Eichler–Selberg relations for singular moduli

Abstract

The Eichler–Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz–Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(\tau )=1$. More generally, we consider the singular moduli for the Hecke system of modular functions

For each $\nu \geq 0$ and $m\geq 1$, we obtain an Eichler–Selberg relation. For $\nu =0$ and $m\in \{1, 2\},$ these relations are Kaneko’s celebrated singular moduli formulas for the coefficients of $j(\tau ).$ For each $\nu \geq 1$ and $m\geq 1,$ we obtain a new Eichler–Selberg trace formula for the Hecke action on the space of weight $2 \nu +2$ cusp forms, where the traces of $j_m(\tau )$ singular moduli replace Hurwitz–Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution L-functions.

1 Introduction and statement of results

Let $j(\tau )$ be the usual modular function for $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ with Fourier expansion

$$ \begin{align*}j(\tau)=q^{-1}+744+196884q+21493760q^2+\cdots, \end{align*} $$

where Its values at imaginary quadratic arguments in the upper-half of the complex plane are examples of singular moduli [34]. They are algebraic integers that generate Hilbert class fields of imaginary quadratic fields, in addition to serving as isomorphism class invariants of elliptic curves with complex multiplication. Well-known examples of these values include

$$ \begin{align*}j\kern1pt\left(\frac{1+\sqrt{-3}}{2}\right)=0,\ \ j(i)=1728, \ \ {\mathrm{and}}\ \ j\kern1pt\left(\frac{1+\sqrt{-15}}{2}\right)=\frac{-191025-85995\sqrt{5}}{2}. \end{align*} $$

We consider the sequence of modular functions that satisfy

$$ \begin{align*}j_m(\tau) = q^{-m} + O(q). \end{align*} $$

Each $j_m(\tau )$ is a monic degree m polynomial in $\mathbb {Z}[j(\tau )],$ and the set $\{j_m(\tau ) : m \ge 0\}$ is a basis of $M^!_0$ , the space of weakly holomorphic modular functions on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ . The first examples are $j_0(\tau ) = 1$ and

$$ \begin{align*} j_1(\tau) &= j(\tau) - 744 = q^{-1} + 196884q + \cdots,\\ j_2(\tau) &= j(\tau)^2 - 1488j(\tau) + 159768 = q^{-2} + 42987520q + \cdots,\\ j_3(\tau) &= j(\tau)^3 - 2232 j(\tau)^2 + 1069956 j(\tau) - 36866976 = q^{-3} + 2592899910q + \cdots. \end{align*} $$

In terms of the Hecke operators $T_m$ (see [28, Ch. VII] and [34]), for positive integers m, we have

(1.1) $$ \begin{align} j_m(\tau)=q^{-m}+\sum_{n=1}^{\infty}c_m(n)q^n= mT_m\left(j(\tau)-744\right). \end{align} $$

We shall derive infinitely many relations for the singular moduli of these functions. To make this precise, for positive integers d with $-d\equiv 0,1\ \pmod 4$ , we let $\mathcal {Q}_d$ be the set of integral positive definite binary quadratic forms with discriminant $-d=B^2-4AC$ . The group acts on $\mathcal {Q}_d$ by

and does so with finitely many orbits, the number of which is the discriminant $-d$ class number. For each $Q \in \mathcal {Q}_d$ , we let $\alpha _Q \in \mathbb {H}$ be a root of $Q(\tau ,1) = 0.$ The numbers $j_m(\alpha _Q)$ are its singular moduli.

We study the weighted traces of these values which are defined as follows. If we let $\Gamma _Q$ be the stabilizer of Q in $\Gamma $ , then it is well known that

$$\begin{align*}\#\Gamma_Q = \begin{cases} 3 &\text{if } Q \sim a(X^2 + XY + Y^2),\\ 2 &\text{if } Q \sim a(X^2 + Y^2),\\ 1 &\text{if otherwise}. \end{cases} \end{align*}$$

Following Zagier [34], the trace functions we consider are

(1.2)

For $m = 0$ , where $j_0(\tau )=1,$ we obtain the Hurwitz–Kronecker class numbers These numbers are prominent in the Eichler–Selberg trace formula (for example, see [32]) for the trace $\mathrm {Tr}(n;2k)$ of the action of the Hecke operators $T_n$ on $S_{2k},$ the complex vector space of weight $2k$ cusp forms on $\operatorname {\mathrm {SL}}_2(\mathbb {Z}).$

Theorem (The Eichler–Selberg trace formula).

