Cycle integrals of meromorphic modular forms and Siegel theta functions

Abstract

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.

1 Introduction

Let $\mathcal {Q}_{d}$ be the set of all (positive definite if $d < 0$) integral binary quadratic forms of discriminant d. For $k \in \mathbb {N}$ with $k \ge 2$, we consider the functions

$$\begin{aligned} f_{k,d}(z) = \frac{|d|^{k-1/2}}{\pi }\sum _{Q \in \mathcal {Q}_{d}}\frac{1}{Q(z,1)^{k}}, \end{aligned}$$

which transform like modular forms of weight 2k for $\Gamma = {{\,\textrm{PSL}\,}}_{2}(\mathbb {Z})$. These functions were first studied by Zagier [12] for $d > 0$, in which case they are cusp forms, and by [6] for $d < 0$, in which case they are meromorphic modular forms. We obtain a refinement $f_{k,P}$ of $f_{k,d}$ by summing only over quadratic forms in the equivalence class of a fixed form P.

In [1], we computed the cycle integrals of the meromorphic modular forms $f_{k,P}$ associated with positive definite quadratic forms P. These cycle integrals are defined by¹

$$\begin{aligned} \mathcal {C}_A(f_{k,P}) = \int _{\Gamma _A \backslash S_A}f_{k,P}(z)A(z,1)^{k-1}dz, \end{aligned}$$

where $A = [a,b,c]$ is an indefinite integral binary quadratic form of positive non-square discriminant $D > 0$, $\Gamma _A$ denotes the stabilizer of A, and $S_A = \{z \in \mathbb {H}\, : \, a|z|^2 + b\Re (z) + c = 0\}$ is the geodesic corresponding to A. We let $\mathcal {C}_D(f_{k,P})$ be the D-th trace of cycle integrals, that is, the sum of $\mathcal {C}_A(f_{k,P})$ over all classes $A \in \mathcal {Q}_D /\Gamma$, and we let $M_{3/2-k}^!$ denote the space of weakly holomorphic modular forms of weight $3/2-k$ for $\Gamma _0(4)$ satisfying Kohnen’s plus space condition $a_g(n) = 0$ unless $(-1)^{k+1} n \equiv 0,1 \ \, \left( \textrm{mod} \, 4 \right)$. Then, the main result of [1] is as follows.

Theorem 1.1

(Theorem 1.1 in [1]) Let $k \ge 2$ be even and let $g \in M_{3/2-k}^!$. Suppose that the coefficients $a_g(-D)$ are rational for $D > 0$ and vanish if D is a square. Then, the linear combinations of traces of cycle integrals

$$\begin{aligned} \sum _{D > 0}a_g(-D)\mathcal {C}_D(f_{k,P}) \end{aligned}$$

of the individual meromorphic modular forms $f_{k,P}$ are rational.

In [11], we gave some refinements of this result and made a conjecture concerning the rationality of the individual cycle integrals of linear combinations of various $f_{k,d}$’s, which can be viewed as a dual version of Theorem 1.1 for odd k. Our goal is to prove this conjecture.

Theorem 1.2

(Conjecture 2.5 in [11]) Let $k\ge 3$ be odd and let $g \in M_{3/2-k}^!$. Suppose that the coefficients $a_g(d)$ for $d < 0$ are rational. Then, the individual cycle integrals

$$\begin{aligned} \mathcal {C}_A\left( \sum _{d < 0}a_g(d)f_{k,d}(z)\right) \end{aligned}$$

of linear combinations of the meromorphic $f_{k,d}$ are rational.

In [11], this result was stated as a conjecture for higher level, even and odd k, and twisted versions of $f_{k,d}$. Our arguments also work in this general setup, see Sect. 5.

The proof of Theorem 1.2 follows a method of Bruinier, Ehlen, and Yang (see the proof of [8, Theorem 5.4]). We use the well-known fact that $f_{k,d}$ can be written as a regularized theta lift. Then, we interchange the cycle integral with the regularized integral, which leads us to study the cycle integrals of the Siegel theta function $\Theta _L(\tau ,z)$ associated with an even lattice L of signature (1, 2). The evaluation of these cycle integrals is the main novelty of this work. Let us describe the result in more detail.

The indefinite quadratic form A defines a vector of negative length in the lattice L, and hence yields sublattices $I = L \cap (\mathbb {Q}A)^\perp$ and $N = L \cap (\mathbb {Q}A)$ of signature (1, 1) and (0, 1), respectively. Hecke [9] constructed an indefinite theta series $\vartheta _I(\tau )$ associated with the lattice I, which is a cusp form of weight 1 for the Weil representation $\rho _I$, see (3.3). Moreover, there is a classical holomorphic weight 3/2 theta series $\Theta _{3/2,N}(\tau )$ associated to the unary lattice N, see (3.1).

