---
abstract: 'We extend a result of Ahlgren and Ono [@ao] on congruences for traces of singular moduli of level $1$ to traces defined in terms of Hauptmodul associated to certain groups of genus 0 of higher levels.'
address: 'Department of Mathematics $\&$ Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6'
author:
- Robert Osburn
title: Congruences for Traces of Singular moduli
---

Introduction
============

Let $j(z)$ denote the usual elliptic modular function on $\operatorname{SL}_2(\mathbb Z)$ with $q$-expansion ($q:=e^{2\pi i
z}$)

$j(z)=q^{-1}+744+196884q+21493760q^2+\cdots.$

The values of $j(z)$ at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli. Singular moduli are important algebraic integers which generate ring class field extensions of imaginary quadratic fields (Theorem 11.1 in [@cox]), are related to supersingular elliptic curves ([@ao]), and to Borcherds products of modular forms ([@b1], [@b2]).

Let $d$ denote a positive integer congruent to 0 or 3 modulo 4 so that $-d$ is the discriminant of an order in an imaginary quadratic field. Denote by $\mathcal{Q}_d$ the set of positive definite integral binary quadratic forms $$Q(x,y)=ax^2+bxy+cy^2$$ with discriminant $-d=b^2-4ac$. To each $Q \in \mathcal{Q}_d$, let $\alpha_Q$ be the unique complex number in the upper half plane which is a root of $Q(x,1)$; the singular modulus $j(\alpha_Q)$ depends only on the equivalence class of $Q$ under the action of $\Gamma:= PSL_2(\mathbb Z)$. Define $\omega_Q\in \{1, 2, 3\}$ by $$\omega_Q:
=\left \{ \begin{array}{l}
2 \quad \mbox{if $Q \sim_{\Gamma} [a,0,a]$},\\
3 \quad  \mbox{if $Q \sim_{\Gamma} [a,a,a]$},\\
1 \quad \mbox{otherwise.}
\end{array}
\right. \\$$ Let $J(z)$ be the Hauptmodul $$J(z):= j(z)-744=q^{-1}+196884q+21493760q^2+\cdots.$$ Zagier [@z] defined the trace of the singular moduli of discriminant $-d$ as $$t(d):=\sum_{Q\in \mathcal{Q}_d/\Gamma}
   \frac{J(\alpha_Q)}{\omega_Q}=\sum_{Q\in \mathcal{Q}_d/\Gamma}
   \frac{j(\alpha_Q)-744}{\omega_Q} \in \mathbb{Z}.$$

Zagier has shown that $t(d)$ has some interesting properties. Namely, the following result (see Theorem 1 in [@z]) shows that the $t(d)$’s are Fourier coefficients of a half-integral weight modular form.

Let $\theta_1(z)$ and $E_4(z)$ be defined by $$\begin{aligned}
&E_4(z):=1+240\sum_{n=1}^{\infty}\frac{n^3q^n}{1-q^n},\\
&\theta_1(z):=
\frac{\eta^2(z)}{\eta(2z)}
=\sum_{n=-\infty}^{\infty}(-1)^nq^{n^2}=1-2q+2q^4-2q^9+\cdots.\\
\end{aligned}$$ and let $g(z)$ be defined by $$\begin{aligned}
g(z): &=-q^{-1}+2+\sum_{0<
d\equiv 0, 3\pmod 4}t(d)q^d \\
\end{aligned}$$

Then

$$\begin{aligned}
g(z) &=-\frac{\theta_1(z)E_4(4z)}{\eta^6(4z)} \\
&=-q^{-1}+2-248q^3+492q^4-4119q^7\cdots \\
\end{aligned}$$

i.e., $g(z)$ is a modular form of weight $\frac{3}{2}$ on $\Gamma_{0}(4)$, holomorphic on the upper half plane and meromorphic at the cusps.

Now what about divisibility properties of $t(d)$ as $d$ varies? In this direction, Ahlgren and Ono [@ao] recently proved the following result which shows that these traces $t(d)$ satisfy congruences based on the factorization of primes in certain imaginary quadratic fields.

