Twisted Borcherds Products Associated to Genus Zero $\Gamma _{0}(N)$

Abstract

In this short note, we realize a product associated to a uniformizer for genus zero $\Gamma _{0}(N)$ twisted with a genus character as a twisted Borcherds product lifted from a weight $\frac{1}{2}$ harmonic weak Maass form for the Weil representation.

1 Introduction

In 1998, Borcherds [1] created a remarkable method for constructing meromorphic modular forms on orthogonal groups of signature (2, n) from weakly holomorphic modular forms, and he showed that the Fourier expansions of such meromorphic modular forms at cusps have incredibly beautiful product representations called the Borcherds products of type (2, n). Such seminal work of Borcherds is now known as the theory of Borcherds products, and since it was established, it has become one of the powerful tools in the study of number theory and algebraic geometry. For example, the celebrated monster denominator formula (see, e.g., [2])

$$\begin{aligned} j(z_{1})-j(z_{2})=(q_{1}^{-1}-q_{2}^{-1})\prod _{m,n>0} \left( 1-q_{1}^{m}q_{2}^{n}\right) ^{c(mn)}, \end{aligned}$$

(1.1)

where $j(\tau )$ is the modular j-invariant and $q_{j}=e^{2\pi iz_{j}}$, indeed simply follows from the fact that the Hilbert modular function $j(z_{1})-j(z_{2})$ on the left-hand side of (1.1) is a Borcherds product of type (2, 2). As a result, the product expansion on the right-hand side of (1.1) can be obtained by computing the Borcherds product of $j(z_{1})-j(z_{2})$ at the cusp $(i\infty ,i\infty )$. Similar results have been recently proved [12] to be true for a uniformizer $j_{N}(\tau )$ for a genus zero Hecke subgroup $\Gamma _{0}(N)$. In addition, it is worth remarking that the fact that a function can be realized as a Borcherds product can also be employed to obtain some subsequent amazing number theoretic or arithmetic geometric properties related to the function. For instance [11, Section 4], following the example above, as $j(z_{1})-j(z_{2})$ is a Borcherds product of type (2, 2), the rational norm of the value of $j(z_{1})-j(z_{2})$ at the point $\left( \frac{d_{1}+\sqrt{d_{1}}}{2},\frac{d_{2}+\sqrt{d_{2}}}{2}\right)$ with $d_{1},\,d_{2}$ coprime negative fundamental discriminants is computable and explicitly expressible applying the so-called big CM value formula [3, Theorem 5.2] due to Bruinier, Kudla, and Yang to its corresponding Borcherds product. This yields the Gross–Zagier CM value formula [7], the trivial case of the famous Gross–Zagier arithmetic formula [6], which indicates the prime factorization of the rational norm as an integer. Also interestingly, such a result can also be obtained by applying Schofer’s small CM value formula [10, Theorem 3.4] to the Borcherds product of type (2, 1)

$$\begin{aligned} \prod _{Q\in {\mathcal {Q}}_{d_{2}}/\mathrm{SL}_{2}({\mathbb {Z}})}\left( j(\tau )-j(\tau _{Q})\right) , \end{aligned}$$

(1.2)

where ${\mathcal {Q}}_{d_{2}}$ is the set of positive definite quadratic forms of fundamental discriminant $d_{2}$, and $\tau _{Q}$ is the imaginary quadratic point defined by the quadratic form $Q=Q(X,Y)$ via $Q(\tau _{Q},1)=0$, and computing its average value over the set of imaginary quadratic points induced by ${\mathcal {Q}}_{d_{1}}/\mathrm{SL}_{2}({\mathbb {Z}})$.

In fact, as noted in the last example, the Borcherds products of type (2, 1) are always of particular interests to number theorists due to their connections with binary quadratic forms. As such, Borcherds’ original work regarding type (2, 1) has been extensively studied by many mathematicians in related areas, and most recently, it has been generalized by Bruinier and Ono [4] (see Sect. 2 for a brief review) to constructing meromorphic modular functions from harmonic weak Maass forms, which are somehow twisted with a genus character $\chi _{\Delta }$ of quadratic forms and have Borcherds products like expansions called twisted Borcherds products. In addition, the usual Borcherds products of type (2, 1) can be interpreted as the case $\Delta =1$ in Bruinier and Ono’s general theory. As one has seen in the last example, product (1.2) involving the modular j-invariant turns out to be a nontwisted Borcherds product, and as similar results [12] hold for a uniformizer $j_{N}(\tau )$ of $\Gamma _{0}(N)$, it will be interesting to see if there is a product analogous to (1.2) involving $j_{N}(\tau )$ that would turn out to be a twisted Borcherds product. Moreover, the modular functions $j_{N}(\tau )$ have many interesting properties related to number theory, for example [5], $j_{N}(\frac{D+\sqrt{D}}{2})$ with D a negative fundamental discriminant generates the ring class field of conductor N over the imaginary quadratic field ${\mathbb {Q}}(\sqrt{D})$. Therefore, the twisted Borcherds product realizations associated to $j_{N}(\tau )$ may be useful for further study of the number theoretic consequences related to $j_{N}(\tau )$. In this short note, we aim to realize a product analogous to (1.2) associated to $j_{N}(\tau )$ as a twisted Borcherds product for $\Delta \ne 1$, and record its product expansion as a reference for further study of the properties of $j_{N}(\tau )$. As a simple consequence, we deduce a closed formula for a sum of special values of a generalized Kloosterman zeta function.