Theorem 1.3

Suppose that $L = I \oplus N$. Then, we have

$$\begin{aligned} \mathcal {C}_A\left( \frac{\partial }{\partial z}\Theta _L(\tau ,z) \right) {\mathop {=}\limits ^{\cdot }} \vartheta _I(\tau ) \otimes \overline{\Theta _{3/2,N}(\tau )}v^{3/2}. \end{aligned}$$

Here, ${\mathop {=}\limits ^{\cdot }}$ means equality up to an explicit non-zero constant factor. For the precise version of this result, we refer to Theorem 3.1. The idea of the proof is that $\Theta _L(\tau ,z)$ essentially splits as a product of two Siegel theta functions $\Theta _I(\tau ,z)$ and $\Theta _{N}(\tau )$ along the geodesic $S_A$, and the cycle integral of $\Theta _I(\tau ,z)$ yields the Hecke theta series $\vartheta _I(\tau )$.

Combining the above evaluation of the cycle integral of the Siegel theta function with the theta lift realization of $f_{k,d}$, we obtain a representation of $\mathcal {C}_A(f_{k,d})$ as a regularized Petersson inner product, which can in turn be evaluated using the theory of harmonic Maass forms. This gives an explicit formula for the cycle integrals appearing in Theorem 1.2 (see Theorem 4.1), from which their rationality can easily be deduced.

Remark 1.4

Following the proof of Theorem 3.1, one can compute the cycle integrals of other Siegel-type theta functions, e.g., of the Kudla-Millson or the Shintani theta functions. However, in all cases that we tried, Hecke’s indefinite theta function $\vartheta _I(\tau )$ was replaced by a non-holomorphic indefinite theta function. This makes the subsequent computation of the regularized Petersson inner product more difficult, that is, we were not able to compute the cycle integrals of the corresponding theta lifts in this way.

This paper is organized as follows. In Sect. 2, we recall the theta lift realization of $f_{k,d}$. In Sect. 3, we evaluate the cycle integrals of the Siegel theta function, which is the technical heart of this work. In Sect. 4, we put everything together and deduce an explicit rational formula for the cycle integrals of $f_{k,d}$, which also implies Theorem 1.2. Finally, in Sect. 5, we sketch the proof of a higher level version of Theorem 1.2.

2 Meromorphic modular forms as theta lifts

In this section, we recall the realization of the meromorphic modular form $f_{k,d}$ as a theta lift of a harmonic Maass form, following [3]. We consider the even lattice

$$\begin{aligned} L = \left\{ X=\begin{pmatrix}b &{} c \\ -a &{} -b \end{pmatrix}: a,b,c \in \mathbb {Z}\right\} \end{aligned}$$

with the quadratic form $q(X) = \det (X)$. It has signature (1, 2), and its dual lattice is given by

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

The group $\Gamma = {{\,\textrm{PSL}\,}}_{2}(\mathbb {Z})$ acts on L via $g.X = gXg^{-1}$. We can view $L'$ as the set of integral binary quadratic forms [a, b, c], where $-4q(X)=b^2-4ac$ corresponds to the discriminant.

Let ${{\,\textrm{Gr}\,}}(L)$ be the Grassmannian of positive definite lines in $V(\mathbb {R}) = L \otimes \mathbb {R}$. We can identify ${{\,\textrm{Gr}\,}}(L)$ with $\mathbb {H}$ by sending $z = x+iy \in \mathbb {H}$ to the positive line generated by

$$\begin{aligned} X(z) = \frac{1}{\sqrt{2}y}\begin{pmatrix}-x &{} |z|^2 \\ -1 &{} x \end{pmatrix}. \end{aligned}$$

This bijection is compatible with the actions of ${{\,\textrm{PSL}\,}}_2(\mathbb {R})$, in the sense that $g.X(z) = X(gz)$ for any $g \in {{\,\textrm{PSL}\,}}_2(\mathbb {R})$. We also consider the vectors

$$\begin{aligned} U_1(z) = \frac{1}{\sqrt{2}y}\begin{pmatrix}x &{} -x^2+y^2 \\ 1 &{} -x \end{pmatrix}, \qquad U_2(z) = \frac{1}{\sqrt{2}y}\begin{pmatrix}y &{} -2xy \\ 0 &{} -y \end{pmatrix}, \end{aligned}$$

which, together with X(z), form an orthogonal basis of $V(\mathbb {R})$. For each fixed $z = x+iy \in \mathbb {H}$, we define the polynomials

$$\begin{aligned} Q_X(z)&= -\sqrt{2}y(X,U_1(z)+iU_2(z)) = az^2 + bz + c, \\ p_X(z)&= \sqrt{2}(X,X(z)) = \frac{1}{y}(a|z|^2 + bx + c), \end{aligned}$$

on $V(\mathbb {R})$. They satisfy the invariances

$$\begin{aligned} p_{X}(g z) = p_{g^{-1}.X}(z), \qquad Q_{X}(gz) = (cz+d)^{-2}Q_{g^{-1}.X}(z), \end{aligned}$$