If $d$ is a positive integer for which an odd prime $l$ splits in $\mathbb Q(\sqrt{-d})$, then $$t(l^2d)\equiv 0\pmod l.$$

Recently, Kim [@kim2] and Zagier [@z] defined an analogous trace of singular moduli by replacing the $j$-function by a modular function of higher level, in particular by the Hauptmodul associated to other groups of genus 0. Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, i.e., $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, there are only finitely many prime values of $N$. For such a prime $p$, let $j_{p}^{*}$ be the corresponding Hauptmodul. For these primes $p$, Kim and Zagier define a trace $t^{(p)}(d)$ (see Section 3 below) in terms of singular values of $j_{p}^{*}$. The goal of this paper is to prove that the same type of congruence holds for $t^{(p)}(d)$, namely

Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0. If $d$ is a positive integer such that $-d$ is congruent to a square modulo $4p$ and for which an odd prime $l \neq p$ splits in $\mathbb
Q(\sqrt{-d})$, then $$t^{(p)}(l^2d)\equiv 0\bmod l.$$

Preliminaries on Modular and Jacobi forms
=========================================

We first recall some facts about half-integral weight modular forms (see [@kob], [@koh]). If $f(z)$ is a function of the upper half-plane, $\lambda \in \frac{1}{2}\mathbb Z$, and $\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in GL_{2}^{+}(\mathbb R)$, then we define the slash operator by

$\displaystyle f(z)|_{\lambda}\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) :=
(ad-bc)^{\frac{\lambda}{2}}(cz+d)^{-\lambda}f\Big(\frac{az+b}{cz+d}\Big)$

Here we take the branch of the square root having non-negative real part. If $\gamma=\left(\begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \Gamma_0(4)$, then define

$\displaystyle j(\gamma, z):=\Big(\frac{c}{d}\Big)\epsilon_d^{-1}\sqrt{cz+d}$,

where

$$\epsilon:=
\left \{ \begin{array}{l}
1 \quad \mbox{if $d \equiv 1 \pmod 4$},\\
i \quad  \mbox{if $d \equiv -1 \pmod 4$}.
\end{array}
\right. \\$$

If $k$ is an integer and $N$ is an odd positive integer, then let $\mathcal{M}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ denote the infinite dimensional vector space of nearly holomorphic modular forms of weight $k+\frac{1}{2}$ on $\Gamma_0(4N)$. These are functions $f(z)$ which are holomorphic on the upper half-plane, meromorphic at the cusps, and which satisfy

$$f({\gamma}z)=j(\gamma, z)^{2k+1}f(z)$$

for all $\gamma \in \Gamma_0(4N)$. Denote by $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$ the “Kohnen plus-spaces” (see [@koh]) of nearly holomorphic forms which transform according to (1) and which have a Fourier expansion of the form

$\displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n$.

We recall some properties of Hecke operators on $\mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$. If $l$ is a prime such that $l \nmid N$, then the Hecke operator $T_{k+\frac{1}{2}, 4N}(l^2)$ on a modular form

$f(z):= \displaystyle \sum_{(-1)^{k}n \equiv 0,1 \pmod 4} a(n)q^n
\in \mathcal{M}^{+}_{k+{\frac{1}{2}}}(\Gamma_0(4N))$

is given by

$f(z)|T_{k+\frac{1}{2}, 4N}(l^2):= \displaystyle \sum_{(-1)^{k}n
\equiv 0,1 \pmod 4} \Bigl ( a(l^{2}n) +
\Biggl(\frac{(-1)^{k}n}{l}\Biggr)l^{k-1}a(n) + l^{2k-1}a(n/l^2)
\Bigr ) q^n$

where $\Bigl(\frac{*}{l}\Bigr)$ is a Legendre symbol. Let us now recall some facts about Jacobi forms (see [@ez]). A Jacobi form on $\operatorname{SL}_2(\mathbb Z)$ is a holomorphic function

