Schneider-Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

TL;DR

This paper extends classical results on the transcendence of special values of the modular j-invariant to a family of values of harmonic weak Maass forms evaluated at Hecke orbits, showing they are transcendental for infinitely many primes under certain conditions.

Contribution

It generalizes Schneider and Siegel's theorem to harmonic weak Maass forms on Hecke orbits, establishing transcendence of their values at algebraic points for infinitely many primes.

Findings

Values of harmonic weak Maass forms at Hecke orbits are transcendental for infinitely many primes.

The result applies to a broad class of harmonic weak Maass forms with algebraic principal parts.

Transcendence holds under mild assumptions on the form and for primes outside a specific finite set.

Abstract

Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $\tau$ is not a CM point, i.e., $[\mathbb{Q}(\tau):\mathbb{Q}]\neq2$, then $j(\tau)$ is transcendental. Let $f$ be a harmonic weak Maass form of weight $0$ on $\Gamma_0(N)$. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of $f$ on Hecke orbits of $\tau$. For a positive integer $m$, let $T_m$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i \infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i \infty$. We prove, under a mild assumption on $f$, that for any fixed $\tau$, if $N$ is a prime such that $ N\geq 23 \text{ and } N \not \in \{23, 29, 31, 41, 47, 59, 71\},$ then $f(T_m.\tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.

Equations

$$f(D):=\sum n_{P}f(P).$$

$${\xi }_{-k}:H_{-k}({\Gamma }_0(N))\to M_{k+2}^{!}({\Gamma }_0(N)),$$

$$D_{\tau }:=Q_{i\infty }-Q_{\tau }\in {\mathrm{Div}}_{X_0(N)}(\mathbb{C}).$$

$$f(T_m.Q_{\tau })-m^{-1}{\lambda }_m{\left({\xi }_0(f),f_{{\psi }_{D_{\tau }}^{alg}}\right)}_{reg}$$

$$T_{Q_{i\infty}}(X_{0}(N)):=\left\{Q\in X_{0}(N)(\overline{\mathbb{Q}})\;\big{|}\;i_{Q_{i\infty}}(Q)\text{ is a torsion point in }J_{\Gamma_{0}(N)}\right\}.$$

$$T_{Q_{i\infty}}(X_{0}(N))=\left\{\begin{array}[]{ll}\{0,i\infty\}&\text{ if }g^{+}>0,\\ \{0,i\infty\}\cup\{\text{hyperelliptic branch points}\}&\text{ if }g^{+}=0,\end{array}\right.$$

$$N\ge 23\text{ and }N\notin \{23,29,31,41,47,59,71\}.$$

$$(f{|}_k\gamma )(z):=(cz+d{)}^{-k}f(\gamma z)$$

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

$${\sigma }_t^{-1}{\Gamma }_0(N{)}_t{\sigma }_t=\left\{\pm \left(\begin{array}{cc}1& \ell {\alpha }_t\\ 0& 1\end{array}\right):\ell \in \mathbb{Z}\right\}.$$

$$f_t^{+}(z)$$

$$f_t^{-}(z)$$

$${\xi }_{-k}(f{|}_{-k}{T}_n)=n^{-k-1}({\xi }_{-k}(f){|}_{k+2}{T}_n)$$

$$(g{|}_2{\sigma }_t)(z)=\sum a_g^t(n)q^{n/{\alpha }_t},$$

$${\mathrm{Res}}_{Q_{\tau }}gdz=\{\begin{array}{ll}\frac{1}{e_{\tau }}{\mathrm{Res}}_{\tau }g& \text{ if }\tau \in \mathbb{H},\\ \frac{1}{2\pi i}{\alpha }_{\tau }{a}_g^{\tau }(0)& \text{ if }\tau \in {\mathcal{C}}_N.\end{array}$$

$$res(\varphi )=\sum _j{m}_j{P}_j.$$

$$B_{\tau}(\varepsilon):=\left\{\begin{array}[]{ll}\{z\in\mathbb{H}\;:\;|z-\tau|<\varepsilon\}&\text{if }\tau\in\mathbb{H},\\ \{z\in\mathcal{F}_{N}\;:\;\mathrm{Im}(\sigma_{\tau}z)>1/\varepsilon\}&\text{if }\tau\in\{i\infty\}\cup\mathbb{Q}.\end{array}\right.$$

$${\mathcal{F}}_N(g,\epsilon ):={\mathcal{F}}_N-{\cup }_{\tau \in \mathrm{Sing}(g)\cup {\mathcal{C}}_N}{B}_{\tau }(\epsilon ).$$

$$(f,g{)}_{reg}:=\underset{\epsilon \to 0}{\lim }{\int }_{{\mathcal{F}}_N(g,\epsilon )}f(z)\overline{g(z)}\frac{dxdy}{y^{k-2}}.$$

$$({\xi }_0(f),g{)}_{reg}=\sum _{m+n=0}\sum _{t\in {\mathcal{C}}_N}{a}_f^t(m)a_g^t(n)+\sum _{\tau \in {\mathcal{F}}_N}\frac{2\pi i}{e_{\tau }}{\mathrm{Res}}_{\tau }(g)f(\tau ),$$

$$(f,g{)}_{reg}=0$$

$$d(f(z)\overline{G(z)}dz)=f(z)\overline{{\partial }_{z}G(z)}d\overline{z}dz=f(z)\overline{g(z)}(2i)dxdy.$$

$$\gamma_{\tau}(\varepsilon):=\left\{\begin{array}[]{ll}\{z\in\mathbb{H}\;:\;|z-\tau|=\varepsilon\}&\text{if }\tau\in\mathbb{H},\\ \{z\in\mathcal{F}_{N}\;:\;\mathrm{Im}(\sigma_{\tau}z)=1/\varepsilon\}&\text{if }\tau\in\{i\infty\}\cup\mathbb{Q}.\end{array}\right.$$

$${\partial }^{\ast }{\mathcal{F}}_N(g,\epsilon )={\cup }_{\tau \in \mathrm{Sing}(g)\cup {\mathcal{C}}_N}{\gamma }_{\tau }(\epsilon ).$$