(2.1)

for each $g = \left( {\begin{matrix}a &{} b \\ c &{} d \end{matrix}} \right) \in {{\,\textrm{PSL}\,}}_2(\mathbb {R})$. For brevity we write $X_{z}$ and $X_{z^\perp }$ for the orthogonal projection of X to the positive line generated by X(z) and to the negative plane $X(z)^\perp$ spanned by $U_1(z),U_2(z)$, respectively. We have the useful relations

$$\begin{aligned} q(X_z) = \frac{1}{4}p_X(z)^2, \qquad q(X_{z^\perp }) = -\frac{1}{4}y^{-2}|Q_X(z)|^2. \end{aligned}$$

(2.2)

Since L has signature (1, 2), the Siegel theta function

$$\begin{aligned} \Theta _{L}(\tau ,z) = v \sum _{X \in L'}e\left( q(X_z)\tau + q(X_{z^\perp }){\overline{\tau }} \right) {{\,\mathrm{\mathfrak {e}}\,}}_X \end{aligned}$$

transforms like a modular form of weight $-1/2$ in $\tau$ for the Weil representation $\rho _L$ associated with L. Moreover, it is $\Gamma$-invariant in z. Here, ${{\,\mathrm{\mathfrak {e}}\,}}_X = {{\,\mathrm{\mathfrak {e}}\,}}_{X + L}$ are the standard basis vectors of the group ring $\mathbb {C}[L'/L]$. The derivative $R_0\Theta _{L}(\tau ,z) = 2i\frac{\partial }{\partial z}\Theta _L(\tau ,z)$ transforms like a modular form of weight 2 in z. It is explicitly given by

$$\begin{aligned} R_0\Theta _L(\tau ,z) = 2\pi v^2 \sum _{X \in L'}p_X(z) y^{-2}\overline{Q_X(z)}e\left( q(X_z)\tau + q(X_{z^\perp }){\overline{\tau }} \right) {{\,\mathrm{\mathfrak {e}}\,}}_X. \end{aligned}$$

(2.3)

Let $A_{k,L}$ denote the space of all $\mathbb {C}[L'/L]$-valued functions on $\mathbb {H}$ which transform like modular forms of weight k for $\rho _L$. If $M \subset L$ is a sublattice of finite index, we have natural maps $A_{k,L} \rightarrow A_{k,M}, f \mapsto f_M$, and $A_{k,M} \rightarrow A_{k,L}, g \mapsto g^L$, given by

$$\begin{aligned} (f_{M})_\gamma = {\left\{ \begin{array}{ll}f_{\gamma } &{} \text { if } \gamma \in L'/M, \\ 0, &{} \text { if }\gamma \notin L'/M, \end{array}\right. } \qquad \text {and} \qquad (g^L)_\gamma = \sum _{\beta \in L/M}g_{\beta +\gamma }. \end{aligned}$$

(2.4)

These maps are adjoint with respect to the bilinear pairings on $\mathbb {C}[L'/L]$ and $\mathbb {C}[M'/M]$. Moreover, the Siegel theta functions for M and L are related by $\Theta _L(\tau ,z) = \Theta _M(\tau ,z)^L$.

For a smooth modular form f of weight 1/2 for ${\overline{\rho }}_L$, the regularized theta lift is defined by

$$\begin{aligned} \Phi _{L}(f,z) = \int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}} f(\tau )\cdot \Theta _{L}(\tau ,z) d\mu (\tau ), \end{aligned}$$

where $\mathcal {F}$ is a fundamental domain for $\Gamma \backslash \mathbb {H}$, the dot denotes the natural bilinear pairing on $\mathbb {C}[L'/L]$, and $d\mu (\tau )$ is the invariant measure. The integral is regularized as explained in [7].

We let $P_{3/2-k,d}(\tau )$ be the unique harmonic Maass form of weight $3/2-k$ for the dual Weil representation ${\overline{\rho }}_L$ with principal part $q^{d}{{\,\mathrm{\mathfrak {e}}\,}}_d+O(1)$.

Proposition 2.1

(Proposition 4.1 in [3]) For odd $k\ge 3$, we have

$$\begin{aligned} f_{k,d}(z) = c_k \cdot R_{0}^k\Phi _{L}\left( R_{3/2-k}^{(k-1)/2}P_{3/2-k,d},z\right) \end{aligned}$$

with the constant $c_k = (2^{k+1}\pi ^{(k+1)/2}(k-1)!)^{-1}$. Here, $R_{\kappa }^n = R_{\kappa + 2n-2} \circ \dots \circ R_{\kappa }$ is an iterated version of the raising operator $R_{\kappa } = 2i\frac{\partial }{\partial z}+\kappa y^{-1}$.