$\phi: \frak{H} \times \mathbb C \to \mathbb C$

satisfying

$\displaystyle \phi\Big(\frac{a{\tau}+b}{c{\tau}+d}, \frac{z}{c{\tau} + d}\Big)=
(c{\tau}+d)^{k} e^{2{\pi}iN\frac{cz^2}{c{\tau}+ d}} \phi(\tau, z)$

$\displaystyle \phi({\tau}, z+{\lambda}{\tau}+ {\mu})=
e^{-2{\pi}iN({\lambda}^2{\tau}+2{\lambda}z)} \phi(\tau, z)$

for all $\left( \begin{matrix} a & b \\
c & d \\ \end{matrix} \right) \in \operatorname{SL}_2(\mathbb Z)$ and $(\lambda,\mu) \in {\mathbb Z}^2$, and having a Fourier expansion of the form ($q=e^{2{\pi}i{\tau}}$, $\zeta=e^{2{\pi}iz}$)

$\phi(\tau, z) = \displaystyle \sum_{n=0}^{\infty}
\sum_{\substack{r\in \mathbb Z \\ r^2 \leq 4Nn}}
c(n,r)q^{n}{\zeta}^{r}$.

Here $k$ and $N$ are the weight and index of $\phi$, respectively. Let $J_{k,N}$ denote the space of Jacobi forms of weight $k$ and index $N$ on $\operatorname{SL}_2(\mathbb Z)$. By Theorem 2.2 in [@ez], the coefficient $c(n,r)$ depends only on $4Nn - r^2$ and $r \bmod 2N$. By definition $c(n,r)=0$ unless $4Nn
- r^2 \geq 0$. If we drop the condition $4Nn - r^2 \geq 0$, we obtain a nearly holomorphic Jacobi form. Let $J^{!}_{k,N}$ be the space of nearly holomorphic Jacobi forms of weight $k$ and index $N$.

Traces
======

Let $\Gamma_0(N)^{*}$ be the group generated by $\Gamma_0(N)$ and all Atkin-Lehner involutions $W_e$ for $e||N$, that is, $e$ is a positive divisor of $N$ for which gcd$(e, N/e)=1$. $W_e$ can be represented by a matrix of the form $\frac{1}{\sqrt{e}}\left( \begin{matrix} ex & y \\
Nz & ew \\ \end{matrix} \right)$ with $x$, $y$, $z$, $w \in
\mathbb Z$ and $xwe-yzN/e=1$. There are only finitely many values of $N$ for which $\Gamma_0(N)^{*}$ is of genus 0 (see [@fricke1], [@fricke2], or [@ogg]). In particular, if we let $\frak{S}$ denote the set of prime values for such $N$, then

$\frak{S}=\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71
\}.$

For $p \in \mathfrak{S}$, let $j_{p}^{*}$ be the corresponding Hauptmodul with Fourier expansion

$q^{-1} + 0 + a_{1}q + a_{2}q^2 + \dots$

Let us now define a trace in terms of the $j_{p}^{*}$’s. Let $d$ be a positive integer such that $-d$ is congruent to a square modulo $4p$. Choose an integer $\beta \bmod 2p$ such that ${\beta}^2 \equiv -d \bmod 4p$ and consider the set

$\mathcal{Q}_{d, p, \beta}= \{[a,b,c] \in \mathcal{Q}_d : a \equiv
0 \bmod p$, $b \equiv \beta \bmod 2p \}$.

Note that $\Gamma_0(p)$ acts on $\mathcal{Q}_{d, p, \beta}$. Assume that $d$ is not divisible as a discriminant by the square of any prime dividing $p$, i.e. not divisible by $p^2$. Then we have a bijection via the natural map between