For integers $k \ge 2$ , we have

(1.3) $$ \begin{align} \mathrm{Tr}(n; 2k) = -\frac{1}{2} \sum_{r \in \mathbb{Z}} p_{2k} (r,n) \mathbf{t}_0(4n-r^2) - \lambda_{2k-1}(n), \end{align} $$

where and

(1.4) $$ \begin{align} p_k(r,n) = \sum_{0 \le j \le \frac{k}{2}-1} (-1)^j \binom{k-2-j}{j} n^j r^{k-2-2j} = \mathrm{Coeff}_{X^{k-2}} \left(\frac{1}{1-rX+nX^2}\right). \end{align} $$

We generalize these formulas to traces of singular moduli, where (1.3) are the $m=0$ cases of a doubly infinite suite of formulas in $m\geq 0$ and $\nu \geq 0.$ The general formulas involve the trace functions $\mathbf {t}_m(4n-r^2)$ . To make this precise, for every $\nu \geq 0$ and $m\geq 0,$ we define the generating function

(1.5)

where for $d\le 0$ , we let

(1.6)

By (1.3), each $\mathrm {Tr}(n;2k)$ is essentially the nth coefficient of $\mathcal {G}_{0,k-1}(\tau ).$ Therefore, we refer to any explicit formula for $\mathcal {G}_{m,\nu }(\tau )$ as an Eichler–Selberg relation for m and $\nu .$

Our first result establishes that these generating functions are weakly holomorphic modular forms, meromorphic modular forms whose poles (if any) are supported at cusps. For convenience, we let $M^!_k$ denote the space of such weight k forms on $\operatorname {\mathrm {SL}}_2(\mathbb {Z}).$

Theorem 1.1. If $\nu \geq 0$ and $m\geq 1,$ then we have that $\mathcal {G}_{m,\nu }(\tau )\in M^!_{2\nu +2}.$

The $\nu =0$ Eichler–Selberg relations only involve derivatives of the $j_m(\tau )$ , as they generate $M^!_2$ due to the absence of holomorphic modular forms. For convenience, we let .

Theorem 1.2. For positive integers $m,$ the following are true.

1) We have

$$\begin{align*}\mathcal{G}_{m,0}(\tau) = -\frac{1}{2}\sum_{\kappa \mid m} \sum_{0 < r < \kappa} \frac{\kappa}{r(\kappa-r)} \cdot Dj_{r(\kappa-r)}(\tau). \end{align*}$$

2) If n is a positive integer, then we have

$$\begin{align*}\sum_{r \in \mathbb{Z}} \mathbf{t}_m(4n-r^2) = n \sum_{\kappa \mid m} \sum_{0 < r < \kappa} \frac{\kappa}{r(\kappa-r)} c_{r(\kappa-r)}(n). \end{align*}$$

Example. Theorem 1.2, with $m\in \{1, 2\}$ , gives Kaneko’s identities [16]

$$ \begin{align*} \sum_{r\in \mathbb{Z}} \mathbf{t}_1(4n-r^2)=0\ \ \ \ {\mathrm{and}}\ \ \ \ \sum_{r\in \mathbb{Z}} \mathbf{t}_2(4n-r^2)=2nc_1(n), \end{align*} $$

which he used to derive his well-known singular moduli formula for the coefficients of $j(\tau )$

$$ \begin{align*} c_1(n)=\frac{1}{n}\left \{ \sum_{r \in \mathbb{Z}} \mathbf{t}_1(n-r^2)+\sum_{r\geq 1\ odd}\left( (-1)^n \mathbf{t}_1(4n-r^2)-\mathbf{t}_1(16n-r^2)\right)\right\}. \end{align*} $$

Such formulas have been extended to higher levels N in subsequent works [18, 19, 25]. Finally, as a different kind of generalization, Theorem 1.2 (2) shows how to express the coefficients of each $j_m(\tau )$ in terms of traces of singular moduli.