Remark 1.1

For the case $\Delta =1$, similar results for $j_{N}^{*}(\tau )$ with N square free, where $j_{N}^{*}(\tau )$ is a uniformizer for the genus zero $\Gamma _{0}(N)^{*}$, the discrete group generated by $\Gamma _{0}(N)$ and all of its Atkin–Lehner involutions, have been obtained and proved in [9] by Pippich et al. The main reason for considering $j_{N}^{*}(\tau )$ instead of $j_{N}(\tau )$ in this case has been explained in [13, Remark 2.6].

Throughout the remainder of this note, denote by ${\mathbb {H}}$ the set of complex numbers with positive imaginary part, and write $\tau =u+iv\in {\mathbb {H}}$. We now conclude this section by summarizing the main results of this work in the following.

Theorem 1.2

Let N be a positive integer such that $X_{0}(N):=\Gamma _{0}(N)\backslash ({\mathbb {H}}\cup {\mathbb {P}}^{1}({\mathbb {Q}}))$ is of genus zero and $j_{N}(\tau )$ be a uniformizer for $X_{0}(N)$. For $\beta ,\gamma \in {\mathbb {Z}}/2N{\mathbb {Z}}$, let $\Delta \ne 1$ be a positive fundamental discriminant (thus, $\Delta \ge 5$) with $\Delta \equiv \gamma ^{2}\pmod {4N}$ and d be a positive integer such that $-d$ is a negative discriminant with $-d\equiv \beta ^{2}\pmod {4N}$. Let ${\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}$ be the set of quadratic forms $AX^{2}-BXY+CY^{2}$ of discriminant $-d\Delta$ with N|A and $B\equiv \beta \gamma \pmod {2N}$ on which $\Gamma _{0}(N)$ has a group action, and let $\chi _{\Delta }(Q)$ be a generalized genus character defined for $Q=AX^{2}-BXY+CY^{2}\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}$ by

$$\begin{aligned} \chi _{\Delta }(Q)= {\left\{ \begin{array}{ll} \left( \frac{\Delta }{n}\right) &{} {if (A,B,C,\Delta )=1 \textit{and Q represents n with} (n,\Delta )=1,}\\ 0&{} {otherwise.} \end{array}\right. } \end{aligned}$$

(1.3)

Let $H_{c}(\alpha ,m,\delta ,n)$ be the generalized Kloosterman sum of weight 1 / 2 defined for $\alpha ,\delta \in {\mathbb {Z}}/2N{\mathbb {Z}}$ and $m,n\in {\mathbb {Z}}$ by

$$\begin{aligned} H_{c}(\alpha ,m,\delta ,n)=\frac{e\left( -\mathrm{sign}(c)/8\right) }{|c|}\sum _{\begin{array}{c} d\in \left( \mathbb {Z}/c\mathbb {Z} \right) ^{\times }\\ \text {M}= \begin{pmatrix}a&{}b\\ c&{}d\end{pmatrix}\in \mathrm{SL}_{2}(\mathbb {Z}) \end{array}}\langle \phi _{\alpha },\rho _{L_{N}}({M})\phi _{\delta }\rangle e\left( \frac{ma+nd}{c}\right) , \end{aligned}$$

where $e(\tau )=e^{2\pi i\tau }$, $\rho _{L_{N}}$ denotes the Weil representation $\rho _{L_{N}}$ associated to the discriminant form $({\mathbb {Z}}/2N{\mathbb {Z}},\frac{x^{2}}{4N})=\{\phi _{r}\}_{r\in {\mathbb {Z}}/2N{\mathbb {Z}}}$, and $\langle \cdot ,\cdot \rangle$ is a pairing defined on the group algebra of the discriminant form $({\mathbb {Z}}/2N{\mathbb {Z}},\frac{x^{2}}{4N})=\{\phi _{r}\}_{r\in {\mathbb {Z}}/2N{\mathbb {Z}}}$ over ${\mathbb {C}}$ by

$$\begin{aligned} \left\langle \sum _{r\in {\mathbb {Z}}/2N{\mathbb {Z}}} a(r)\phi _{r},\sum _{r\in {\mathbb {Z}}/2N{\mathbb {Z}}}c(r)\phi _{r} \right\rangle =\sum _{r\in {\mathbb {Z}}/2N{\mathbb {Z}}} a(r)\overline{c(r)}. \end{aligned}$$

Moreover, define the generalized Kloosterman zeta function associated to $H_{c}(\alpha ,m,\delta ,n)$ by

$$\begin{aligned} Z(s;\alpha ,m,\delta ,n)=\sum _{c\ne 0}|c|^{1-2s}H_{c}(\alpha ,m,\delta ,n) \end{aligned}$$

for $s\in {\mathbb {C}}$ with $\mathrm{Re}(s)>1$. Finally, define ${\tilde{b}}(\alpha ,m,\delta ,n)$ for $m,n\in {\mathbb {Q}}$ by

$$\begin{aligned} {\tilde{b}}(\alpha ,m,\delta ,n)=\left( \frac{2\pi }{|n|}\right) ^{ \frac{1}{2}}\sum _{j=0}^{\infty }\frac{(-4\pi ^{2}|n|m)^{j}}{j!\Gamma \left( \frac{1}{2}+j\right) }Z\left( \frac{1}{4}+j;\alpha ,m,\delta ,n\right) , \end{aligned}$$