$\mathcal{Q}_{d, p, \beta} \diagup \Gamma_0(p)$

and

$\mathcal{Q}_d \diagup \Gamma$

as the image of the root $\alpha_{Q}$, $Q \in
\mathcal{Q}_{d, p, \beta}$, in $\Gamma_0(p) \diagup \mathfrak{H}$ corresponds to a Heegner point. We could then define a trace $t^{(p, \beta)}(d)$ as the sum of the values of $j_{p}^{*}$ with $Q$ running over a set of representatives for $\mathcal{Q}_{d, p,
\beta} \diagup \Gamma_0(p)$. As $t^{(p, \beta)}(d)$ is independent of $\beta$, we define the trace $t^{(p)}(d)$ (see Section 8 of [@z] or Section 1 of [@kim2])

$t^{(p)}(d) = \displaystyle \sum_{Q}
\frac{j_{p}^{*}(\alpha_{Q})}{\omega_{Q}} \in \mathbb{Z}$

where the sum is over $\Gamma_0(p)^{*}$ representatives of forms $Q=[a,b,c]$ satisfying $a \equiv 0 \bmod p$.

For $p=2$, we have $t^{(2)}(4)=\frac{1}{2}
j_{2}^{*}(\frac{1+i}{2})=-52$, $t^{(2)}(7)=
j_{2}^{*}(\frac{1+\sqrt{-7}}{4})=-23$, $t^{(2)}(8)=
j_{2}^{*}(\frac{\sqrt{-2}}{2})=152$. For $p=3$, we have $t^{(3)}(3)=\frac{1}{3} j_{3}^{*}(\frac{-3+\sqrt{-3}}{6})=-14$, $t^{(3)}(11)= j_{3}^{*}(\frac{1+\sqrt{-11}}{6})=22$. Moreover by the table in Section 8 of [@z], we have:

[c|ccc]{}\
$d$ & $t^{(2)}(d)$ & $t^{(3)}(d)$ & $t^{(5)}(d)$\
3 & & $-14$ &\
4 & $-52$ & & $-8$\
7 & $-23$ & &\
8 & 152 & $-34$ &\
11 & & 22 & $-12$\
12 & $-496$ & 52 &\
15 & $-1$ & $-138$ & $-38$\
16 & 1036 & & $-6$\
19 & & & 20\
20 & $-2256$ & $-116$ & 12\
23 & $-94$ & 115 &\
24 & 4400 & 348 & $-44$\
27 & & $-482$ &\
28 & $-8192$ & &\

The empty entries correspond to $-d$ which are not congruent to squares modulo $4p$.

By the discussion in Section 8 of [@z] or Section 2.2 in [@kim2], there exist forms $\phi_{p} \in J^{!}_{2,p}$ uniquely characterized by the condition that their Fourier coefficients $c(n,r)=B(4pn-r^2)$ depend only on $r^2-4pn$ and where $B(-1)=1$, $B(d)=0$ if $d=4pn-r^2<0$, $\neq -1$ and $B(0)=-2$. Define $g_p(z)$ as

$g_p(z):= q^{-1} + \displaystyle \sum_{d \geq 0} B(d)q^d$.

By the correspondence between Jacobi forms and half-integral weight forms (Theorem 5.6 in [@ez]), $g_p(z) \in
\mathcal{M}^{+}_{\frac{3}{2}}(\Gamma_0(4p))$. As the dimension of $J_{2,p}$ is zero, we have that for every integer $d \geq 0$ such that $-d$ is congruent to a square modulo $4p$, there exists a unique $f_{d,p} \in \mathcal{M}^{+}_{\frac{1}{2}}(\Gamma_0(4p))$ with Fourier expansion

$f_{d,p}(z)= q^{-d} + \displaystyle \sum_{0<D\equiv 0,1 \pmod 4}
A(D,d) q^D$.

An explicit construction of $f_{d,p}$ can be found in the appendix of [@kim1] and the uniqueness of $f_{d,p}$ follows from the discussion at the end of Section 2 in [@kim1]. The following result relates the Fourier coefficients $A(1,d)$ and $B(d)$ and shows that the traces $t^{(p)}(d)$ are Fourier coefficients of a nearly holomorphic Jacobi form of weight 2 and index $p$ (see Theorem 8 in [@z] or Lemma 3.5 and Corollary 3.6 in [@kim2]).