For $\nu> 0$ , there are holomorphic modular forms, and so the relations have richer structure. To make this precise, we recall the weight $2k$ modular Poincaré series [4, Ch. 6.3]

(1.7)

where $|_{2k}$ is the slash operator, $\Gamma =\operatorname {\mathrm {PSL}}_2(\mathbb {Z})$ , and $\Gamma _{\infty }$ is the stabilizer for the cusp infinity. The usual Eisenstein series is $P_{2k,0}(\tau ) = E_{2k}(\tau ),$ and for negative integers $-h$ , we have the weakly holomorphic

(1.8) $$ \begin{align} P_{2k,-h}(\tau)=q^{-h}+\sum_{n=1}^{\infty}c_{2k,-h}(n)q^n. \end{align} $$

For small $\nu ,$ when there are no cusp forms, we obtain the following Eichler–Selberg relations.

Theorem 1.3. If $\nu \in \{1, 2, 3, 4, 6\},$ then for every positive integer m, the following are true.

1) We have that

$$\begin{align*}\mathcal{G}_{m,\nu}(\tau) = \sum_{\kappa \mid m} \sum_{0 < r \le \kappa} r^{2\nu+1} P_{2\nu+2, -r(\kappa-r)}(\tau). \end{align*}$$

2) If n is a positive integer, then we have

$$\begin{align*}\sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) \mathbf{t}_m(4n-r^2) = -2 \sum_{\kappa \mid m} \sum_{0 < r \le \kappa} r^{2\nu+1} c_{2\nu+2, -r(\kappa-r)}(n). \end{align*}$$

Remark. The Poincaré series in Theorem 1.3 are easily described in terms of the Eisenstein series

$$ \begin{align*}E_4(\tau)=1+240\sum_{n=1}^{\infty}\sigma_3(n)q^n \ \ \ {\mathrm{and}}\ \ \ E_6(\tau)=1-504\sum_{n=1}^{\infty} \sigma_5(n)q^n. \end{align*} $$

For $k\in \{4, 6, 8, 10, 14\},$ we have

$$ \begin{align*}P_{k,-1}(\tau) = \begin{cases} E_4(\tau)\cdot (j(\tau)-984) &\mathrm{if}\ k=4,\\ E_6(\tau)\cdot (j(\tau)-240) &\mathrm{if}\ k=6,\\ E_4^2(\tau)\cdot (j(\tau)-1224) &\mathrm{if}\ k=8,\\ E_4(\tau)E_6(\tau)\cdot (j(\tau)-480) &\mathrm{if}\ k=10,\\ E_4^2(\tau)E_6(\tau)\cdot (j(\tau)-720) &\mathrm{if}\ k=14. \end{cases} \end{align*} $$

Generalizing (1.1), for $m>1,$ we have the Hecke formula

$$ \begin{align*}P_{k,-m}(\tau) = m^{-k+1} \cdot T_m P_{k,-1}(\tau). \end{align*} $$

Example. For positive integers $n,$ Theorem 1.3 with $\nu = 1$ and $m=1$ implies that

$$ \begin{align*}\sum_{r\in \mathbb{Z}} r^2 \mathbf{t}_1(4n-r^2)=-480\sigma_3(n). \end{align*} $$

Cusp forms arise in the general case. Special values of symmetrized shifted convolution L-functions, and Petersson norms control these cusp forms in these Eichler–Selberg relations. Throughout, we let $d_{2k}$ denote the dimension of $S_{2k}$ , the space of weight $2k$ cusp forms on $\operatorname {\mathrm {SL}}_2(\mathbb {Z}).$

Theorem 1.4. If $\nu \geq 1$ and $m \ge 1$ , then we have

$$ \begin{align*} \mathcal{G}_{m,\nu}(\tau)= &\sum_{\kappa \mid m} \sum_{0 < r \le \kappa} r^{2\nu+1} P_{2\nu+2, -r(\kappa-r)}(\tau) -\sum_{j=1}^{d_{2\nu+2}} \left(24\sigma_1(m) - \frac{\Gamma(2\nu+1)}{(4\pi)^{2\nu+1}} \frac{\widehat{L}(f_j, m; 2\nu+1)}{\|f_j\|^2} \right) f_j, \end{align*} $$

where the $f_j$ ’s are normalized Hecke eigenforms of $S_{2\nu +2}$ and

Example. Example 1 of [22] gives $\widehat {L}(\Delta , 1; 11)=-33.383\dots $ and $\widehat {L}(\Delta , 2; 11)=266.439\dots ,$ which arise in Theorem 1.4 when $\nu = 5$ and $m\in \{1, 2\}.$ By brute force computation, we have