3 Cycle integrals of Siegel theta functions

In this section, we compute the cycle integrals of the derivative of the Siegel theta function $\Theta _L(\tau ,z)$. Throughout this section, we let

$$\begin{aligned} A = \begin{pmatrix}b/2 &{} c \\ -a &{} -b/2 \end{pmatrix} \in L', \qquad D = -4q(A) = b^2-4ac > 0, \end{aligned}$$

be an indefinite integral binary quadratic form of positive non-square discriminant D. We let $S_A = \{z \in \mathbb {H}\, : \, a|z|^2 + bx + c = 0\}$ be the corresponding geodesic semi-circle in $\mathbb {H}$. Changing A to $-A$ only changes the sign of the cycle integral, so we will assume $a > 0$ from now on. Then, $S_A$ is oriented counter-clockwise. Since A is as a vector of negative norm in $L'$, we can define two sublattices

$$\begin{aligned} I = L \cap (\mathbb {Q}A)^\perp , \qquad N = L \cap (\mathbb {Q}A), \end{aligned}$$

of signature (1, 1) and (0, 1), respectively. Note that I is anisotropic since A has non-square discriminant, and that $N^- = (N,-q)$ is a positive definite one-dimensional lattice. We define the weight 3/2 unary theta function corresponding to N (or rather $N^-$) by

$$\begin{aligned} \Theta _{3/2,N}(\tau ) = \sum _{W \in N'}(W,A)e(-q(W)\tau ) {{\,\mathrm{\mathfrak {e}}\,}}_W. \end{aligned}$$

(3.1)

It is a cusp form of weight 3/2 for the dual Weil representation ${\overline{\rho }}_N$.

Let $w = \frac{-b-\sqrt{D}}{2a}$ be the real endpoint of the geodesic $S_A$, and consider the vector

$$\begin{aligned} Y_0 = \begin{pmatrix} -w &{} w^2 \\ -1 &{} w \end{pmatrix} \in I \otimes \mathbb {R}. \end{aligned}$$

(3.2)

We define the Hecke theta series associated with the signature (1, 1) lattice I by

$$\begin{aligned} \vartheta _I(\tau ) = \sum _{\begin{array}{c} Y \in \Gamma _I \backslash I' \\ q(Y) > 0 \end{array}}\frac{{{\,\textrm{sgn}\,}}(Y,Y_0)}{|\Gamma _Y|}e(q(Y)\tau ){{\,\mathrm{\mathfrak {e}}\,}}_Y. \end{aligned}$$

(3.3)

It is a cusp form of weight 1 for $\rho _I$, which was first constructed by Hecke [9] as a theta lift.

Theorem 3.1

We have

$$\begin{aligned} \mathcal {C}_A\big (R_{0}\Theta _{L}(\tau ,\,\cdot \,) \big ) = -\frac{4\pi }{\sqrt{D}}\left( \vartheta _I(\tau ) \otimes \overline{\Theta _{3/2,N}(\tau )}v^{3/2}\right) ^L, \end{aligned}$$

with the map $f^L$ defined in (2.4).

Proof

Note that $I \oplus N$ is a sublattice of finite index in L, and we have

$$\begin{aligned} \mathcal {C}_A(R_{0}\Theta _{L}) = \mathcal {C}_A(R_{0}\Theta _{I \oplus N})^L, \end{aligned}$$

so we can assume $L = I \oplus N$ for now. We first split $R_{0}\Theta _{I \oplus N}$ along the geodesic $S_A$ into tensor products of theta functions associated with I and N . We can write $X \in (I\oplus N)'$ uniquely as $X = Y + W$ with $Y \in I'$ and $W \in N'$. Note that $W = \frac{(W,A)}{(A,A)}A$. Hence, for $z \in S_A$, we have

$$\begin{aligned} p_W(z) = 0, \qquad W_z = 0, \qquad y^{-2}\overline{Q_W(z)} = -2(W,A) Q_A(z)^{-1}. \end{aligned}$$

Plugging this into the explicit formula (2.3) for $R_{0}\Theta _{I \oplus N}$, we obtain for $z \in S_A$ the splitting

$$\begin{aligned} R_0\Theta _{I \oplus N}(\tau ,z)&= 2\pi \left( \Theta _I^*(\tau ,z)\otimes \overline{\Theta _{1/2,N}(\tau )}v^{1/2}\right) \\&\quad - 4\pi \left( Q_A(z)^{-1}\Theta _I(\tau ,z) \otimes \overline{\Theta _{3/2,N}(\tau )}v^{3/2}\right) , \end{aligned}$$

with the Siegel theta functions

$$\begin{aligned} \Theta _I^*(\tau ,z)&= v^{3/2}\sum _{Y \in I'}p_{Y}(z)y^{-2}\overline{Q_Y(z)}e(q(Y_{z})\tau + q(Y_{z^{\perp }}){\overline{\tau }}){{\,\mathrm{\mathfrak {e}}\,}}_{Y}, \\ \Theta _I(\tau ,z)&= v^{1/2}\sum _{Y \in I'}p_{Y}(z)e(q(Y_{z})\tau + q(Y_{z^{\perp }}){\overline{\tau }}){{\,\mathrm{\mathfrak {e}}\,}}_{Y}, \end{aligned}$$

and the holomorphic weight 1/2 and weight 3/2 theta functions associated with N.