where $\Gamma (s)$ is the usual Gamma function. Then near the cusp $i\infty$,

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/{\Gamma _{0}(N)}}\left( j_{N}(\tau )-j_{N}(\tau _{Q}) \right) ^{\chi _{\Delta }(Q)}=\prod _{n=1}^{\infty }\prod _{\ell =1}^{\Delta -1} \left( 1-q^{n}\zeta _{\Delta }^{\ell }\right) ^{\left( \frac{\Delta }{\ell } \right) {\tilde{b}}\left( \beta ,-\frac{d}{4N},\gamma n,\frac{\Delta n^{2}}{4N}\right) },\nonumber \\ \end{aligned}$$

(1.4)

where $\zeta _{\Delta }=e^{\frac{2\pi i}{\Delta }}$, and is a twisted Borcherds product of type (2, 1).

Remark 1.3

Similar results hold for $\Delta <0$ by considering the dual Weil representation ${\overline{\rho }}_{L_{N}}$.

Remark 1.4

The minus sign in $AX^{2}-BXY+CY^{2}$ of ${\mathcal {Q}}_{-dD,\beta \gamma ,N}$ follows from the identification given in (2.3).

As a simple but interesting consequence, we obtain a closed formula for a sum of special values of the generalized Kloosterman zeta function as follows.

Corollary 1.5

Following the notation defined in Theorem 1.2, one has

$$\begin{aligned}&\left( \frac{8\pi N}{\Delta }\right) ^{\frac{1}{2}}\sum _{j=0}^{\infty }\left( \frac{-\pi ^{2} d\Delta }{4N^{2}}\right) ^{j}\frac{1}{j!\Gamma \left( \frac{1}{2}+j \right) }Z\left( \frac{1}{4}+j;\beta ,-\frac{d}{4N},\gamma ,\frac{\Delta }{4N}\right) \\&\quad =\frac{1}{\sqrt{\Delta }}\left( \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)j_{N}(\tau _{Q})\right) . \end{aligned}$$

2 Review of Twisted Borcherds Products

In this section, we briefly review the theory of twisted Borcherds products due to Bruinier and Ono [4].

2.1 The Weil Representation and Harmonic Weak Maass Forms

Write $\mathrm{Mp}_{2}({\mathbb {R}})$ for the metaplectic twofold cover of $\mathrm{SL}_{2}({\mathbb {R}})$. The elements of this group are the pairs $(M,\psi (\tau ))$, where $M=\begin{pmatrix}a&{}b\\ c&{}d\end{pmatrix}\in \mathrm{SL}_{2}({\mathbb {R}})$ and $\psi :{\mathbb {H}}\rightarrow {\mathbb {C}}$ is a holomorphic function with $\psi (\tau )^{2}=c\tau +d$. The multiplication operation of $\mathrm{Mp}_{2}({\mathbb {R}})$ is given by

$$\begin{aligned} (M,\psi (\tau ))(M',\psi '(\tau ))=(MM',\psi (M'\tau )\psi '(\tau )). \end{aligned}$$

We write M for $(M,\psi (\tau ))$ when there is no confusion, and write $\mathrm{Mp}_{2}({\mathbb {Z}})$ for the inverse image of $\mathrm{SL}_{2}({\mathbb {Z}})$ under the covering map, which is known to be generated by $T=\left( \begin{pmatrix}1&{}1\\ 0&{}1\end{pmatrix},1\right)$ and $S=\left( \begin{pmatrix}0&{}-1\\ 1&{}0\end{pmatrix},\sqrt{\tau }\right)$.

Let (V, Q) be a nondegenerate rational quadratic space of signature $(r^{+},r^{-})$, and write $(x,y)=Q(x+y)-Q(x)-Q(y)$ for the bilinear form on $V \times V$ induced by the quadratic form $Q(\cdot )$. Let L be an even ${\mathbb {Z}}$-lattice of V with dual lattice $L'$. The finite abelian group $L'/L$ together with the ${\mathbb {Q}}/{\mathbb {Z}}$-valued quadratic form induced by Q is called the discriminant form $(L'/L,Q)$ associated to the lattice L. The Weil representation $\rho _{L}$ associated with the discriminant form $(L'/L,Q)$ is a representation of $\mathrm{Mp}_{2}({\mathbb {Z}})$ on the group algebra ${\mathbb {C}}[L'/L]=\langle \{\phi _{\mu }\}_{\mu \in L'/L}\rangle$ where $\phi _{\mu }$ is the standard basis element associated to $\mu \in L'/L$, and it is defined via

$$\begin{aligned} \rho _{L}(T)\phi _{\mu }= & {} e(Q(\mu ))\phi _{\mu }, \end{aligned}$$

(2.1)

$$\begin{aligned} \rho _{L}(S)\phi _{\mu }= & {} \frac{e((r^{-}-r^{+})/8)}{\sqrt{|L'/L|}}\sum _{\nu \in L'/L}e(-(\nu ,\mu ))\phi _{\nu }. \end{aligned}$$

(2.2)