$$ \begin{align*} \mathcal{G}_{1,5}(\tau) &= E_{12}(\tau) -\frac{82104}{691} \Delta(\tau),\\ \mathcal{G}_{2,5}(\tau) &= P_{12,-1}(\tau) + 2049 E_{12}(\tau) - \left(\alpha - \frac{1746612}{691} \right) \Delta(\tau), \end{align*} $$

where

$$ \begin{align*} P_{12,-1}(\tau) &= \Delta(\tau) (j_2(\tau) + 24 j_1(\tau) + 324 + \alpha) = q^{-1} + \alpha q + \cdots, \end{align*} $$

with $\alpha = 1842.894\ldots $ . Using $\|\Delta \|^2 = \langle \Delta , \Delta \rangle = 0.0000010353\ldots ,$ these numerics illustrate Theorem 1.4

$$ \begin{align*} \frac{82104}{691} &= 24 + \frac{65520}{691} = 24 - \frac{\Gamma(11)}{(4\pi)^{11}} \frac{(-33.383\ldots)}{\|\Delta\|^2},\\ \alpha - \frac{1746612}{691} &= 24 \cdot 3 - \frac{\Gamma(11)}{(4\pi)^{11}} \frac{(266.439\ldots)}{\|\Delta\|^2}. \end{align*} $$

Theorem 1.4 gives a doubly infinite family of modified Eichler–Selberg trace formulas, where Hecke eigenvalues are weighted by shifted convolution L-values and where traces of singular moduli $\mathbf {t}_m(4n-r^2)$ replace the Hurwitz–Kronecker class numbers $\mathbf {t}_0(4n-r^2)=H(4n-r^2).$ To make this precise, we let

(1.9)

where, as above, $c_{f_j}(n)$ is the eigenvalue of $T_n$ for the Hecke eigenform $f_j\in S_{2k}.$

Corollary 1.5. If $2k \in 2\mathbb {Z}^+\setminus \{2, 4, 6, 8, 10, 14\}$ and m is a positive integer, then we have

$$ \begin{align*}\mathrm{Tr}(n; 2k) =\frac{1}{ 24 \sigma_1(m)}\cdot\left( \mathrm{Tr}_m(n; 2k) +\frac{1}{2} \sum_{r \in \mathbb{Z}} p_{2k}(r,n) \mathbf{t}_m(4n-r^2) + \sum_{\kappa \mid m} \sum_{0 < r \le \kappa} r^{2k-1} c_{2k, -r(\kappa-r)}(n)\right). \end{align*} $$

To obtain these results, we adapt Zagier’s novel (unpublished) proof [32] of the Eichler–Selberg trace formula. In Section 2, we recall his proof and his work on traces of singular moduli, and we prove Theorems 1.1–1.3. The proof of Theorem 1.4 is more involved, as we make use of the theory of vector-valued Poincaré series, the arithmetic of half-integral weight Kloosterman sums, Rankin–Cohen bracket operators and symmetrized shifted convolution L-functions. In Section 3, we recall important formalities regarding vector-valued modular forms that transform according to the Weil representation. In Section 4, we relate the Fourier coefficients of half-integral weight Maass–Poincaré series to traces of singular moduli, and finally, in Section 5, we assemble these facts to prove Theorem 1.4.

2 Zagier’s work and the proofs of Theorems 1.1–1.3

In unpublished notes [32], Zagier gave a novel proof of the Eichler–Selberg trace formula using harmonic Maass forms (see [7] or [4] for background on harmonic Maass forms). Saad and the third author [26] obtained further such formulas by modifying his argument. We adapt his argument in a different aspect.

2.1 Zagier’s Proof

We begin by sketching his proof, which relies on the following theorem.

Theorem (Zagier [33]).

We have that

is a harmonic Maass form of weight $3/2$ on $\Gamma _0(4)$ , where and $\Gamma (s; x)$ is the incomplete Gamma function. Its holomorphic part is the Fourier series

Zagier uses a sequence of modular forms he constructs from $\mathcal {H}(\tau )$ and Jacobi’s weight 1/2 theta function

(2.1)

To define these modular forms, he requires Atkin’s U-operator defined by

(2.2)

and the Rankin–Cohen bracket differential operators. For modular forms f and g (possibly non-holomorphic), with weights k and l, respectively, these operators are defined by

(2.3)

where . These functions are weight $2\nu +k+l$ (possibly non-holomorphic) modular forms, which one can project to obtain a holomorphic modular form via an integral map $\pi _{\mathrm {hol}}.$