$${\int }_{{\mathcal{F}}_N(g,\epsilon )}f(z)\overline{g(z)}dxdy={\int }_{{\partial }^{\ast }{\mathcal{F}}_N(g,\epsilon )}\frac{1}{2i}f(z)\overline{G(z)}dz=\sum _{\tau \in \mathrm{Sing}(g)\cup {\mathcal{C}}_N}{\int }_{{\gamma }_{\tau }(\epsilon )}\frac{1}{2i}f(z)\overline{G(z)}dz.$$

$$\left|{\int }_{{\gamma }_{\tau }(\epsilon )}\frac{1}{2i}f(z)\overline{G(z)}dz\right|$$

$$\le {\int }_{{\gamma }_{\tau }(\epsilon )}\frac{1}{2}|f(z)||\overline{G(z)}||dz|$$

$$\le \max \{|G(z)|:z\in {\gamma }_{\tau }(\epsilon )\}{M}_1{\int }_{{\gamma }_{\tau }(\epsilon )}|dz|(\text{some constant }{M}_1)$$

$$\le \max \{|G(z)|:z\in {\gamma }_{\tau }(\epsilon )\}{M}_1(2\pi \epsilon ).$$

$$G(z)=-{\log }_e(|z-\tau |(z-\tau ))+G_0(z),$$

$$|G(z)|$$

$$\le |{\log }_e|z-\tau ||+|{\log }_e(z-\tau )|+M_2(\text{some fixed constant }{M}_2)$$

$$\le 2|{\log }_e\epsilon |+\pi +M_2.$$

Full text

For a positive integer $m$, let $T_{m}$ denote the $m$-th Hecke operator. Suppose that the coefficients of the principal part of $f$ at the cusp $i\infty$ are algebraic, and that $f$ has its poles only at cusps equivalent to $i\infty$. We prove, under a mild assumption on $f$, that for any fixed $\tau$, if $N$ is a prime such that $N\geq 23\text{ and }N\not\in\{23,29,31,41,47,59,71\},$ then $f(T_{m}.\tau)$ are transcendental for infinitely many positive integers $m$ prime to $N$.

Keywords: harmonic weak Maass form, CM point, meromorphic differential

1. Introduction

Let $j(z)$ be the modular $j$-invariant function on the complex upper half plane $\mathbb{H}$. Let $\tau$ be an algebraic number in $\mathbb{H}$. It was proved by Kronecker [11] and Weber [17] that if $\tau$ is a CM point, i.e., $[\mathbb{Q}(\tau):\mathbb{Q}]=2$, then $j(\tau)$ is algebraic. Schneider [14] and Siegel [16] proved that if $\tau$ is not a CM point, then $j(\tau)$ is transcendental. By combining these two results, we state the following.

Theorem A** (Kronecker, Schneider, Siegel, Weber).**

Assume that $\tau$ is an algebraic number in $\mathbb{H}$. Then, $\tau$ is a CM point if and only if $j(\tau)$ is algebraic.

Let $m$ be a positive integer, and $T_{m}$ denote the $m$-th Hecke operator. The operators $T_{m}$ act on both of modular forms $f$ and divisors $D$ of a modular curve, and they are denoted by $f|T_{m}$ and $T_{m}.D$, respectively. Then, $j(T_{m}.\tau)=(j|T_{m})(\tau)$, and $(j|T_{m})(z)$ is a polynomial of $j(z)$ with rational coefficients. Thus, $j(T_{m}.\tau)$ is algebraic for every $m$ if and only if $j(\tau)$ is algebraic. Therefore, Theorem A is equivalent to the following theorem.

Theorem B** (Kronecker, Schneider, Siegel, Weber).**

Assume that $\tau$ is an algebraic number in $\mathbb{H}$. Then, $\tau$ is a CM point if and only if $j(T_{m}.\tau)$ is algebraic for every positive integer $m$ .

In this vein, we consider an extension of the results of Kronecker, Schneider, Siegel, and Weber to a family of values of a harmonic weak Maass form $f$ on Hecke orbits of $\tau$. Let $N$ be a positive integer, and $f$ be a harmonic weak Maass form of weight [math] on $\Gamma_{0}(N)$. In contrast to the case for the $j$-invariant function, the value of $f$ at a CM point $\tau$ is not algebraic in general. Thus, first we obtain the period of $f(\tau)$ for a CM point $\tau$, which is expressed as the regularized Petersson inner product of a cusp form and a meromorphic modular form. Next, by using this result, we obtain an extension of the results of Kronecker, Schneider, Siegel, and Weber to a family of values of $f$ on Hecke orbits of $\tau$.

Let $Y_{0}(N)$ be the modular curve of level $N$ defined by $\Gamma_{0}(N)\backslash\mathbb{H}$, and $X_{0}(N)$ denote the compactification of $Y_{0}(N)$ by adjoining the cusps. Let us note that $X_{0}(N)$ is a curve defined over $\mathbb{Q}$. We fix an algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$. Let $K$ be a subfield of $\overline{\mathbb{Q}}$ or the field $\mathbb{C}$ of complex numbers. Let $C$ be a curve defined over $K$. For an extension $E$ of $K$, we denote by $\mathrm{Div}_{C}(E)$ the group of divisors of $C$ defined over $E$. Let $f$ be a function on $C(\mathbb{C})\setminus S$ for a finite subset $S$ of $C(\mathbb{C})$. If $D=\sum_{P\in C}n_{P}P\in\mathrm{Div}_{C}(\mathbb{C})$ and the support of $D$ does not contain any point in $S$, then we define

[TABLE]

The $m$-th Hecke operator $T_{m}$ acts on $\mathrm{Div}_{X_{0}(N)}(\mathbb{C})$, and it is denoted by $T_{m}.D$ for $D\in\mathrm{Div}_{X_{0}(N)}(\mathbb{C})$.