Let $p$ be a prime for which $\Gamma_0(p)^{*}$ is of genus 0.\
(i) Let $d=4pn-r^2$ for some integers $n$ and $r$. Let $A(1,d)$ be the coefficient of $q$ in $f_{d,p}$ and $B(d)$ be the coefficient of ${q^n}{\zeta^{r}}$ in $\phi_{p}$. Then $A(1,d)=-B(d)$.\
(ii) For each natural number $d$ which is congruent to a square modulo $4p$, let $t^{(p)}(d)$ be defined as above. We also put $t^{(p)}(-1)=-1, t^{(p)}(d)=0$ for $d<-1$. Then $t^{(p)}(d)=-B(d)$.

Proof of Theorem 1.3
====================

The proof requires the study of Hecke operators $T_{k+\frac{1}{2},
4p}(l^2)$ on the forms $g_p(z)$ and $f_{d,p}(z)$. Define integers $A_{l}(d)$ and $B_{l}(d)$ by

$A_{l}(d):=$ the coefficient of $q$ in $f_{d,p}|T_{\frac{1}{2},
4p}(l^2)$,

$B_{l}(d):=$ the coefficient of $q^{d}$ in $g_p(z)|T_{\frac{3}{2},
4p}(l^2)$.

From equation (19) of [@z], we have

$A_{l}(d)=A(1,d) + lA(l^2,d)$.

Also note that we have

$g_p(z)|T_{\frac{3}{2}, 4p}(l^2)= q^{-1} + lq^{-{l^2}} +
\displaystyle \sum_{0<d \equiv 0, 3 \pmod 4} \Bigl ( B(l^{2}d) +
\Biggl(\frac{-d}{l}\Biggr)B(d) + lB(d/l^2) \Bigr ) q^d$

and so $B_{l}(d) = B(l^{2}d) +
\Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^2)$. Now suppose $p$ is in $\frak{S}$ and $d$ is a positive integer such that $-d$ is a square modulo $4p$ and for which an odd prime $l\neq p$ splits in $\mathbb Q(\sqrt{-d})$. Then $\Bigl(\frac{-d}{l}\Bigr)=1$. By Theorem 3.2 and the above calculations, we have $$\begin{aligned}
t^{(p)}(l^{2}d)&= -B(l^{2}d) \\
&= -B_{l}(d) + \Bigl(\frac{-d}{l}\Bigr)B(d) + lB(d/l^{2}) \\
&\equiv -B_{l}(d) + B(d) \bmod l \\
&\equiv A_{l}(d) + B(d) \bmod l \\
&\equiv A(1,d) + lA(l^{2},d) + B(d) \bmod l \\
&\equiv -B(d) + B(d) \bmod l \\
&\equiv 0 \bmod l.\\
\end{aligned}$$

We now illustrate Theorem 1.3. If $p=2$ and $l=3$, then for every non-negative integer $s$, we have

$t^{(2)}(3^2(24s+23)) \equiv 0 \bmod 3$.

In particular, if we want to compute $t^{(2)}(207)$, then we are interested in $\phi_{2} \in J^{!}_{2,2}$. By Theorem 9.3 in [@ez] and the discussion preceding Table 8 in [@z], $J^{!}_{2,2}$ is the free polynomial algebra over

$\mathbb C[E_4(\tau), E_6(\tau),
{\Delta}^{-1}]\diagup({E_4(\tau)}^{3} - {E_6(\tau)}^{2})$

on two generators $a$ and $b$ where $\displaystyle \Delta=\frac{{E_4(\tau)}^{3} - {E_6(\tau)}^{2}}{1728}$. The Fourier expansions of $a$ and $b$ begin

$$\begin{aligned}
a&=(\zeta - 2 + {\zeta}^{-1}) + (-2{\zeta}^{2} + 8{\zeta} - 12 +
8{\zeta}^{-1} - 2{\zeta}^{-2})q + ({\zeta}^{3} -12{\zeta}^2 +
39{\zeta} - 56 + \cdots)q^2 \\ &+ (8{\zeta}^3 - 56{\zeta}^2 +
152{\zeta} - 208 + \cdots)q^3 + \cdots.
\end{aligned}$$