Zagier studies the resulting sequence of modular forms $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ , where $\nu \geq 1$ . He computes them in two ways. The first method is combinatorial, and it uses the identity (for example, see [20, 21])

$$\begin{align*}\pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu|U_4) = [\mathcal{H}^+, \theta]_\nu|U_4 + 2 \binom{2\nu}{\nu} \sum_{n=1}^\infty \lambda_{2\nu+1}(n) q^n. \end{align*}$$

A straightforward brute force calculation with (2.3) gives

(2.4) $$ \begin{align} [\mathcal{H}^+, \theta]_\nu |U_4 = \binom{2\nu}{\nu} \sum_{n=0}^\infty \left(\sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) H(4n-r^2) \right) q^n. \end{align} $$

Therefore, the nth coefficient of $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is

(2.5) $$ \begin{align} \binom{2\nu}{\nu} \left(\sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) H(4n-r^2) + 2\lambda_{2\nu+1}(n) \right). \end{align} $$

As an alternate calculation, Zagier combines (for example, see [13, Theorem 5.5]) the Rankin–Cohen bracket operators with Hecke–Petersson theory. As each $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is a cusp form, we have

$$ \begin{align*} \pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu |U_4) = \sum_{j=1}^{d_{2\nu+2}} a_j f_j, \end{align*} $$

where the $f_j$ ’s form a basis of Hecke eigenforms for $S_{2\nu +2}$ . In particular, we have $T_n f_j = c_{f_j}(n) f_j,$ where

$$ \begin{align*}f_j(\tau)=q+\sum_{n\geq 2} c_{f_j}(n)q^n.\end{align*} $$

To compute the $a_j,$ he expresses $\mathcal {H}(\tau )$ in terms of Eisenstein series (see [26, Section 2.2] or [14, Ch. 2]), which allows him to use the method of unfolding and the Rankin–Selberg method to derive the Petersson inner product identity (for example, see [4, Ch. 6.3])

$$ \begin{align*} a_j \langle f_j, f_j \rangle = \langle \pi_{\mathrm{hol}}([\mathcal{H}, \theta]_\nu |U_4), f_j \rangle = -2 \binom{2\nu}{\nu} \langle f_j, f_j \rangle. \end{align*} $$

For each j, this gives $a_j=-2\binom {2\nu }{\nu }.$ Therefore, the nth coefficient of $\pi _{\mathrm {hol}}([\mathcal {H}, \theta ]_\nu |U_4)$ is $-2 \binom {2\nu }{\nu }\cdot \mathrm {Tr}(n; 2\nu +2),$ which when equated with (2.5) gives the Eichler–Selberg trace formula.

2.2 Proofs of Theorems 1.1–1.3

Zagier’s proof begins with the fact that $\mathcal {H}^+(\tau )$ is the holomorphic part of a weight 3/2 harmonic Maass form. In 2002, Zagier [34] greatly generalized this fact.

Theorem 5 of [34].

For positive integers m, we have that

(2.6)

is a weakly holomorphic modular form of weight $3/2$ on $\Gamma _0(4)$ .

Proof of Theorem 1.1.

Emulating Zagier’s proof of the Eichler–Selberg trace formula, we replace $\mathcal {H}^+(\tau )$ in (2.4) with the $g_m(\tau )$ . Namely, we define

By the combinatorial calculation that gave (2.4), we obtain the earlier definition (1.5)

$$ \begin{align*} \mathcal{G}_{m,\nu}(\tau) = -\frac{1}{2} \sum_{n \gg -\infty} \sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) \mathbf{t}_m(4n-r^2) q^n. \end{align*} $$

Furthermore, the theory of Rankin–Cohen brackets in this setting (see [13, Theorem 5.5]) implies that $\mathcal {G}_{m,\nu }(\tau )$ is a weakly holomorphic modular form in $M_{2\nu +2}^!.$

Proof of Theorem 1.2.

The space of weight 2 holomorphic modular forms is $M_2 = \{0\}$ and

$$ \begin{align*}Dj_{-n}(\tau) = n q^n + O(q) \in M_2^!. \end{align*} $$

Therefore, we have

$$ \begin{align*} \mathcal{G}_{m,0}(\tau) +\frac{1}{2} \sum_{-\frac{m^2}{4} \le n < 0} \frac{1}{n} \left(\sum_{r \in \mathbb{Z}} \mathbf{t}_m(4n-r^2)\right) Dj_{-n}(\tau) = 0. \end{align*} $$

The first claim follows from (1.6). By comparing the nth coefficients, the second claim is obtained.

