C=-2.732921...,C'=3.7329211...,\alpha =- 0.019847,\alpha '= 2.543165...$. Note that the coefficients given above agree, apart from the constant term, with those of the Hauptmodul for the group $\Gamma _0(23)^+$ given by

  $$\begin{aligned} T_{23+}(\tau )=t(\tau )+4\frac{t(\tau )}{(t(\tau )-1)},\quad t(\tau )=\frac{\eta (\tau )\eta (23\tau )}{\eta (2\tau )\eta (46\tau )} \end{aligned}$$

  (see, e.g., [18, Tables 3 and 4a], correcting an error in loc.cit.).

4.4 Level 256

We consider the newform

$$\begin{aligned} f(\tau )=q+2\sqrt{2} q+5q^9-2\sqrt{2}q^{11}+6q^{17}-6\sqrt{2}q^{19}+O(q^{21})\in S_2(256), \end{aligned}$$

which has CM by $\mathbb {Q}(\sqrt{-2})$. This is the smallest level for which there exists a CM newform with non-rational coefficients. As in the case of level 27 in Sect. 4.2, the coefficients of the Kleinian mock modular form are algebraic. As before, we denote the Galois conjugate of f by $f^\sigma$.

In this case, one may check (e.g., by going through a list of generators of $\Gamma _0(256)$) that the associated period lattice in $\mathbb {C}^2$ is generated by

$$\begin{aligned} c_1&=-2\pi i\int _{49/512}^\infty f(t)\partial t=\begin{pmatrix} 2.767505- 2.767505i \\ 1.146338+ 1.146338i\end{pmatrix},\\ c_2&=-2\pi i\int _{55/512}^\infty f(t)\partial t=\begin{pmatrix} 1.956921- 1.956921i \\ -0.810583- 0.810583i \end{pmatrix},\\ c_3&=-2\pi i\int _{29/256}^\infty f(t)\partial t=\begin{pmatrix} 0.810583+ 1.956921i\\ 1.956921+ 0.810583i \end{pmatrix},\\ c_4&=-2\pi i\int _{39/256}^\infty f(t)\partial t=\begin{pmatrix} 1.956921+ 1.956921i \\ -0.810583+ 0.810583i \end{pmatrix}. \end{aligned}$$

A suitable basis as described in Sect. 4.1 is given as follows

$$\begin{aligned} \omega&=(c_1,c_2-c_4)=\begin{pmatrix} 2.767505- 2.767505i &{} - 3.913843i \\ 1.146338+ 1.146338i &{} -1.621166i \end{pmatrix},\\ \omega '&=(c_2+c_3+2c_4,c_2+c_3+c_4)\\&=\begin{pmatrix} 6.681348+ 3.913843i &{} 4.724426+ 1.956921i\\ -0.474828+ 1.621166i &{} 0.335754+ 0.810583i \end{pmatrix}, \end{aligned}$$

yielding

$$\begin{aligned} \Omega =\omega ^{-1}\omega '=i\begin{pmatrix} 1.414213 &{} 0.707106 \\ 0.707106 &{} 0.707106 \end{pmatrix}=i\frac{\sqrt{2}}{2}\begin{pmatrix} 2 &{} 1 \\ 1 &{} 1 \end{pmatrix}\in \mathfrak {H}_2. \end{aligned}$$

Fig. 1

Plot of $|{\mathfrak {z}}_V(x+iy_1)|$ for $x\in [-1/2,1/2]$

The latter equality is to be understood within computational precision (100 digits), and the authors are not aware of a rigorous proof for this observation. However, this observation is reminiscent of the well-known fact that for an elliptic curve E (i.e. $r=1$) with complex multiplication by some quadratic order ${\mathcal {O}}$, the corresponding period lattice $\Lambda _E$ is invariant under multiplication by elements in ${\mathcal {O}}$, so up to homothety, $\Lambda _E$ contains ${\mathcal {O}}$ as a sublattice.

The sum of the components ${\mathfrak {z}}_V$ of $\mathfrak {Z}_V$ is then given by

$$\begin{aligned} {\mathfrak {z}}_V(\tau )&=q^{-1}+\sum _{n=1}^\infty a(n)q^n\\&=q^{-1}+(1.828427+ 0.585786i)q + (-0.828427 - 3.313708i)q^3\\&\qquad + (-5.254833 - 7.484271i)q^5 + (-7.487074 + 45.254833i)q^7 \\&\qquad + (27.456710 + 44.496608i)q^9+O(q^{11})\\&=q^{-1}+[(-1+2\sqrt{2})+i(2-\sqrt{2})]q +[(2-2\sqrt{2})+i(8-8\sqrt{2})]q^3\\&\qquad +\frac{1}{5}[(200-160\sqrt{2})+i(104+100\sqrt{2})]q^5 +\frac{1}{7}[(1441-1056\sqrt{2})+i224\sqrt{2}]q^7\\&\quad \quad +\frac{1}{9}[(5211-3510\sqrt{2})+i(-5238+3987\sqrt{2})]q^9+O(q^{11}). \end{aligned}$$

Again, the last equality is to be understood within computational accuracy.

Using the standard formula for the radius of convergence of a power series due to Cauchy–Hadamard (for a convenient formulation in the context of meromorphic modular forms see, e.g., [9, Corollary 1.4]), we can numerically compute the quantity

$$\begin{aligned} \frac{\log |a(n)|}{2\pi n} \end{aligned}$$

for the first few 100 values of n for which $a(n)\ne 0$. This suggests that ${\mathfrak {z}}_V$ has poles with imaginary part

$$\begin{aligned} y_1=\limsup _{n\rightarrow \infty }\frac{\log |a(n)|}{2\pi n} \approx 0.0627 \end{aligned}$$

and is holomorphic for ${\text {Im}}(\tau )>y_1$. Evaluating ${\mathfrak {z}}_V$ numerically on the line segment $\{x+iy_1\,:\, x\in [-1/2,1/2]\}$ suggests that there are two poles with imaginary part $y_1$, namely $\tau _1=x_1+iy_1\approx 0.375+i0.0627$ and $\tau _2=x_2+iy_1\approx 0.125+i0.0627$ (see Fig. 1).

We leave numerically finding further poles of this function for the interested reader.