Let $k\in \frac{1}{2}{\mathbb {Z}}$. A harmonic weak Maass form of weight k with respect to the Weil representation $\rho _{L}$ is a twice continuously differentiable function $f:{\mathbb {H}}\rightarrow {\mathbb {C}}[L'/L]$ satisfying

• (i) $f(M\tau )=\psi (\tau )^{2k}\rho _{L}(M)f(\tau )$ for all $M\in \mathrm{Mp}_{2}({\mathbb {Z}})$,

• (ii) there is a constant $C>0$ such that $f(\tau )=O(e^{Cv})$ as $v\rightarrow \infty$,

• (iii) $\Delta _{k}(f)=0$, where $\Delta _{k}$ is the weight k hyperbolic Laplacian defined by

  $$\begin{aligned} \Delta _{k}=-v^{2}\left( \frac{\partial ^{2}}{\partial u^{2}}+\frac{\partial ^{2}}{\partial v^{2}}\right) +ikv\left( \frac{\partial }{\partial u}+\frac{\partial }{\partial v}\right) . \end{aligned}$$

2.2 A Rational Quadratic Space of Type (2, 1) and a Lattice Related to $\Gamma _{0}(N)$

Let N be a positive integer. Let V be the rational quadratic space defined by

$$\begin{aligned} V=\{x\in \mathrm{M}_{2}({\mathbb {Q}})|\,\mathrm{tr}(x)=0\} \end{aligned}$$

with the quadratic form $Q(x)=-N\det (x)$ of signature (2, 1) whose corresponding bilinear form is given by $(x,y)=N\mathrm{tr}(xy)$. The even Clifford algebra $C^{0}(V)$ of V can be identified with $\mathrm{M}_{2}({\mathbb {Q}})$, and the Clifford norm on $C^{0}(V)$ is identified with the determinant on $\mathrm{M}_{2}({\mathbb {Q}})$ (see, e.g., [8, Chapter 1.5]). Then one has $\mathrm{GSpin}(V)\cong \mathrm{GL}_{2}$ as an algebraic group over ${\mathbb {Q}}$, which is a central extension of $\mathrm{SO}(V)$ satisfying the exact sequence

$$\begin{aligned} 1\rightarrow {\mathbb {G}}_{m}\rightarrow \mathrm{GL}_{2}\rightarrow \mathrm{SO}(V)\rightarrow 1 \end{aligned}$$

and acts on V via $g\cdot x=gxg^{-1}$ for $x\in V$.

Let $L_{N}$ be the ${\mathbb {Z}}$-lattice

$$\begin{aligned} L_{N}=\left\{ \left. \begin{pmatrix}b&{}-c/N\\ a&{}-b \end{pmatrix}\right| \,a,b,c\in {\mathbb {Z}}\right\} \end{aligned}$$

of V. Then its dual lattice is given by

$$\begin{aligned} L'_{N}=\left\{ \left. \begin{pmatrix}b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix} \right| \,a,b,c\in {\mathbb {Z}}\right\} . \end{aligned}$$

One can identify the discriminant form $(L'_{N}/L_{N},-N\det (x))$ with $({\mathbb {Z}}/2N{\mathbb {Z}},\frac{x^{2}}{4N})$ via

$$\begin{aligned} \begin{pmatrix}r/2N&{}0\\ 0&{}-r/2N\end{pmatrix}\rightarrow r \end{aligned}$$

for $r\in {\mathbb {Z}}/2N{\mathbb {Z}}$. When there is no confusion, we also write r for the former as an element of $L_{N}$. For $r\in {\mathbb {Z}}/2N{\mathbb {Z}}$ and $D\in {\mathbb {Z}}$ with $D\equiv r^{2}\pmod {4N}$, define

$$\begin{aligned} L_{D,r,N}=\{x\in L'_{N}|\,-N\det (x)=D/4N\quad \text{ and }\quad x\equiv r\pmod {L_{N}}\}. \end{aligned}$$

It is known [4, Proposition 2.2] that $\Gamma _{0}(N)$ as a subgroup of $\mathrm{GSpin}(V)$ acts on $L_{N}$ and $L_{D,r, N}$ and acts trivially on $L_{N}'/L_{N}$.

Remark 2.1

One can note that $L_{D,r,N}$ can be identified with ${\mathcal {Q}}_{D,r,N}$ the set of integral binary quadratic forms $NaX^{2}-bXY+cY^{2}$ with $D=b^{2}-4Nac$ and $b\equiv r\pmod {2N}$ via

$$\begin{aligned} \begin{pmatrix}b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix}\rightarrow&\begin{pmatrix}X&Y \end{pmatrix}\begin{pmatrix}0&{}-1\\ 1&{}0\end{pmatrix}^{t}\begin{pmatrix} b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix}\begin{pmatrix}0&{}-1\\ 1&{}0\end{pmatrix} \begin{pmatrix}X\\ Y\end{pmatrix}\nonumber \\&\quad =NaX^{2}-bXY+cY^{2}. \end{aligned}$$

(2.3)

With the notation defined as above, we conclude this subsection with the following lemma due to von Pippich et al. [9, Lemma 3.7].