Proof of Theorem 1.3.

For $\nu> 0$ , we note that

(2.7) $$ \begin{align} \mathcal{G}_{m,\nu}(\tau) + \frac{1}{2} \sum_{-\frac{m^2}{4} \le n \le 0} \sum_{r \in \mathbb{Z}} p_{2\nu+2}(r,n) \mathbf{t}_m(4n-r^2) P_{2\nu+2, n}(\tau) \end{align} $$

is a cusp form. We are merely cancelling the poles at infinity with Poincaré series that satisfy (1.8), and we capture the constant term with Eisenstein series $P_{2\nu +2,0}(\tau )=E_{2\nu +2}(\tau )=1+\cdots .$ For $\nu \in \{1, 2, 3, 4, 6\}$ , the space of cusp forms $S_{2\nu +2}=\{0\}$ is trivial. Therefore, the theorem follows from the identity

(2.8) $$ \begin{align} p_{2\nu+2} \left(r, \frac{r^2-\kappa^2}{4}\right) = \frac{(\kappa - r)^{2\nu+1} + (\kappa + r)^{2\nu+1}}{2^{2\nu+1} \kappa}. \end{align} $$

3 Vector-valued modular forms

The proof of Theorem 1.4 is much more involved than the proofs of Theorems 1.2 and 1.3. Nevertheless, its proof is still based on Theorem 1.1, and the aim is to understand the Fourier expansion of $\mathcal {G}_{m,\nu }(\tau )$ arithmetically in terms of traces of Hecke operators and shifted convolution L-functions. These calculations shall depend on the arithmetic of half-integral weight vector-valued modular forms that transform with respect to the Weil representation. To this end, here we recall essential preliminaries.

3.1 The Weil representation

Let $\mathcal {O}(\mathbb {H})$ be the set of all holomorphic functions $\phi : \mathbb {H} \to \mathbb {C}$ . For $z \in \mathbb {C} \setminus \{0\}$ , we take the principal branch of $z^{1/2}$ as $\arg (z^{1/2}) \in (-\pi /2, \pi /2]$ . For an integer $k \in \mathbb {Z}$ , we put $z^{k/2} = (z^{1/2})^k$ . For $n \in \mathbb {Z}_{\ge 0}$ , we put , and .

The metaplectic group $\operatorname {\mathrm {Mp}}_2(\mathbb {R})$ is a group defined by

where the group operation is

As usual, we have , and for any $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {R})$ , we define $j(\gamma , \tau ) = c\tau +d$ and $\widetilde {\gamma } = \left (\bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ), j(\gamma , \tau )^{1/2} \right ) \in \operatorname {\mathrm {Mp}}_2(\mathbb {R})$ . Let $\operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ be the inverse image of $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ under the projection $\operatorname {\mathrm {Mp}}_2(\mathbb {R}) \to \operatorname {\mathrm {SL}}_2(\mathbb {R})$ . As usual, we let and It is well known that $\operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ is generated by $\widetilde {T}$ and $\widetilde {S}$ , (see [6, p.16]) and its center is generated by

$$\begin{align*}\widetilde{-I} = \widetilde{S}^2 = (\widetilde{S} \widetilde{T})^3 = \left(\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}, i \right). \end{align*}$$

Moreover, we let , representing the metaplectic stabilizer for the cusp at infinity.

We recall the Weil representation,¹ the unitary representation $\rho : \operatorname {\mathrm {Mp}}_2(\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {C})$ defined by

(3.1)

We note that $\rho (\widetilde {-I}) = \rho (\widetilde {S}^2) = -i I$ . We let $\rho ^* : \operatorname {\mathrm {Mp}}_2(\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {C})$ be the dual representation of $\rho $

We recall an explicit formula for $\rho (\widetilde {\gamma })$ , which is easily derived from work of both Shintani [29, Proposition 1.6] and Bruinier [6, Proposition 1.1], where for odd integers d, we let

(3.2)