Let $k$ be a non-negative even integer. Let $S_{k}(\Gamma_{0}(N))$ denote the space of cusp forms of weight $k$ on $\Gamma_{0}(N)$. We denote by $H_{k}(\Gamma_{0}(N))$ the space of harmonic weak Maass forms of weight $k$ on $\Gamma_{0}(N)$. For the differential operator $\xi_{-k}$ defined by $\xi_{-k}(f)(z):=2iy^{-k}\overline{\frac{\partial}{\partial\bar{z}}f(z)}$, the assignment $f(z)\mapsto\xi_{-k}(f)(z)$ gives an anti-linear mapping

[TABLE]

where $M^{!}_{k}(\Gamma_{0}(N))$ denotes the space of weakly holomorphic modular forms of weight $k$ on $\Gamma_{0}(N)$. Here, $y$ denotes the imaginary part of $z\in\mathbb{H}$. Let $H^{*}_{-k}(\Gamma_{0}(N))$ be the inverse image of the space $S_{k+2}(\Gamma_{0}(N))$ of cusp forms under the mapping $\xi_{-k}$.

Definition 1.1**.**

Let $f$ be a harmonic weak Maass form of weight [math] on $\Gamma_{0}(N)$. We say that $f$ is arithmetic if $f$ satisfies the following conditions:

1. (1)

the principal part of $f$ at the cusp $i\infty$ belongs to $\overline{\mathbb{Q}}[q^{-1}]$, and its constant term is zero, 2. (2)

the principal part of $f$ at each cusp not equivalent to $i\infty$ is constant.

Here, $q:=e^{2\pi iz}$ for a complex number $z\in\mathbb{H}$.

This definition is similar with that of being good in [6]; however, the conditions in this definition are weaker than those in the definition of being good. Furthermore, a harmonic weak Maass form $f$ is called an arithmetic Hecke eigenform if $f$ is arithmetic and $\xi_{0}(f)$ is a Hecke eigenform.

For $\tau\in\mathbb{H}\cup\{i\infty\}\cup\mathbb{Q}$, let $Q_{\tau}$ be the image of $\tau$ under the canonical map from $\mathbb{H}\cup\{i\infty\}\cup\mathbb{Q}$ to $X_{0}(N)$ and

[TABLE]

If $\tau$ is a CM point, then $D_{\tau}$ is defined over $\overline{\mathbb{Q}}$. Thus, there exists a differential $\psi_{D_{\tau}}^{alg}$ of the third kind associated to $D_{\tau}$ such that $\psi_{D_{\tau}}^{alg}$ is defined over $\overline{\mathbb{Q}}$. Note that $\psi_{D_{\tau}}^{alg}$ can be written as $\psi_{D_{\tau}}^{alg}=2\pi if_{\psi_{D_{\tau}}^{alg}}(z)dz$ for some meromorphic modular form $f_{\psi_{D_{\tau}}^{alg}}$ of weight $2$ on $\Gamma_{0}(N)$ (see Section 2.3 for details). Let $\left(\xi_{0}(f),f_{\psi_{D_{\tau}}^{alg}}\right)_{reg}$ be the regularized Petersson inner product of $\xi_{0}(f)$ and $f_{\psi_{D_{\tau}}^{alg}}$ (see Section 3 for the definition of the regularized Petersson inner product). The following theorem shows that, for each positive integer $m$ prime to $N$, the period of $f(T_{m}.Q_{\tau})$ can be expressed as the multiplication of $\left(\xi_{0}(f),f_{\psi_{D_{\tau}}^{alg}}\right)_{reg}$ and the eigenvalue of $\xi_{0}(f)$ for $T_{m}$.

Theorem 1.2**.**

Let $N$ be a prime, and $f$ be a harmonic weak Maass form of weight [math] on $\Gamma_{0}(N)$. Assume that $f$ is an arithmetic Hecke eigenform, and that $\tau$ is a CM point. Let $\psi_{D_{\tau}}^{alg}:=2\pi if_{\psi_{D_{\tau}}^{alg}}(z)dz$ be a differential of the third kind associated to $D_{\tau}$ defined over $\overline{\mathbb{Q}}$. Then,

[TABLE]

is algebraic for every positive integer $m$ prime to $N$, where $\lambda_{m}$ is the eigenvalue of $\xi_{0}(f)$ for $T_{m}$.

Theorem 1.2 is applied to study the transcendence of $f(T_{m}.Q_{\tau})$ for an algebraic number $\tau$ in $\mathbb{H}$. Then, we have the following theorem concerning to an extension of the above results of Kronecker, Schneider, Siegel, and Weber to a family of values of $f$ on Hecke orbits of $\tau$.

Theorem 1.3**.**

Let $N$ and $f$ be given as in Theorem 1.2. Assume that $(g,\xi_{0}(f))\neq 0$ for each Hecke eigenform $g\in S_{2}(\Gamma)$. Let $\tau$ be an algebraic number in $\mathbb{H}$. Then, $f(T_{m}.Q_{\tau})$ is algebraic for every positive integer $m$ prime to $N$ if and only if $\tau$ is a CM point and $nD_{\tau}$ is rational on $X_{0}(N)$ for some positive integer $n$.

Remark 1.4**.**

*Assume that $f$ and $\tau$ are given as in Theorem 1.3. In fact, if there is a positive integer $m$ prime to $N$ such that $f(T_{m}.Q_{\tau})$ is transcendental, then there are infinitely many such positive integers $m$ prime to $N$ (see the proof of Theorem 1.3 in Section 4). *

Let $J_{\Gamma_{0}(N)}$ be the Jacobian variety of the $X_{0}(N)$ defined over $\mathbb{Q}$. Then, an Albanese embedding $i_{Q_{i\infty}}:X_{0}(N)\rightarrow J_{\Gamma_{0}(N)}$ can be defined by sending $Q$ to $Q_{i\infty}-Q$. Let us note that $m(Q_{i\infty}-Q)$ is rational on $X_{0}(N)$ for some positive integer $m$ if and only if $i_{Q_{i\infty}}(Q)$ is a torsion point in $J_{\Gamma_{0}(N)}$. Let

[TABLE]

If the genus of $X_{0}(N)$ is larger than or equal to $2$, then, by the Mumford-Manin conjecture (proved by Raynaud), $T_{Q_{i\infty}}(X_{0}(N))$ is a finite set.