Lemma 2.2

(von Pippich et al.) Following the notation defined in Theorem 1.2 and preceding subsections, let $\beta \in L'_{N}/L_{N}$ and $m\in {\mathbb {Z}}+Q(\beta )$. For $m<0$, define

$$\begin{aligned} P_{\beta ,m,N}(\tau )&=e(m\tau )\left( \phi _{\beta }+\phi _{-\beta }\right) +\sum _{\gamma \in L_{N}'/L_{N}}\sum _{\begin{array}{c} n\in {\mathbb {Z}}+ Q(\gamma )\\ n>0 \end{array}}{\tilde{b}}(\beta ,m,\gamma ,n)e(n\tau )\phi _{\gamma }\\&\quad +\sum _{\gamma \in L'_{N}/L_{N}}\sum _{\begin{array}{c} n\in {\mathbb {Z}}+ Q(\gamma )\\ n<0 \end{array}}{\tilde{b}}(\beta ,m,\gamma ,n)\Gamma (1/2,4\pi |n|v)e(n\tau )\phi _{\gamma }, \end{aligned}$$

where ${\tilde{b}}(\beta ,m,\gamma ,n)$ is defined as in Theorem 1.2 for $n>0$ and is defined by

$$\begin{aligned} {\tilde{b}}(\beta ,m,\gamma ,n)=\left( \frac{2\pi }{|n|}\right) ^{ \frac{1}{2}}\lim _{s\rightarrow 0}\frac{Z(1/4+s;\beta ,m,\gamma ,n)}{\Gamma (s)} \end{aligned}$$

for $n<0$, and $\Gamma (s,x)$ is the truncated Gamma function defined by

$$\begin{aligned} \Gamma (s,x)=\int _{x}^{\infty }e^{-t}t^{s}\frac{\mathrm{d}t}{t}. \end{aligned}$$

Then $P_{\beta ,m,N}(\tau )$ is a harmonic weak Maass form of weight 1 / 2 with respect to the Weil representation $\rho _{L_{N}}$.

2.3 The Grassmannian of Positive 2-Plane of $V \otimes _{{\mathbb {Q}}}{\mathbb {R}}$

Let V and $L_{N}$ be defined as in the preceding subsection. Let $\mathrm{Gr}(V)$ be the Grassmannian of positive 2-plane of $V\otimes _{{\mathbb {Q}}}{\mathbb {R}}$, i.e.,

$$\begin{aligned} \mathrm{Gr}(V)=\left\{ {\mathbb {R}}x+{\mathbb {R}}y|\,x,y\in V\otimes _{{\mathbb {Q}}}{\mathbb {R}},\,Q(x)=Q(y)>0,\,(x,y)=0\right\} , \end{aligned}$$

which naturally possesses a complex structure and can be identified with a fixed connected component ${\mathcal {L}}^{+}$ of the complex projective space

$$\begin{aligned} {\mathcal {L}}=\left\{ Z\in V\otimes _{{\mathbb {Q}}}{\mathbb {C}}|\,(Z,Z)=0,\,(Z,{\overline{Z}})>0 \right\} /{\mathbb {C}}^{\times } \end{aligned}$$

via ${\mathbb {R}}x+{\mathbb {R}}y\rightarrow x+iy$. If we take $\ell =\begin{pmatrix}0&{}1/N\\ 0&{}0\end{pmatrix}$ and $\ell '=\begin{pmatrix}0&{}0\\ 1&{}0\end{pmatrix}$, then we have

$$\begin{aligned} K:=V\cap ({\mathbb {Q}}\ell +{\mathbb {Q}}\ell ')^{\perp }=\left\{ \left. \begin{pmatrix}b&{}0\\ 0&{}-b\end{pmatrix}\right| \,b\in {\mathbb {Q}}\right\} , \end{aligned}$$

and thus when there is no ambiguity, we write b for $\begin{pmatrix}b&{}0\\ 0&{}-b\end{pmatrix}$ as an element of K. Then one can further realize ${\mathcal {L}}^{+}$ as a connected component ${\mathcal {H}}^{+}$ of the so-called tube domain

$$\begin{aligned} {\mathcal {H}}=\left\{ z=u+iv\in K\otimes _{{\mathbb {Q}}}{\mathbb {C}}|\,Q(v)>0\right\} \end{aligned}$$

via $z\rightarrow \ell '-Q(z)\ell +z=\begin{pmatrix}z&{}-z^{2}\\ 1&{}-z\end{pmatrix}$. Moreover, one can see that ${\mathcal {H}}$ is isomorphic to ${\mathbb {H}}\cup \overline{{\mathbb {H}}}$ via

$$\begin{aligned} \begin{pmatrix}\tau &{}0\\ 0&{}-\tau \end{pmatrix}\rightarrow \tau , \end{aligned}$$

and we fix ${\mathcal {H}}^{+}\cong {\mathbb {H}}$.