It remains to compute the cycle integrals of $\Theta _I^*(\tau ,z)$ and $Q_A(z)^{-1}\Theta _I(\tau ,z)$. Note that the geodesic $S_A$ can be identified with the Grassmannian ${{\,\textrm{Gr}\,}}(I)$ of positive lines in $I \otimes \mathbb {R}$. Hence, the cycle integrals can be computed as done by Hecke [9]. We give the details for $\Theta _I(\tau ,z)$.

Recall that we assume $a > 0$. The two real endpoints $w < w'$ of $S_A$ are given by $w = \frac{-b-\sqrt{D}}{2a}$ and $w' = \frac{-b+\sqrt{D}}{2a}$. The matrix

$$\begin{aligned} \sigma = \frac{a^{\frac{1}{2}}}{D^{\frac{1}{4}}}\begin{pmatrix}w' &{} w \\ 1 &{} 1 \end{pmatrix} \in {{\,\textrm{PSL}\,}}_{2}(\mathbb {R}) \end{aligned}$$

maps 0 to w and $i\infty$ to $w'$ and hence maps the imaginary axis (oriented from $i\infty$ to 0) to $S_{A}$ (oriented counter-clockwise). Note that $\sigma ^{-1}. A = [0,-\sqrt{D},0]$. Modding out the stabilizer $\Gamma _A = \Gamma _I$ in the sum over Y, and using the parametrization $z = \sigma .it$ of $S_A$, we obtain

$$\begin{aligned} \int _{\Gamma _A \backslash S_A}\Theta _I(\tau ,z)\frac{dz}{Q_A(z)}\\ {}{} & {} = \frac{v^{1/2}}{\sqrt{D}}\int _{0}^\infty \sum _{Y \in \Gamma _I \backslash I'}\frac{1}{|\Gamma _Y|}p_{Y}(\sigma .it)e(q(Y_{\sigma .it})\tau \\{} & {} \quad + q(Y_{\sigma .it^{\perp }}){\overline{\tau }}){{\,\mathrm{\mathfrak {e}}\,}}_{Y}\frac{dt}{t}. \end{aligned}$$

If we write $Y = \left( {\begin{matrix}\beta /2 &{} \gamma \\ -\alpha &{} -\beta /2 \end{matrix}} \right) \in I'$, a short computation shows that

$$\begin{aligned} \sigma ^{-1}.Y = \begin{pmatrix}0 &{} \lambda \\ \lambda ' &{} 0 \end{pmatrix}, \qquad \lambda = \frac{a}{\sqrt{D}}(\alpha w^2 + \beta w + \gamma ) \in \mathbb {Q}\big (\sqrt{D}\big ), \end{aligned}$$

where $\lambda '$ is the conjugate of $\lambda$ in $\mathbb {Q}(\sqrt{D})$. Using (2.1) and (2.2), we find

$$\begin{aligned} p_{Y}(\sigma .it)&= p_{\sigma ^{-1}.Y}(it) = \lambda t^{-1}-\lambda ' t, \\ q(Y_{\sigma .it^\perp })&= q((\sigma ^{-1}.Y)_{it^\perp }) = -\frac{1}{4}(\lambda t^{-1}+\lambda ' t)^2. \end{aligned}$$

Hence, using $q(Y_z)\tau + q(Y_{z^\perp }){\overline{\tau }} = q(Y)\tau - 2iv q(Y_{z^{\perp }})$, we get

$$\begin{aligned} \int _{\Gamma _A \backslash S_A}\Theta _I(\tau ,z)\frac{dz}{Q_A(z)}\\ {}&= \frac{v^{1/2}}{\sqrt{D}}\sum _{Y \in \Gamma _I \backslash I'}\frac{1}{|\Gamma _Y|}\\&\quad \left( \int _{0}^{\infty }(\lambda t^{-1}-\lambda ' t) e^{-\pi v (\lambda t^{-1} + \lambda ' t)^2}\frac{dt}{t}\right) e(q(Y) \tau ){{\,\mathrm{\mathfrak {e}}\,}}_{Y}. \end{aligned}$$

A standard computation² shows that the integral is given by

$$\begin{aligned} \int _{0}^\infty (\lambda t^{-1}-\lambda ' t) e^{-\pi v (\lambda t^{-1} + \lambda ' t)^2}\frac{dt}{t} = \frac{1}{2}v^{-1/2}({{\,\textrm{sgn}\,}}(\lambda )-{{\,\textrm{sgn}\,}}(\lambda '))e^{-2\pi v|\lambda \lambda '|}e^{-2\pi v \lambda \lambda '}, \end{aligned}$$

which equals $v^{-1/2}{{\,\textrm{sgn}\,}}(\lambda )$ if $q(Y) = -\lambda \lambda ' > 0$ and vanishes otherwise. Using $\sqrt{D}\lambda =a(\alpha w^2 + \beta w + \gamma ) = a(Y,Y_0)$, with $Y_0$ as in (3.2), we get ${{\,\textrm{sgn}\,}}(\lambda )={{\,\textrm{sgn}\,}}(Y,Y_0)$. Summarizing, we obtain

$$\begin{aligned} \int _{\Gamma _A \backslash S_A}\Theta _I(\tau ,z)\frac{dz}{Q_A(z)}&= \frac{1}{\sqrt{D}} \sum _{\begin{array}{c} Y \in \Gamma _I \backslash I' \\ q(Y) > 0 \end{array}}\frac{{{\,\textrm{sgn}\,}}(Y,Y_0)}{|\Gamma _Y|}e(q(Y) \tau ){{\,\mathrm{\mathfrak {e}}\,}}_{Y} = \frac{1}{\sqrt{D}}\vartheta _I(\tau ). \end{aligned}$$