Let $X^{+}_{0}(N)$ denote the quotient of $X_{0}(N)$ by the Atkin-Lehner involution $w_{N}$. For primes $N$, Coleman, Kaskel, and Ribet [8] conjectured the following statement: for all prime numbers $N\geq 23$,

[TABLE]

where $g^{+}$ denotes the genus of $X_{0}(N)^{+}$. Baker [2] proved this conjecture. Furthermore, for $N\geq 23$, $g^{+}$ is zero if and only if $N\in\{23,29,31,41,47,59,71\}$. Thanks to these results on torsion points on the Jacobian of a modular curve, we obtain the following theorem from Theorem 1.3.

Theorem 1.5**.**

Under the assumption as in Theorem 1.3, assume that

[TABLE]

Then, $f(T_{m}.Q_{\tau})$ are transcendental for infinitely many positive integers $m$ prime to $N$.

The remainder of this paper is organized as follows. In Section 2, we introduce some preliminaries for harmonic weak Maass forms, residues of meromorphic differentials on a modular curve, and differentials of the third kind on a complex curve. In Section 3, we review the definition of a regularized Petersson inner product and prove that the regularized Petersson inner product of a meromorphic modular form, associated with a canonical differential of the third kind of some divisor on $X_{0}(N)$, with every cusp form of weight $2$ on $\Gamma_{0}(N)$ is zero. In Section 4, we prove Theorem 1.2 and 1.3.

2. Preliminaries

In this section, we recall definitions and basic facts about harmonic weak Maass forms, residues of meromorphic differentials on a modular curve, and properties for differentials of the third kind on a complex curve.

2.1. Harmonic weak Maass forms

For details of harmonic weak Maass forms, we refer to [4] and [13]. Let $k$ be an even integer. We recall the weight $k$ slash operator

[TABLE]

for any function $f$ on $\mathbb{H}$ and $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$. Let $\Delta_{k}$ denote the weight $k$ hyperbolic Laplacian defined by

[TABLE]

where $x$ (resp. $y$) denotes the real (resp. imaginary) part of $z$. Now, we give the definition of a harmonic weak Maass form.

Definition 2.1**.**

Let $N$ be a positive integer. A smooth function $f$ on $\mathbb{H}$ is a harmonic weak Maass form of weight $k$ on $\Gamma_{0}(N)$ if it satisfies the following conditions:

1. (1)

$f|_{k}\gamma=f$* for all $\gamma\in\Gamma_{0}(N)$,* 2. (2)

$\Delta_{k}f=0$, 3. (3)

a linear exponential growth condition in terms of $y$ at every cusp of $\Gamma_{0}(N)$.

We denote by $H_{k}(\Gamma_{0}(N))$ the space of harmonic weak Maass forms of weight $k$ on $\Gamma_{0}(N)$.

Assume that $t$ is a cusp of $\Gamma_{0}(N)$. Let $\sigma_{t}\in\mathrm{SL}_{2}(\mathbb{Z})$ be a matrix such that $\sigma_{t}(i\infty)=t$, and $\Gamma_{0}(N)_{t}$ denote the stabilizer of cusp $t$ in $\Gamma_{0}(N)$. We define a positive integer $\alpha_{t}$ by

[TABLE]

Recall that $f|_{k}\sigma_{t}$ has the Fourier expansion of the form $f|_{k}\sigma_{t}=f^{+}_{t}+f^{-}_{t}$, where

[TABLE]

where $\Gamma(x,s)$ denotes the incomplete gamma function defined as an analytic continuation of the function $\int_{x}^{\infty}t^{s-1}e^{-t}dt$. This Fourier series is called the Fourier expansion of $f$ at a cusp $t$. The function $\sum_{n\leq 0}a^{t}_{f}(n)e^{2\pi inz/\alpha_{t}}$ is called the principal part of $f$ at the cusp $t$.

For a positive integer $n$, let $T_{n}$ denote the $n$-th Hecke operator. Then, the Hecke operator $T_{n}$ commutes with the differential operator $\xi_{-k}$ in the following way

[TABLE]

for a harmonic weak Maass form $f$ of weight $-k$.

2.2. Residues of a meromorphic differential on $X_{0}(N)$

Let $\psi$ be a meromorphic differential on $X_{0}(N)$. Then, there exists a unique meromorphic modular form $g$ of weight $2$ on $\Gamma_{0}(N)$ such that $\psi=g(z)dz$. Assume that $t$ is a cusp of $\Gamma_{0}(N)$. Assume that, for each cusp $t$, $g$ has the Fourier expansion of the form

[TABLE]

where $q:=e^{2\pi iz}$ for $z\in\mathbb{H}$.

For $\tau\in\mathbb{H}\cup\{i\infty\}\cup\mathbb{Q}$, let $Q_{\tau}$ be the image of $\tau$ under the canonical map from $\mathbb{H}\cup\{i\infty\}\cup\mathbb{Q}$ to $X_{0}(N)$. Let $\mathrm{Res}_{Q_{\tau}}gdz$ denote the residue of the differential $g(z)dz$ at $Q_{\tau}$ on $X_{0}(N)$, and $\mathrm{Res}_{\tau}g$ be the residue of $g$ at $\tau$ on $\mathbb{H}$. We describe $\mathrm{Res}_{Q_{z}}gdz$ in terms of $\mathrm{Res}_{\tau}g$ as follows. Let $\mathcal{C}_{N}$ be the set of inequivalent cusps of $\Gamma_{0}(N)$. For $\tau\in\mathbb{H}$, let $e_{\tau}$ be the order of the isotropy subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ at $\tau$. Then, we have

[TABLE]

2.3. Differentials of the third kind

In this subsection, we review properties for differentials of the third kind on a complex curve. For details, we refer to [5], [10], and [15]. A differential of the third kind on $X_{0}(N)$ means a meromorphic differential on $X_{0}(N)$ such that its poles are simple and its residues are integers. Let $\phi$ be a differential of the third kind on $X_{0}(N)$ such that $\phi$ has a pole at $P_{j}$ with residue $m_{j}$ and is holomorphic elsewhere.

Then, we define a linear map $res$ for the space of differentials of the third kind on $X_{0}(N)$ to $\mathrm{Div}_{X_{0}(N)}(\mathbb{C})$ by

[TABLE]

The image of $\phi$ under the map $res$ is called the residue divisor of $\phi$. By the residue theorem, the residue divisor $res(\phi)$ has degree zero.