2.4 Twisted Siegel Theta Functions and Twisted Borcherds Theta Integrals

As before, for $\gamma \in {\mathbb {Z}}/2N{\mathbb {Z}}$, let $\Delta$ be a positive fundamental discriminant with $\Delta \equiv \gamma ^{2}\pmod {4N}$. Under the identification between the elements of $L_{N}'$ and quadratic forms, we can extend the generalized genus character $\chi _{\Delta }$ to be defined on a subset of $L_{N}'$. Then the so-called twisted Siegel theta function is defined as follows. For a coset $\beta \in L'/L$, and variables $\tau =u+iv\in {\mathbb {H}}$, $z\in \mathrm{Gr}(V)\cong {\mathbb {H}}$, define

$$\begin{aligned} \theta _{\Delta ,\gamma }(\tau ,z;\beta )=v^{\frac{1}{2}}\sum _{\begin{array}{c} \lambda \in \beta \gamma +L_{N}\\ Q(\lambda )\equiv \Delta Q(\beta )\pmod {\Delta } \end{array}}\chi _{\Delta }(\lambda )e\left( \frac{Q(\lambda _{z})}{ \Delta }\tau +\frac{Q(\lambda _{z^{\perp }})}{\Delta }{\bar{\tau }}\right) , \end{aligned}$$

and a ${\mathbb {C}}[L'_{N}/L_{N}]$-valued twisted Siegel theta function is defined by

$$\begin{aligned} \Theta _{\Delta ,\gamma }(\tau ,z)=\sum _{\beta \in L_{N}'/L_{N}}\theta _{\Delta ,\gamma }(\tau ,z;\beta )\phi _{\beta }. \end{aligned}$$

Then as a function in z, $\Theta _{\Delta ,\gamma }(\tau ,z)$ is invariant under $\Gamma _{0}(N)$, and as a function in $\tau$, it is a nonholomorphic modular form of weight 1 / 2 for $\rho _{L_{N}}$.

Then for a harmonic weak Maass form f of weight $\frac{1}{2}$ with respect to the Weil representation $\rho _{L_{N}}$, and for $z\in \mathrm{Gr}(V)\cong {\mathbb {H}}$, the twisted Borcherds theta integral of f is defined by

$$\begin{aligned} \Phi _{\Delta ,\gamma }(z,f)=\int _{{\mathcal {F}}}^{reg}\langle f(\tau ),\Theta _{\Delta ,\gamma }(\tau ,z)\rangle v^{\frac{1}{2}}\frac{\mathrm{d}u\mathrm{d}v}{v^{2}} \end{aligned}$$

(2.4)

where ${\mathcal {F}}$ denotes the standard fundamental domain for the action of $\mathrm{SL}_{2}({\mathbb {Z}})$, and the integral has to be regularized as in [1].

2.5 Twisted Heegner Divisors

For any $\lambda \in V({\mathbb {R}})$ with negative norm, a prime Heegner divisor $Z(\lambda )$ associated to $\lambda$ is defined to be the point $[\lambda ^{\perp }]\in \Gamma _{0}(N)\backslash {{\mathbb {H}}}$ induced from $\mathrm{Gr}(\lambda ^{\perp })\subset \mathrm{Gr}(V)\cong {\mathbb {H}}$, which can be identified with the orthogonal complement $\lambda ^{\perp }$, via the projection ${\mathbb {H}}\twoheadrightarrow \Gamma _{0}(N)\backslash {{\mathbb {H}}}$. For $\beta \in L_{N}'/L_{N}$ and a negative rational number $m\in {\mathbb {Z}}+ Q(\beta )$, we define the twisted Heegner divisor

$$\begin{aligned} Z_{\Delta ,\gamma }(\beta ,m)=\sum _{\begin{array}{c} \lambda \in L_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N) \end{array}}\frac{\chi _{ \Delta }(\lambda )}{2}Z(\lambda )\in \mathrm{Div}(X_{0}(N))_{{\mathbb {Q}}}, \end{aligned}$$

where $-d=4Nm\in {\mathbb {Z}}$. The twisted Heegner divisor associated to a harmonic weak Maass form f of weight $\frac{1}{2}$ with respect to the Weil representation $\rho _{L_{N}}$ with

$$\begin{aligned} f(\tau )=\sum _{\beta \in L_{N}'/L_{N}}\sum _{m\in {\mathbb {Z}}+ Q(\beta )}c^{+}(m,\beta )q^{m}\phi _{\beta }+\text{ nonholomorphic } \text{ part } \end{aligned}$$

is defined by

$$\begin{aligned} Z_{\Delta ,\gamma }(f)=\sum _{\beta \in L'_{N}/L_{N}}\sum _{\begin{array}{c} m\in {\mathbb {Z}}+ Q(\beta )\\ m<0 \end{array}}c^{+}(\beta ,m)Z_{\Delta ,\gamma }(\beta ,m)\in \mathrm{Div}(X_{0}(N))_{{\mathbb {R}}}. \end{aligned}$$

Remark 2.3

By the definition, the prime Heegner divisor $Z(\lambda )$ associated to $\lambda =\begin{pmatrix}b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix}\in L_{-d\Delta ,\beta \gamma ,N}$ defines a point $z=\begin{pmatrix}z&{}-z^{2}\\ 1&{}-z\end{pmatrix}$ of $\mathrm{Gr}(V)$ such that $(\lambda ,z)=0$, that is,

$$\begin{aligned} \left( \begin{pmatrix}b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix},\begin{pmatrix}z&{}-z^{2}\\ 1&{}-z\end{pmatrix}\right)&=N\mathrm{tr}\left( \begin{pmatrix}b/2N&{}-c/N\\ a&{}-b/2N\end{pmatrix}\begin{pmatrix}z&{}-z^{2}\\ 1&{}-z\end{pmatrix}\right) \\&=-Naz^{2}+bz-c=0. \end{aligned}$$

Therefore, the prime Heegner divisor $Z(\lambda )$ can be identified with the imaginary quadratic point $\tau _{Q}$ defined by the quadratic form $Q_{\lambda }$ associated to $\lambda$ under the identification between $L_{-d\Delta ,\beta \gamma ,N}$ and ${\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}$ given in (2.3).