The computation of the cycle integral of $\Theta _I^*(\tau ,z)$ is very similar. Using

$$\begin{aligned} \int _{0}^\infty (\lambda ^2 t^{-2}-\lambda '^2 t^2)e^{-\pi v (\lambda t^{-1} + \lambda ' t)^2} \frac{dt}{t} = \left[ \frac{1}{2\pi v}e^{-\pi v (\lambda t^{-1} + \lambda ' t)^2} \right] _0^{\infty } = 0, \end{aligned}$$

we see that we in fact have $\mathcal {C}_A(\Theta _I^*) = 0$. This finishes the proof. $\square$

Remark 3.2

One can compute the cycle integrals of $R_0 \Theta _{L}(\tau ,z)$ along infinite geodesics (if D is a square) in a similar way. In this case, the lattice I is isotropic, and $\vartheta _I$ has to be replaced with a weight 1 Hecke Eisenstein series.

4 Cycle integrals of meromorphic modular forms: the proof of Theorem 1.2

We are now ready to give an explicit evaluation of the cycle integrals appearing in Theorem 1.2. As before, we let $A \in L'$ be an indefinite integral binary quadratic form of non-square discriminant $D = -4q(A) > 0$ with $a > 0$, and we let $I = L \cap (\mathbb {Q}A)^\perp$ and $N = L \cap (\mathbb {Q}A)$ be the corresponding indefinite and negative definite sublattices of L. Moreover, we let $\vartheta _I$ be the Hecke theta series of weight 1 for $\rho _I$ defined in (3.3) and $\Theta _{3/2,N}$ the holomorphic theta function of weight 3/2 for ${\overline{\rho }}_{N}$ as in (3.1).

We choose a harmonic Maass form ${\widetilde{\Theta }}_{3/2,N}$ of weight 1/2 for $\rho _{N}$ with

$$\begin{aligned} \xi _{1/2}{\widetilde{\Theta }}_{3/2,N} = \frac{1}{\sqrt{D}}\Theta _{3/2,N}, \end{aligned}$$

and we let ${\widetilde{\Theta }}_{3/2,N}^+$ be its holomorphic part. Moreover, we require the Rankin-Cohen bracket

$$\begin{aligned} \left[ f,g \right] _{n} = \sum _{s = 0}^n (-1)^s \left( {\begin{array}{c}\kappa +n-1\\ s\end{array}}\right) \left( {\begin{array}{c}\ell + n-1\\ n-s\end{array}}\right) f^{(n-s)}g^{(s)}, \qquad \left( f^{(s)} = \frac{1}{(2\pi i )^s}\frac{\partial ^s}{\partial \tau ^s}f \right) , \end{aligned}$$

which maps modular forms f of weight $\kappa$ and g of weight $\ell$ to a form of weight $\kappa + \ell + 2n$.

Theorem 4.1

Let $k \ge 3$ be odd and let $g \in M_{3/2-k,{\overline{\rho }}_L}^!$ be a weakly holomorphic modular form of weight $3/2-k$ for ${\overline{\rho }}_L$. Then, we have

$$\begin{aligned} \mathcal {C}_A\left( \sum _{d < 0}a_g(d)f_{k,d}\right) = -(4D)^{(k-1)/2}{{\,\textrm{CT}\,}}\left( g_{I \oplus N} \cdot \left[ \vartheta _I,{\widetilde{\Theta }}_{3/2,N}^+ \right] _{(k-1)/2} \right) , \end{aligned}$$

with the Rankin-Cohen bracket in weights $\kappa = 1$ and $\ell = 1/2$, and $g_{I \oplus N}$ defined in (2.4).

Proof

We follow the idea of the proof of [8, Theorem 5.4]. For brevity we put $f_{k,g} = \sum _{d < 0}a_g(d)f_{k,d}$. Using Proposition 2.1, we can write $f_{k,g}$ as a theta lift,

$$\begin{aligned} f_{k,g}(z) = c_k\cdot R_{0}^{k}\Phi _L\left( R_{3/2-k}^{(k-1)/2}g,z\right) , \end{aligned}$$

with the constant $c_k = (2^{k+1}\pi ^{(k+1)/2}(k-1)!)^{-1}$. Now we take the cycle integral $\mathcal {C}_A$ on both sides. As a preliminary step, we will reduce the power of the outer raising operator $R_0^k$. It is well known that $\Phi _L = \Phi _L(R_{3/2-k}^{(k-1)/2}g,z)$ is an eigenform of the Laplacian $\Delta _0$ with eigenvalue $k(1-k)$, see [7, Theorem 4.6]. Hence, a repeated application of [5, Theorem 1.1] gives