Conversely, if $D$ is a divisor on $X_{0}(N)$ whose degree is zero, then there is a differential $\psi_{D}$ of the third kind with $res(\psi_{D})=D$ by the Riemann-Roch theorem and the Serre duality. The differential $\psi_{D}$ is unique up to addition of a cusp form of weight $2$ on $\Gamma_{0}(N)$.

Let $D$ be a divisor of $X_{0}(N)$ with degree zero. Then, there is a unique differential $\Phi_{D}$ of the third kind such that $res(\Phi_{D})=D$ and $\Phi_{D}=\partial_{z}h$, where $h$ is a real harmonic function on $X_{0}(N)$ with some singularities. Here, the differential $\Phi_{D}$ of the third kind is called the canonical differential of the third kind associated to $D$ (for example, see Section 1 in [15] for more details). Scholl proved the following theorem by using Waldschmidt’s result on the transcendence of periods of differentials of the third kind.

Theorem 2.2**.**

[15, Theorem 1]** With the above notation, assume that $D$ is defined over a number field $F$. Then, $\Phi_{D}$ is defined over $\overline{\mathbb{Q}}$ if and only if some non-zero multiple of $D$ is a principal divisor.

For a differential $\psi$ of the third kind, we may write $\psi=2\pi if(z)dz$, where $f$ is a meromorphic modular form of weight $2$ on $\Gamma_{0}(N)$. All poles of $f$ are simple poles and lie on $Y_{0}(N)$, and their residues are integers. The residue of $\psi$ at the cusp $t$ is a constant term of the Fourier expansion of $f$ at the cusp $t$. By the $q$-expansion principle, $\psi$ is defined over a number field $F$ if and only if all Fourier coefficients of $f$ at the cusp $i\infty$ are contained in $F$. Therefore, the following theorem [5, Theorem 3.3] was followed from Theorem 2.2.

Theorem 2.3**.**

[5, Theorem 3.3]** Let $F$ be a number field. Let $D$ be a divisor of degree [math] on $X_{0}(N)$ defined over $F$. Let $\Phi_{D}$ be the canonical differential of the third kind associated to $D$ and write $\Phi_{D}=2\pi if(z)dz$. If some non-zero multiple of $D$ is a principal divisor, then all the coefficients $a(n)$ of $f$ at the cusp $i\infty$ are contained in $F$. Otherwise, there exists an integer $n$ such that $a(n)$ is transcendental.

3. Regularized Petersson inner product

Petersson introduced an inner product on the space of cusp forms, which is called the Petersson inner product. Borcherds [3] used a regularized integral to extend the Petersson inner product to the case that one of two modular forms is a weakly holomorphic modular form. In this section, we recall the definition of a regularized Petersson inner product of a cusp form and a meromorphic modular form with the same weight by following [3] and [7]. Furthermore, we prove that if $g$ is a meromorphic modular form on $\Gamma_{0}(N)$ such that $2\pi ig(z)dz$ is the canonical differential of the third kind associated to some divisor, then the regularized Petersson inner product of $g$ with every cusp form of weight $2$ on $\Gamma_{0}(N)$ is zero.

Let $g$ be a meromorphic modular form of weight $k$ on $\Gamma_{0}(N)$. Let $\mathrm{Sing}(g)$ be the set of singular points of $g$ on $\mathcal{F}_{N}$, where $\mathcal{F}_{N}$ denotes the fundamental domain for the action of $\Gamma_{0}(N)$ on $\mathbb{H}$. For a positive real number $\varepsilon$, an $\varepsilon$-disk $B_{\tau}(\varepsilon)$ at $\tau$ is defined by

[TABLE]

Let $\mathcal{F}_{N}(g,\varepsilon)$ be a punctured fundamental domain for $\Gamma_{0}(N)$ defined by

[TABLE]

Let $f$ be a cusp form of weight $k$ on $\Gamma_{0}(N)$. The regularized Petersson inner product $(f,g)_{reg}$ of $f$ and $g$ is defined by

[TABLE]

By using the Stokes’ theorem, we obtain the following theorem (for example, see [4, Proposition 3.5] or [7, Lemma 3.1]).

Lemma 3.1**.**

Suppose that $f$ is a harmonic weak Maass form in $H^{*}_{k}(\Gamma_{0}(N))$ with singularities only at cusps equivalent to $i\infty$, and that $g$ is a meromorphic modular form. Then

[TABLE]

where $a_{g}^{t}(n)$ and $a_{f}^{t}(n)$ are $n$-th Fourier coefficients of $g$ and $f$ at the cusp $t$, respectively.

By using this lemma, we prove the following theorem, which states that the regularized Petersson inner product of a meromorphic modular form associated with a canonical differential of the third kind of some divisor on $X_{0}(N)$ with every cusp form of weight $2$ on $\Gamma_{0}(N)$ is zero.

Proposition 3.2**.**

Let $g$ be a meromorphic modular form of weight $2$ on $\Gamma_{0}(N)$ associated with a canonical differential of the third kind. Then, for every cusp form $f$ of weight $2$ on $\Gamma_{0}(N)$,

[TABLE]

Proof.

Let $G$ be a harmonic function on $\mathbb{H}$ with $\log$-type singularities such that $\partial_{z}G=g$. Then, we have

[TABLE]

To apply the Stokes’ theorem, we give the description of the boundary of $\mathcal{F}_{N}(g,\epsilon)$. For a subset $D$ of $\mathbb{C}$, let $\partial D$ denote the boundary of $D$. For a positive real number $\varepsilon$, we define

[TABLE]

Assume that $\varepsilon$ is sufficiently small. If we let $\partial^{*}\mathcal{F}_{N}(g,\varepsilon)$ be the closure of the set $\partial\mathcal{F}_{N}(g,\varepsilon)-\partial\mathcal{F}_{N}$ in $\mathbb{C}$, then

[TABLE]

From (3.1), the Stokes’ theorem implies

[TABLE]

For each $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$, the absolute value $|(f|_{2}{\gamma})(z)|$ exponentially decays as $\mathrm{Im}(z)\rightarrow\infty$, since $f$ is a cusp form. Thus, if $\tau\in\mathcal{C}_{N}$, then $\lim_{\varepsilon\rightarrow 0}\int_{\gamma_{\tau}(\varepsilon)}\frac{1}{2i}f(z)\overline{G(z)}d{z}=0$.