Proposition 3.1. For $c \ge 0$ , we have

$$ \begin{align*} \rho(\widetilde{\gamma}) = \begin{cases} \displaystyle{\frac{\epsilon_c}{1+i} \left(\frac{a}{c}\right) \begin{pmatrix}1 & i^{cd} \\ i^{ac} & -i^{(a+d)c}\end{pmatrix}} &\text{if } c \equiv 1\quad \pmod{2},\\ \displaystyle{\epsilon_a^{-1} \left(\frac{c}{a}\right) \begin{pmatrix}0 & i^{ab} \\ 1 & 0\end{pmatrix}} &\text{if } c \equiv 2\quad \pmod{4},\\ \displaystyle{\epsilon_a^{-1} \left(\frac{c}{a}\right) \begin{pmatrix}1 & 0 \\ 0 & i^{ab}\end{pmatrix}} &\text{if } c \equiv 0\quad \pmod{4}. \end{cases} \end{align*} $$

We now give the definition of a vector-valued modular form that transforms under the Weil representation. If $k \in \frac {1}{2} \mathbb {Z}$ and $f: \mathbb {H} \to \mathbb {C}^2$ . For $(\gamma , \phi (\tau )) \in \operatorname {\mathrm {Mp}}_2(\mathbb {Z}),$ then we define the slash operator

We say that $f: \mathbb {H} \to \mathbb {C}^2$ is a weight k (vector-valued) modular form with respect to $\rho $ if

$$\begin{align*}f|_{k,\rho} (\gamma, \phi) = f \end{align*}$$

for every $(\gamma , \phi ) \in \operatorname {\mathrm {Mp}}_2(\mathbb {Z})$ . We define them for $\rho ^*$ in a similar manner.

3.2 Jacobi’s theta functions

For later use, we recall the Jacobi theta functions (for example, see [13, Section 5]) in this context. If we set , where we have

(3.3)

and The specialization $\Theta (\tau , 0)$ is a weight 1/2 vector-valued modular form with respect to $\rho $ , and in general is a (vector-valued) Jacobi form, which for $(\gamma , \phi )\in \operatorname {\mathrm {Mp}}_2(\mathbb {Z}),$ in this case, means that

(3.4)

4 Maass–Poincaré series and traces of singular moduli

The proof of Theorem 1.4 relies on Maass–Poincaré series that transform with respect to the Weil representation. We construct these series following [6], and we relate them to traces of singular moduli. The goal of this section, Theorem 4.5, can be immediately derived as a special case of Alfes’ result [1, Theorem 4.3], which applies the Kudla–Millson theta lift ([7]) to integer weight Poincaré series. However, we will also provide a direct proof that requires minimal advanced prior knowledge below.

4.1 The Whittaker functions

Let $M_{\mu , \nu }(z)$ and $W_{\mu , \nu }(z)$ be the Whittaker functions (for example, see [30, Ch. 16] and [17, 24]). The next two lemmas are crucial for constructing Maass–Poincaré series.

Lemma 4.1 [17, 7.2.1], [24, 13.15.19].

For positive integers $n,$ we have

$$\begin{align*}\frac{\mathrm{d}^n}{\mathrm{d} z^n} \left(e^{-z/2} z^{-\nu-1/2} M_{\mu, \nu}(z) \right) = (-1)^n \frac{(\mu + \nu + 1/2)^{\overline{n}}}{(2\nu+1)^{\overline{n}}} e^{-z/2} z^{-\nu - n/2-1/2} M_{\mu+n/2, \nu+n/2}(z). \end{align*}$$

Lemma 4.2 [17, 7.5.1], [24, 13.23.1].

For $\operatorname {\mathrm {Re}}(\nu + \alpha + 1/2)> 0$ and $2\operatorname {\mathrm {Re}}(z)> \beta > 0$ , we have

$$ \begin{align*} \int_0^\infty e^{-zt} t^{\alpha-1} M_{\mu, \nu}(\beta t) \mathrm{d} t &= \frac{\beta^{\nu+1/2} \Gamma\left(\nu+\alpha+\frac{1}{2}\right)}{\left(z+\frac{\beta}{2}\right)^{\nu+\alpha+1/2}}\cdot {}_2F_1 \left(\nu-\mu+\frac{1}{2}, \nu+\alpha+\frac{1}{2}; 2\nu+1; \frac{2\beta}{\beta+2z}\right), \end{align*} $$

where ${}_2F_1(a,b; c; z)$ is the Gaussian hypergeometric function.