2.6 Twisted Borcherds Products

In [4, Theorem 6.1], Bruinier and Ono computed twisted Borcherds theta integral (2.4) and showed that there is a meromorphic modular function associated with such an integral with twisted divisor determined by the principal part of the theta integral input f. We now conclude this section by stating their celebrated result as follows.

Theorem 2.4

Let f be a harmonic weak Maass form of weight 1 / 2 for $\rho _{L_{N}}$ with

$$\begin{aligned} f(\tau )=\sum _{\beta \in L_{N}'/L_{N}}\sum _{m\in {\mathbb {Z}}+ Q(\beta )}c^{+}(m,\beta )q^{m}\phi _{\beta }+\text{ nonholomorphic } \text{ part } \end{aligned}$$

and real coefficients $c^{+}(\beta ,m)$ for all $m\in {\mathbb {Q}}$ and $\beta \in L_{N}'/L_{N}$. Moreover, assume that $c^{+}(\beta ,m)\in {\mathbb {Z}}$ for $m\le 0$. Then there is a meromorphic modular function $\Psi _{\Delta ,\gamma }(z,f)$ called a twisted Borcherds product for $\Gamma _{0}(N)$ such that

• (1) $\Phi _{\Delta ,\gamma }(z,f)=2\sqrt{\Delta }c^{+}(0,0)L(1, \left( \frac{\Delta }{\cdot }\right) )-2\log |\Psi _{\Delta ,\gamma }(z,f)|^{2}$,

• (2) the divisor of $\Psi _{\Delta ,\gamma }(z,f)$ on $X_{0}(N)$ is given by $Z_{\Delta ,\gamma }(f)$,

• (3) the Fourier expansion of $\Psi _{\Delta ,\gamma }(z,f)$ at the cusp $i\infty$ has an infinite product representation

  $$\begin{aligned} \Psi _{\Delta ,\gamma }(z,f)=\prod _{\begin{array}{c} \lambda \in K'\\ \lambda >0 \end{array}}\prod _{n=1}^{\Delta -1}\left( 1-e\left( (\lambda ,z)+b/\Delta \right) \right) ^{\left( \frac{\Delta }{n}\right) c^{+}(\Delta Q(\lambda ),\lambda \gamma )}. \end{aligned}$$

3 Proofs of Theorem 1.2 and Corollary 1.5

In this section, we conclude this work with the proofs of Theorem 1.2 and Corollary 1.5. First, recall that the normalized uniformizer $j_{N}(\tau )$ for the genus zero modular curve $X_{0}(N)$ is a conformal map from $X_{0}(N)$ to ${\mathbb {P}}^{1}({\mathbb {C}})={\mathbb {C}}\cup \{\infty \}$ with Fourier expansion $j_{N}(\tau )=\frac{1}{q}+O(q)$ at the cusp $i\infty$, where $q=e^{2\pi i\tau }$. Now we begin this section with the proof of Theorem 1.2 as follows.

Proof

(Proof of Theorem 1.2) By Lemma 2.2, writing $P_{\beta ,m,N}=P_{\beta ,m,N}(\tau )$, one can easily see that

$$\begin{aligned}&Z_{\Delta ,\gamma }\left( P_{\beta ,-\frac{d}{4N},N}\right) =Z_{\Delta ,\gamma } \left( \beta ,-\frac{d}{4N}\right) +Z_{\Delta ,\gamma }\left( -\beta ,-\frac{d}{4N} \right) \\&\quad =\sum _{\begin{array}{c} \lambda \in L_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N) \end{array}}{\chi _{\Delta }(\lambda )}Z(\lambda ), \end{aligned}$$

which can be identified with

$$\begin{aligned} \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)[\tau _{Q}] \end{aligned}$$

by Remarks 2.1 and 2.3. Then by Theorem 2.4, there is a twisted Borcherds product $\Psi \left( z,P_{\beta ,-\frac{d}{4N},N}\right)$ associated to $P_{\beta ,-\frac{d}{4N},N}(\tau )$ defined on $X_{0}(N)$ with

$$\begin{aligned} \mathrm{Div}\left( \Psi \left( z,P_{\beta ,-\frac{d}{4N},N}\right) \right) =\sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)[\tau _{Q}] \end{aligned}$$

on $X_{0}(N)$. Moreover, since

$$\begin{aligned} \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)=0, \end{aligned}$$

and $j_{N}(\tau )$ is a uniformizer for $X_{0}(N)$, then the divisor of

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\left( j_{N}(\tau )-j_{N}(\tau _{Q})\right) ^{\chi _{\Delta }(Q)} \end{aligned}$$

on $X_{0}(N)$ is clearly

$$\begin{aligned} \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)[\tau _{Q}] \end{aligned}$$

and is the same as $\mathrm{Div}\left( \Psi \left( z,P_{\beta ,-\frac{d}{4N},N}\right) \right)$. Therefore, $\Psi \left( z,P_{\beta ,-\frac{d}{4N},N}\right)$ and

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)} \left( j_{N}(\tau )-j_{N}(\tau _{Q})\right) ^{\chi _{\Delta }(Q)} \end{aligned}$$

as meromorphic functions on the compact Riemann surface $X_{0}(N)$ agree everywhere, and thus the product