To complete the proof, we assume that $\tau\in\mathrm{Sing}(g)$. Then,

[TABLE]

The function $G$ can be expressed around $\tau$ as

[TABLE]

where $G_{0}(z)$ is a smooth function around $\tau$. In the definition of $\log_{e}$, we use the principal branch. If $\varepsilon$ is sufficiently small, then, for any $z\in\gamma_{\tau}(\varepsilon)$, we have

[TABLE]

Thus, for sufficiently small $\varepsilon$, we obtain

[TABLE]

This implies that, for $\tau\in\mathrm{Sing}(g)$,

[TABLE]

Thus, we complete the proof. ∎

4. Proofs

In this section, we prove Theorem 1.2 and 1.3. In this section, we always assume that $N$ is a prime. Let $D$ be a divisor of $X_{0}(N)$. Let $\psi_{D}$ be a differential of the third kind associated to $D$. Let $A_{D}$ be the space of meromorphic differentials, with only simple poles, such that their poles are only at the support of $D$. If $D$ is defined over $\overline{\mathbb{Q}}$, then there is a basis of $A_{D}$ consisting of meromorphic differentials defined over $\overline{\mathbb{Q}}$. Let us note that $D$ is defined over $\overline{\mathbb{Q}}$ if and only if there exists $\psi_{D}$ defined over $\overline{\mathbb{Q}}$. Thus, for a divisor $D$ defined over $\overline{\mathbb{Q}}$, let $\psi_{D}^{alg}$ be a differential of the third kind associated to $D$ defined over $\overline{\mathbb{Q}}$. Let $\Phi_{D}$ be the canonical differential of the third kind associated to $D$ and

[TABLE]

Note that $F_{D}$ (resp. $\Phi_{D}$ and $\psi_{D}^{alg}$) can be written as $2\pi if_{F_{D}}(z)dz$ (resp. $2\pi if_{\Phi_{D}}(z)dz$ and $2\pi if_{\psi_{D}^{alg}}(z)dz$) for some meromorphic modular forms $f_{F_{D}},f_{\Phi_{D}}$, and $f_{\psi_{D}^{alg}}$ of weight $2$ on $\Gamma_{0}(N)$.

Let us consider special divisors $D_{\tau}$, where

[TABLE]

for $\tau\in\mathbb{H}$. Let us note that $Q_{i\infty}$ is defined over $\bar{\mathbb{Q}}$. A modular curve $Y_{0}(N)$ is defined by an equation $\Phi_{N}(X,Y)=0$ such that $\Phi_{N}(X,Y)\in\mathbb{Q}[X,Y]$ and $\Phi_{N}(j(Nz),j(z))=0$ for all $z\in\mathbb{H}$. Thus, by Theorem A, $\tau\in\mathbb{H}$ is a CM point if and only if $Q_{\tau}$ is defined over $\bar{\mathbb{Q}}$. Thus, there exists $\psi_{D_{\tau}}^{alg}$ for $D_{\tau}$ if and only if $\tau$ is a CM point.

With these notations, we prove the following lemma.

Lemma 4.1**.**

With the above notation, assume that $f$ is an arithmetic harmonic weak Maass form in $H^{*}_{k}(\Gamma_{0}(N))$, and that $D:=Q_{i\infty}-Q_{\tau}$ is defined over $\overline{\mathbb{Q}}$. Let $m$ be a positive integer prime to $N$. Then, the following statements are true.

1. (1)

For each $m$,

[TABLE]

is algebraic. Especially, if $f$ is a Hecke eigenform, then

[TABLE]

is algebraic. Here, $\lambda_{m}$ is the eigenvalue of $\xi_{0}(f)$ for $T_{m}$. 2. (2)

For each $m$, $f(T_{m}.Q_{\tau})$ is algebraic if and only if $(\xi_{0}(f|T_{m}),f_{F_{D}})$ is algebraic.

Proof.

(1) By the definition of $\psi_{D}^{alg}$, the constant term of $f_{\psi_{D}^{alg}}$ at each cusp is an integer. Let

[TABLE]

where $E_{2}(z)=1-24\sum_{n=1}^{\infty}\sigma_{1}(d)q^{n}$ is the Eisenstein series of weight $2$. Here, $\sigma_{1}(n)$ is a function over $\mathbb{Q}$ defined by

[TABLE]

Since $N$ is a prime, $\Gamma_{0}(N)$ has only two inequivalent cusps. Thus, there is a rational number $c_{0}$ such that the constant term of

[TABLE]

is zero at each cusp inequivalent to $i\infty$. Let us note that $E_{N}$ is orthogonal to $S_{2}(\Gamma_{0}(N))$ with respect to the Petersson inner product, and that the principal part of $f_{\psi_{D}^{alg}}$ at the cusp $i\infty$ is constant since poles of $\psi_{D}^{alg}$ are simple poles. By Lemma 3.1, we have

[TABLE]

where $a_{f|T_{m}}(n)$ (resp. $c_{\psi_{D}^{alg}}$) denotes the $n$-th Fourier coefficient of $(f|T_{m})^{+}$ (resp. $f_{\psi_{D}^{alg}}$) at the cusp $i\infty$. Note that the principal part of $f|T_{m}$ at the cusp $i\infty$ belongs to $\bar{\mathbb{Q}}[q^{-1}]$ since the principal part of $f$ at the cusp $i\infty$ does. This proves the first part. The second part follows from (2.2).

(2) Let us note that $f_{F_{D}}=f_{\Phi_{D}}-f_{\psi_{D}^{alg}}$ is a cusp form. Thus, by Proposition 3.2 we have

[TABLE]

From (1), we get the desired result. ∎

Theorem 1.2 comes from Lemma 4.1.

Proof of Theorem 1.2.