$$\begin{aligned}
b&=(\zeta + 10 + {\zeta}^-1) + (10{\zeta}^2 -64{\zeta} + 108 -
64{\zeta} + 10{\zeta}^2)q + ({\zeta}^3 + 108{\zeta}^2 - 513{\zeta}
\\ &+ 808  - \cdots)q^2  + (-64{\zeta}^3 + 808{\zeta}^2 - 2752\zeta +
4016 - \cdots)q^3 + \cdots.
\end{aligned}$$

The coefficients for $a$ and $b$ can be obtained using Table 1 or the recursion formulas on page 39 of [@ez]. The representation of $\phi_{2}$ in terms of $a$ and $b$ is:

$\displaystyle \phi_{2}=\frac{1}{12}a(E_4(\tau)b-E_6(\tau)a)$.

By Theorem 3.2, we have $t^{(2)}(207)=-B(207)$. As $8n-r^2=207$ has a solution $n=29$ and $r=5$, then $B(207)$ is the coefficient of $q^{29}{\zeta}^5$ which is $-113643$. Thus

$t^{(2)}(207)=113643 \equiv 0 \bmod 3$.

\(1) Zagier actually defined $t^{(N)}(d)$ and proved part (ii) of Theorem 3.2 for all $N$ such that $\Gamma_0(N)^{*}$ is of genus 0 (see Section 8 in [@z]). One might be able to prove part (i) of Theorem 3.2 in the case $N$ is squarefree. If so, then a congruence, similar to Theorem 1.3, should hold for $t^{(N)}(d)$, $N$ squarefree. If $N$ is not squarefree, then C. Kim has kindly pointed out part (i) of Theorem 3.2 does not hold. For example, if $N=4$ and $d=7$, one can construct $f_{7,4}$ (see the appendix in [@kim2]) and compute that

$f_{7,4}=q^{-7} -55q + 0q^{4} + 220q^{9} + \cdots.$

Thus $A(1,7)=-55$. But $B(7)=23$ (see Remark 3.1).

\(2) We should note that Theorem 1.3 is an extension of the simplest case of Theorem 1 in [@ao]. Ono and Ahlgren have also proven congruences for $t(d)$ which involve ramified or inert primes in quadratic fields. In fact, they prove that a positive proportion of primes yield congruences for $t(d)$ (see parts (2) and (3) of Theorem 1 in [@ao]). It would be interesting to see if such congruences hold for $t^{(p)}(d)$ or $t^{(N)}(d)$.

\(3) The Monster $\mathbb{M}$ is the largest of the sporadic simple groups of order

$2^{46}3^{20}5^{9}7^{6}11^{2}13^{3}17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71$

Ogg [@ogg] noticed that the primes dividing the order of $\mathbb{M}$ are exactly those in the set $\frak{S}$. The monster $\mathbb{M}$ acts on a graded vector algebra $V=V_{-1}
\bigoplus_{n\geq 1} V_n$ (see Frenkel, Lepowsky, and Meurman [@flm] for the construction). For any element $g \in \mathbb{M}$, let $Tr(g|V_n)$ denote the trace of $g$ acting on $V_n$ for each $n$. Then $Tr(g|V_{-1})=1$ and $Tr(g|V_n) \in \mathbb Z$ for every $n \geq 1$. The Thompson series is defined by:

$T_{g}(z)= q^{-1} + \displaystyle \sum_{n \geq 1} Tr(g|V_n)q^n$.

The authors in [@cy] study Thompson series evaluated at imaginary quadratic arguments, i.e. “singular moduli” of Thompson series. It is possible to define a trace of singular moduli of Thompson series. A natural question is “do we have congruences for these traces?”

Acknowledgments {#acknowledgments .unnumbered}
===============

The author would like to thank Imin Chen, Chang Heon Kim, Ken Ono, and Noriko Yui for their valuable comments.

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