$$\begin{aligned} \mathcal {C}_A\big ( R_{0}^k\Phi _L \big ) = c_k'\, \mathcal {C}_A\left( R_{0}\Phi _L \right) , \end{aligned}$$

with the constant $c_k' = D^{(k-1)/2} (k-1)!^2/(\frac{k-1}{2})!^2$. Interchanging the cycle integral and the regularized integral gives

$$\begin{aligned} \mathcal {C}_A\big (f_{k,g}\big ) = c_k c_k' \int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}}R_{3/2-k}^{(k-1)/2}g(\tau )\cdot \mathcal {C}_A\big (R_0\Theta _L(\tau ,\, \cdot \,)\big )d\mu (\tau ). \end{aligned}$$

Now we plug in the evaluation of the cycle integral from Theorem 3.1 to get

$$\begin{aligned} \mathcal {C}_A\big (f_{k,g}\big ) = c_k c_k'c_k''\int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}}R_{3/2-k}^{(k-1)/2}g_{I \oplus N}(\tau )\cdot \vartheta _I(\tau )\otimes \overline{\Theta _{3/2,N}(\tau )} v^{3/2}d\mu (\tau ), \end{aligned}$$

with the constant $c_k'' = -4\pi /\sqrt{D}$. Here, we used that the maps $f^L$ and $g_{I \oplus N}$ are adjoint to each other. Next, we use the “self-adjointness” of the raising operator to compute

$$\begin{aligned} \mathcal {C}_A\big (f_{k,g}\big )&= (-1)^{(k-1)/2}c_k c_k'c_k''\int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}}g_{I \oplus N}(\tau )\cdot R_{-1/2}^{(k-1)/2}\left( \vartheta _I(\tau )\otimes \overline{\Theta _{3/2,N}(\tau )} v^{3/2}\right) d\mu (\tau ) \\&= (-1)^{(k-1)/2}c_k c_k'c_k''\int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}}g_{I \oplus N}(\tau )\cdot \left( R_{1}^{(k-1)/2}\vartheta _I(\tau )\right) \otimes \overline{\Theta _{3/2,N}(\tau )} v^{3/2}d\mu (\tau ). \end{aligned}$$

In the last step, we used that $\Theta _{3/2,N}$ is holomorphic. By [8, Proposition 3.6], we have

$$\begin{aligned} \left( R_{1}^{(k-1)/2}\vartheta _I(\tau )\right) \otimes \overline{\Theta _{3/2,N}(\tau )} v^{3/2} = c_k'''L_{k+1/2}\left[ \vartheta _I, {\widetilde{\Theta }}_{3/2,N}\right] _{(k-1)/2}, \end{aligned}$$

with the constant $c_k''' = 4^{k-1}(-\pi )^{(k-1)/2}\sqrt{D}(\frac{k-1}{2})!^2/(k-1)!$. We arrive at

$$\begin{aligned} \mathcal {C}_A\big (f_{k,g}\big ) = (-1)^{(k-1)/2}c_k c_k'c_k''c_k'''\int _{\mathcal {F}}^{{{\,\textrm{reg}\,}}}g_{I \oplus N}(\tau )\cdot L_{k+1/2}\left[ \vartheta _I, {\widetilde{\Theta }}_{3/2,N}\right] _{(k-1)/2} d\mu (\tau ). \end{aligned}$$

Note that the constants in front simplify to $(-1)^{(k-1)/2}c_k c_k'c_k''c_k''' = -(4D)^{(k-1)/2}$. Now a standard application of Stokes’ Theorem as in [8, Theorem 5.4] finishes the proof. $\square$

We remark that there is a similar formula for the linear combination of cycle integrals of $f_{k,P}$ from Theorem 1.1, see [2, Theorem 1.1]. We can now prove Theorem 1.2.

Proof of Theorem 1.2

We show that the right-hand side in Theorem 4.1 is rational. Note that g and $\vartheta _I$ have rational Fourier coefficients. Moreover, by [10, Theorem 1.1], we can choose ${\widetilde{\Theta }}_{3/2,N}$ such that its holomorphic part has rational Fourier coefficients. Here, we use that the lattice N is of the form $(\mathbb {Z},q(x) = -Dr^2x^2)$ for some rational number r. It remains to note that the Rankin-Cohen brackets preserve the rationality of the Fourier coefficients.

5 Generalization to higher level

In [11, Conjecture 2.5], Theorem 1.2 was stated as a conjecture for level $\Gamma _0(N)$ and twisted versions of $f_{k,d}$. Here, we sketch the proof of this general conjecture.