It is immediately implied by Lemma 4.1. ∎

For a harmonic weak Maass form $f$, we define

[TABLE]

where $a_{f}(n)$ is the $n$-th Fourier coefficient of $f^{+}$ at the cusp $i\infty$. For a positive integer $m$, we define the operators $U_{m}$ and $V_{m}$ for $F$, which are meromorphic functions on $\mathbb{H}$, by

[TABLE]

and

[TABLE]

If $F=\sum_{n}a_{F}(n)q^{n}$ is a meromorphic modular form of weight $k$ on $\Gamma_{0}(N)$, then

1. (1)

$F|U_{m}$ and $F|V_{m}$ are meromorphic modular forms of weight $k$ on $\Gamma_{0}(mN)$, 2. (2)

$(F|U_{m})(z)=\sum_{n}a_{F}(mn)q^{n}$ and $(F|V_{m})(z)=\sum_{n}a_{F}(n)q^{mn}$

(for example, see Section 2.4 in [12] for more details on the operators $U_{m}$ and $V_{m}$). To prove Theorem 1.3, we need the following lemmas.

Lemma 4.2**.**

Let $f\in H^{*}_{0}(\Gamma_{0}(N))$ be an arithmetic Hecke eigenform. Let $\tau$ be an algebraic number on $\mathbb{H}$. Assume that $D:=Q_{i\infty}-Q_{\tau}$. If $\tau$ is not a CM point, then there are infinitely many positive integers $m$ prime to $N\cdot\mathrm{Prin}(f)$ such that

[TABLE]

is transcendental, where $a_{f}(n)$ (resp. $c_{\Phi_{D}}(n)$) is the $n$-th Fourier coefficient of $f^{+}$ (resp. $f_{\Phi_{D}}$) at the cusp $i\infty$.

Proof.

Let $A$ be the set of positive integers $n$ such that $a_{f}(-n)\neq 0$. Note that $A$ is a finite set. Then, we define

[TABLE]

The $m$-th Fourier coefficient $a_{F}(m)$ of $F$ at the cusp $i\infty$ is equal to (4.1). Thus, we need to show that $F$ has infinitely many transcendental Fourier coefficients $a_{F}(m)$ with $(m,N\cdot\mathrm{Prin}(f))=1$.

Let $B=\{p_{1},\ldots,p_{k}\}$ be the set of primes $p$ such that $p|N\cdot\mathrm{Prin}(f)$. By using the operators $U_{n}$ and $V_{n}$, we will remove Fourier coefficients $a_{F}(m)$ with $(m,N\cdot\mathrm{Prin}(f))\neq 1$. We define $F_{0}:=F$ and $F_{i+1}:=F_{i}-F_{i}|U_{p_{i+1}}|V_{p_{i+1}}$ for $0\leq i\leq k-1$. Then, it is enough to show that $F_{k}$ has infinitely many transcendental Fourier coefficients at the cusp $i\infty$.

By the definition of $f_{\Phi_{D}}$, we see that $f_{\Phi_{D}}$ has a singularity at the non-CM point $\tau$ in $\mathbb{H}$. We will prove that $F_{k}$ also has a singularity at a non-CM point in $\mathbb{H}$. We define

[TABLE]

for $n\in A,0\leq i\leq n-1,0\leq\mu_{1}\leq p_{1}-1,\ldots,0\leq\mu_{k}\leq p_{k}-1$. Note that

[TABLE]

for some nonzero constants $\alpha_{n,i,\mu_{1},\ldots,\mu_{k}}$. We fix $n_{0}\in A$. Let

[TABLE]

Then, $\beta$ is not a CM point since $\tau$ is not a CM point, and $\beta$ is a singular point of the function $f_{n_{0},0,1,\ldots,1}$.

To prove that $F_{k}$ has a singularity at $\beta$, it is enough to show that $\beta$ is not a singular point of $f_{n,i,\mu_{1},\ldots,\mu_{k}}$ if $(n,i,\mu_{1},\ldots,\mu_{k})\neq(n_{0},0,1,\ldots,1)$. Note that the set of singular points of $f_{n,i,\mu_{1},\ldots,\mu_{k}}$ is a subset of

[TABLE]

Suppose that $\beta\in T_{n,i,\mu_{1},\ldots,\mu_{k}}$. Then, for some $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(N)$, we have

[TABLE]

This implies that

[TABLE]

If $c\neq 0$, then $\tau$ satisfies a quadratic equation; this is not possible since $\tau$ is not a CM point. Thus, $c=0$. Then, we may assume that $a=d=1$. From this, we have

[TABLE]

Since $\tau$ is not a rational number, we see that $n=n_{0}$. Then, we obtain

[TABLE]

This holds only if $\mu_{1}=\cdots=\mu_{k}=1$ and $b=i=0$. Thus, $\beta$ is a singular point only for $f_{n_{0},0,1,\ldots,1}$.

Let $p$ be a prime. In a similar argument, we see that $F_{k}|U_{p}$ also has a singularity at a non-CM point for every prime $p$. By the Siegel-Schneider theorem, $F_{k}|U_{p}$ is not defined over $\overline{\mathbb{Q}}$. Then, the $q$-expansion principle implies there is a Fourier coefficient of $F_{k}|U_{p}$ at the cusp $i\infty$ which is transcendental. Therefore, $F_{k}$ has infinitely many transcendental Fourier coefficients. ∎

Lemma 4.3**.**

Let $f\in H^{*}_{0}(\Gamma_{0}(N))$ be an arithmetic Hecke eigenform. Assume that $(g,\xi_{0}(f))\neq 0$ for every Hecke eigenform $g\in S_{2}(\Gamma_{0}(N))$. Let $D$ be a divisor of $X_{0}(N)$ defined over $\overline{\mathbb{Q}}$. Assume that $N$ is a prime, and $\Phi_{D}$ is not defined over $\overline{\mathbb{Q}}$. Then, there exist infinitely many positive integers $m$ prime to $N$ such that $f(T_{m}.D)$ are transcendental.

Proof.