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\left( j_{N}(\tau )-j_{N}(\tau _{Q})\right) ^{\chi _{\Delta }(Q)} \end{aligned}$$

is a twisted Borcherds product. Finally, by Theorem 2.4 (3), one can compute and obtain the product expansion for the Fourier expansion of

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\left( j_{N}(\tau )-j_{N}(\tau _{Q})\right) ^{\chi _{\Delta }(Q)} \end{aligned}$$

at the cusp $i\infty$, namely,

$$\begin{aligned} \prod _{ Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/{\Gamma _{0}(N)}}\left( j_{N}(\tau )-j_{N}(\tau _{Q})\right) ^{\chi _{ \Delta }(Q)}=\prod _{n=1}^{\infty }\prod _{\ell =1}^{\Delta -1}\left( 1-q^{n} \zeta _{\Delta }^{\ell }\right) ^{\left( \frac{\Delta }{\ell }\right) {\tilde{b}}\left( \beta ,-\frac{d}{4N},\gamma n,\frac{\Delta n^{2}}{4N}\right) }. \end{aligned}$$

$\square$

We now conclude this note with the proof of Corollary 1.5 as follows.

Proof of Corollary 1.5

Taking logarithmic derivatives with respect to $\tau$ on the both sides of (1.4), one has

$$\begin{aligned}&\sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\frac{\chi _{\Delta }(Q)}{2\pi i}\frac{-j_{N}'(\tau )}{j_{N}(\tau )-j_{N}(\tau _{Q})}\nonumber \\&\qquad =\sum _{n=1}^{\infty } \sum _{\ell =1}^{\Delta -1}\left( \frac{\Delta }{\ell }\right) {\tilde{b}} \left( \beta ,-\frac{d}{4N},\gamma n,\frac{\Delta n^{2}}{4N}\right) \frac{nq^{n}\zeta _{\Delta }^{\ell }}{1-q^{n}\zeta _{\Delta }^{\ell }}. \end{aligned}$$

(3.1)

By the Fourier expansion of $j_{N}(\tau )$ at the cusp $i\infty$, i.e., $j_{N}(\tau )=\frac{1}{q}+O(q)$ with $q=e^{2\pi i\tau }$, one has

$$\begin{aligned} \frac{1}{2\pi i}\frac{-j_{N}'(\tau )}{j_{N}(\tau )-j_{N}( \tau _{Q})}&=\frac{q^{-1}+O(q)}{q^{-1}-j_{N}(\tau _{Q})+O(q)}\\&=\frac{1+O(q^{2})}{1-j_{N}(\tau _{Q})q+O(q^{2})}\\&=1+j_{N}(\tau _{Q})q+O(q^{2}). \end{aligned}$$

Then together with the fact that

$$\begin{aligned} \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)=0, \end{aligned}$$

the Fourier expansion of the left-hand side of (3.1) is equal to

$$\begin{aligned} \left( \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)j_{N}(\tau _{Q})\right) q+O(q^{2}). \end{aligned}$$

By the geometric series expansion, the right-hand side of (3.1) gives

$$\begin{aligned} \sum _{\ell =1}^{\Delta -1}\left( \frac{\Delta }{\ell }\right) \zeta _{\Delta }^{ \ell }{\tilde{b}}\left( \beta ,-\frac{d}{4N},\gamma ,\frac{\Delta }{4N}\right) q+O(q^{2}). \end{aligned}$$

Thus, equating the coefficients of the term q on both sides of (3.1) gives

$$\begin{aligned} \left( \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)j_{N}(\tau _{Q})\right) =\sum _{ \ell =1}^{\Delta -1}\left( \frac{\Delta }{\ell }\right) \zeta _{\Delta }^{\ell }{ \tilde{b}}\left( \beta ,-\frac{d}{4N},\gamma ,\frac{\Delta }{4N}\right) . \end{aligned}$$

By the standard Gauss sum identity, one has that

$$\begin{aligned} \sum _{\ell =1}^{\Delta -1}\left( \frac{\Delta }{\ell }\right) \zeta _{ \Delta }^{\ell }=\sqrt{\Delta }, \end{aligned}$$

and thus

$$\begin{aligned} {\tilde{b}}\left( \beta ,-\frac{d}{4N},\gamma ,\frac{\Delta }{4N}\right)&=\frac{1}{\sqrt{\Delta }}\left( \sum _{Q\in {\mathcal {Q}}_{-d\Delta ,\beta \gamma ,N}/\Gamma _{0}(N)}\chi _{\Delta }(Q)j_{N}(\tau _{Q})\right) . \end{aligned}$$

This completes the proof.

$\square$

Remark 3.1

Using Corollary 1.5 and following the arguments given in [14, Section 7], one can further deduce the integrality of

$$\begin{aligned} \left( \frac{8\pi N}{\Delta }\right) ^{\frac{1}{2}}\sum _{j=0}^{\infty }\left( \frac{-\pi ^{2} d\Delta }{4N^{2}}\right) ^{j}\frac{1}{j!\Gamma \left( \frac{1}{2}+j\right) }Z \left( \frac{1}{4}+j;\beta ,-\frac{d}{4N},\gamma ,\frac{\Delta }{4N}\right) \end{aligned}$$

for $d,\Delta$ coprime.