Let $N \in \mathbb {N}$. We let $\Delta \in \mathbb {Z}$ be a fundamental discriminant (possibly 1) and $\rho \in \mathbb {Z}/2N\mathbb {Z}$ such that $\Delta \equiv \rho ^2 \ \, \left( \textrm{mod} \, 2N \right)$. Moreover, we let $d < 0$ be a negative integer and $r \in \mathbb {Z}/2N\mathbb {Z}$ with $d \equiv {{\,\textrm{sgn}\,}}(\Delta )r^2 \ \, \left( \textrm{mod} \, 2N \right)$. We consider the set $\mathcal {Q}_{N,d|\Delta |,r\rho }$ of (not necessarily positive definite) integral binary quadratic forms $Q = [aN,b,c]$ of discriminant $d|\Delta |$ with $b \equiv r\rho \ \, \left( \textrm{mod} \, 2N \right)$. We let $\chi _\Delta$ be the usual generalized genus character on $\mathcal {Q}_{N,d|\Delta |,r\rho }$. For $k \in \mathbb {N}$ with $k \ge 2$, we define the function

$$\begin{aligned} f_{k,d,r,\Delta ,\rho }(z) = \frac{|d\Delta |^{k-1/2}}{2\pi }\sum _{Q \in \mathcal {Q}_{N,d|\Delta |,r\rho }}\frac{{{\,\textrm{sgn}\,}}(Q)\chi _\Delta (Q)}{Q(z,1)^k}, \end{aligned}$$

where we put ${{\,\textrm{sgn}\,}}(Q) = {{\,\textrm{sgn}\,}}(a)$. It defines a meromorphic modular form of weight 2k for $\Gamma _0(N)$. For $N = 1$ and odd k, we recover $f_{k,d}$ as $f_{k,d,d,1,1}$.

We consider the signature (1, 2) lattice

$$\begin{aligned} L_N = \left\{ X=\begin{pmatrix}b &{} c/N \\ -a &{} -b \end{pmatrix}: a,b,c \in \mathbb {Z}\right\} \end{aligned}$$

with the quadratic form $q_N(X) = N\det (X)$. We have $L_N'/L_N \cong \mathbb {Z}/2N\mathbb {Z}$, so we will write ${{\,\mathrm{\mathfrak {e}}\,}}_r$ with $r \in \mathbb {Z}/2N\mathbb {Z}$ for the standard basis elements of $\mathbb {C}[L_N'/L_N]$. Note that the elements of $L_N'$ correspond to binary quadratic forms [aN, b, c], with the discriminant being given by $-4Nq_N(X)$. We let ${\widetilde{\rho }}_{L_N}$ be equal to ${\overline{\rho }}_{L_N}$ if $\Delta > 0$, and to $\rho _{L_N}$ if $\Delta < 0$.

We have the following higher level version of Theorem 1.2.

Theorem 5.1

Let $k \in \mathbb {N}$ with $k \ge 2$ and let g be a weakly holomorphic modular form of weight $3/2-k$ for ${\widetilde{\rho }}_{L_N}$. Suppose that the Fourier coefficients $a_g(d,r)$ for $d < 0$ and $r \in \mathbb {Z}/2N\mathbb {Z}$ are rational. Then, the cycle integrals

$$\begin{aligned} \mathcal {C}_A\left( \sum _{r \in \mathbb {Z}/2N\mathbb {Z}}\sum _{d < 0}a_{g}(d,r)f_{k,d,r,\Delta ,\rho }\right) \end{aligned}$$

are rational.

Proof

This can be proved analogously to Theorem 1.2, so we only give a sketch. First, by [3, Proposition 4.1], the meromorphic modular form $f_{k,d,r,\Delta ,\rho }$ can be written as a twisted theta lift on the lattice $L_N$. We can reduce to the non-twisted case using an intertwining operator for the involved Weil representations as explained in [4]. Then it remains to evaluate the cycle integrals of certain Siegel theta functions associated with a general even lattice L of signature (1, 2). However, note that we did not use the precise shape of L in the proof of Theorem 3.1, so the arguments work exactly the same for arbitrary L, at least for odd k. For even k, the theta lift involves a Siegel theta function of the shape

$$\begin{aligned} \Theta _{L}^*(\tau ,z) = v\sum _{X \in L'}p_z(X)e\left( q(X_z)\tau + q(X_{z^\perp }){\overline{\tau }} \right) {{\,\mathrm{\mathfrak {e}}\,}}_X. \end{aligned}$$

A computation similar to the proof of Theorem 3.1 shows that we have

$$\begin{aligned} \mathcal {C}_A\big (\Theta _{L}^*(\tau ,z) \big ) {\mathop {=}\limits ^{\cdot }} \left( \vartheta _I(\tau ) \otimes \overline{\Theta _{1/2,N}(\tau )}v^{1/2}\right) ^L, \end{aligned}$$

with the holomorphic weight 1/2 theta function $\Theta _{1/2,N}(\tau ) = \sum _{X \in N'}e(-q(X)\tau ){{\,\mathrm{\mathfrak {e}}\,}}_X$. Apart from this, the proofs for even and odd k are analogous. $\square$

Finally, one might speculate that the method from Theorem 3.1 could be used to compute cycle integrals of Siegel theta functions (and the corresponding theta lifts) on lattices of other signatures, e.g., signature (1, n) or (2, n). We hope to come back to this problem in the future.