Let $\{f_{1},\ldots,f_{k}\}$ be the set of all normalized Hecke eigenforms in $S_{2}(\Gamma_{0}(N))$. Assume that $f_{i}$ has a Fourier expansion of the form

[TABLE]

Then, we have

[TABLE]

where $a_{f}(n)$ is the $n$-th Fourier coefficient of $f^{+}$ at the cusp $i\infty$. Thus, $(\xi_{0}(f),f_{i})$ is algebraic

Note that $f_{F_{D}}$ is a cusp form in $S_{2}(\Gamma_{0}(N))$. Assume that $f_{F_{D}}=\sum_{i=1}^{k}\beta_{i}f_{i}$ for some $\beta_{i}$, and that $\xi_{0}(f)=\sum_{i=1}^{k}\alpha_{i}f_{i}$ for some $\alpha_{i}$. By the assumption, $\alpha_{i}\neq 0$ for all $i$. Let $m$ be a positive integer prime to $N$. Then, we have

[TABLE]

where $\lambda_{i,m}$ is the eigenvalue of $f_{i}$ for $T_{m}$. From this, we obtain

[TABLE]

We define

[TABLE]

for $i=1,\ldots,k$.

Since $\Phi_{D}$ is not defined over $\overline{\mathbb{Q}}$, we see that $F_{D}$ is not defined over $\overline{\mathbb{Q}}$. This implies that at least one of $\beta_{i}$ is transcendental. Note that $\alpha_{i}(f_{i},f_{i})$ is a non-zero algebraic number for all $i$ since $\alpha_{i}(f_{i},f_{i})=(\xi_{0}(f),f_{i})$ and $\alpha_{i}\neq 0$ for all $i$. Thus, at least one of $\tilde{\beta}_{i}$ is transcendental. As a vector space over $\overline{\mathbb{Q}}$, let $W$ be the subspace of $\mathbb{C}$ generated by $\tilde{\beta}_{1},\ldots,\tilde{\beta}_{k}$. Let $\{w_{1},\ldots,w_{\ell}\}$ be a basis of $W$, where $w_{1}=1$. We may assume that

[TABLE]

for $i=1,\ldots,k$ and for some $\beta_{ij}\in\overline{\mathbb{Q}}$. From this, we have

[TABLE]

By Lemma 4.1 (2) and (4.3), $f(T_{m}.D)$ is algebraic if and only if

[TABLE]

for every $j\geq 2$.

Suppose that $f(T_{m}.D)$ is algebraic for all positive integers $m$ prime to $N$. Since at least one of $\tilde{\beta}_{i}$ is transcendental, at least one of $\beta_{i,j}$ with $j\geq 2$ is non-zero by (4.2). From this, there is a positive integer $j_{0}\geq 2$ such that $(\beta_{1,j_{0}},\ldots,\beta_{\ell,j_{0}})\neq(0,\ldots,0)$. We define a cusp form $g$ in $S_{2}(\Gamma_{0}(N))$ by

[TABLE]

Note that by (4.4), $a_{g}(m)=0$ for all positive integers $m$ prime to $N$, where $a_{g}(m)$ denotes the $m$-th Fourier coefficient of $g$. This implies that

[TABLE]

since $N$ is a prime. By Lemma 16 in [1], $g|U_{N}$ is a cusp form in $S_{2}(\Gamma_{0}(1))$. Since $S_{2}(\Gamma_{0}(1))=\{0\}$, this implies that $g|U_{N}=0$. From this, we have $g=0$ since $a_{g}(m)\neq 0$ only when $N\mid m$. This is a contradiction due to the fact that $\{f_{1},\ldots,f_{k}\}$ is a basis of $S_{2}(\Gamma_{0}(N))$ and $(\beta_{1,j_{0}},\ldots,\beta_{\ell,j_{0}})\neq(0,\ldots,0)$. Thus, there is a positive integer $m_{0}$ prime to $N$ such that $f(T_{m_{0}}.D)$ is transcendental.

By (4.4), there exists $j_{0}\geq 2$ such that $\sum_{i=1}^{k}\beta_{i,j_{0}}\lambda_{i,m_{0}}\neq 0$. If we define $g$ by (4.5), then $a_{g}(m_{0})\neq 0$. Thus, the function

[TABLE]

is a non-zero cusp form of weight $2$. Therefore, there exist infinitely many positive integer $m$ prime to $N$ such that $a_{g}(m)\neq 0$. With (4.4), this completes the proof. ∎

Now we prove Theorem 1.3.

Proof of Theorem 1.3.

First, we prove that if $\tau$ is not a CM point or $nD$ is not rational on $X_{0}(N)$ for any positive integer $n$, then $f(T_{m}.Q_{\tau})$ is not algebraic for some positive integer $m$ prime to $N$. Suppose that $\tau$ is not a CM point. Then, by Lemma 4.2, there exists a positive integer $m$ prime to $N\cdot\mathrm{Prin}(f)$ such that (4.1) is transcendental. Since we have

[TABLE]

for positive integers $m$ prime to $N\cdot\mathrm{Prin}(f)$ by Lemma 3.1 and Proposition 3.2, we see that $f(T_{m}.Q_{\tau})$ is transcendental for some positive integer $m$ prime to $N$.

Suppose that $\tau$ is a CM point and that $nD$ is not rational on $X_{0}(N)$ for any positive integer $n$. This implies that $D$ is defined over $\overline{\mathbb{Q}}$ and that $D$ is not a principal divisor. By Theorem 2.2, $\Phi_{D}$ is not defined over $\overline{\mathbb{Q}}$. Therefore, there exists a positive integer $m$ prime to $N$ such that $f(T_{m}.Q_{\tau})$ is not algebraic by Lemma 4.3.

Conversely, suppose that $\tau$ is a CM point and $nD$ is rational on $X_{0}(N)$ for some positive integer $n$. Then, $D$ is defined over $\overline{\mathbb{Q}}$ and $D$ is a principal divisor. By Theorem 2.2, $\Phi_{D}$ is defined over $\overline{\mathbb{Q}}$. Thus, all the Fourier coefficients of $f_{F_{D}}$ at the cusp $i\infty$ is algebraic. Let $m$ be a positive integer prime to $N$. Then, by Lemma 3.1, we have

[TABLE]

where $a_{f|T_{m}}(n)$ (resp. $c_{F_{D}}(n))$ is the $n$-th Fourier coefficient of $(f|T_{m})^{+}$ (resp. $f_{F_{D}}$) at the cusp $i\infty$. This implies that $(\xi_{0}(f|T_{m}),f_{F_{D}})$ is algebraic. Therefore, by Lemma 4.1 (2), $f(T_{m}.Q_{\tau})$ is algebraic. ∎