For $n \in \mathbb {Z}$ , $k \in \frac {1}{2} \mathbb {Z}$ , $y> 0$ , and $s \in \mathbb {C}$ , we define the modified Whittaker functions

(4.1)

(4.2)

The special values of these functions at $s = k/2$ play a crucial role in the construction of the Maass–Poincaré series. To this end, for $n < 0$ , we have

(4.3) $$ \begin{align} \mathcal{M}_{k,n} \left(y, \frac{k}{2}\right) &= \Gamma(k)^{-1} e^{-2\pi ny}. \end{align} $$

As for the $\mathcal {W}$ -function, we have

(4.4) $$ \begin{align} \mathcal{W}_{k,n} \left(y, \frac{k}{2}\right) = \begin{cases} \Gamma(k)^{-1} n^{k-1} e^{-2\pi ny} &\text{if } n> 0,\\ 0 &\text{if } n \le 0 \end{cases} \end{align} $$

(see [17, 7.2.4]). Moreover, we note that, ([17, 7.6.1], [24, 13.14]),

(4.5) $$ \begin{align} \begin{split} W_{\mu, \nu}(y) &\sim e^{-y/2} y^\mu \quad (y \to \infty),\\ M_{\mu, \nu}(y) &= y^{\nu+1/2} (1+ O(y)) \quad (y \to 0). \end{split} \end{align} $$

4.2 Kloosterman sums

The Fourier expansions of the Maass–Poincaré series require Kloosterman sums, which we recall here. For $k \in \frac {1}{2}\mathbb {Z} \setminus \mathbb {Z}$ , $m, n \in \mathbb {Z}$ , and $c>0$ with $c \equiv 0\ \pmod {4}$ , we define the half-integral weight Kloosterman sum by

(4.6)

where $\overline {d} \in \mathbb {Z}/c\mathbb {Z}$ satisfies that $d \overline {d} \equiv 1\ \pmod {c}$ . The condition $d\ (c)^*$ means that d runs over $d \in \mathbb {Z}/c\mathbb {Z}$ such that $(c,d) = 1$ . We note that the Kloosterman sums satisfy

(4.7) $$ \begin{align} K_{k+2}(m,n,c) = K_k(m,n,c) \quad \text{ and } \quad K_{3/2}(m,n,c) = -i K_{1/2}(-m,-n,c). \end{align} $$

We now relate the Weil representation to such Kloosterman sums. For notational convenience, we let

$$\begin{align*}\rho(\widetilde{\gamma}) = \begin{pmatrix}\rho(\widetilde{\gamma})_{00} & \rho(\widetilde{\gamma})_{01} \\ \rho(\widetilde{\gamma})_{10} & \rho(\widetilde{\gamma})_{11}\end{pmatrix}. \end{align*}$$

Then the following sum formula holds for each entry of $\rho (\widetilde {\gamma })$ .

Proposition 4.3. If $\alpha , \beta \in \{0,1\}$ and m and n satisfy $m \equiv -\alpha\ \pmod {4}$ and $n \equiv -\beta\ \pmod {4},$ then for every positive integer c, we have

$$ \begin{align*} \frac{1}{4} \left(1 + \left(\frac{4}{c}\right) \right) K_{3/2}(m,n,4c)= \sum_{d\ (c)^*} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e}\left(\frac{ma + nd}{4c}\right), \end{align*} $$

where we take any $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ for which $(c,d)$ forms its bottom row.

Proof. First, we check that the right-hand side is well defined. Let $R_{\alpha \beta }(\gamma )$ denote its summand. It suffices to show that $R_{\alpha \beta }(T^j \gamma T^l) = R_{\alpha \beta }(\gamma )$ holds for any $j, l \in \mathbb {Z}$ . Since $\widetilde {T^j \gamma T^l} = \widetilde {T}^j \widetilde {\gamma } \widetilde {T}^l$ holds, we have

$$\begin{align*}R_{\alpha \beta}(T^j \gamma T^l) = i^{\alpha j + \beta l} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e} \left(\frac{ma+nd}{4c}\right) \mathbf{e} \left(\frac{mj+nl}{4}\right) = R_{\alpha \beta}(\gamma). \end{align*}$$

Next, for each $\gamma = \bigl (\begin {smallmatrix}a & b \\ c & d\end {smallmatrix}\bigr ) \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ with $c> 0$ , we prove the refined equation

(4.8) $$ \begin{align} \rho(\widetilde{\gamma})_{\alpha \beta} = \frac{1}{4} \left(1 + \left(\frac{4}{c}\right) \right) \sum_{\substack{\delta\ (4c) \\ \delta \equiv d\ (c) \\ \delta \equiv 1\ (2)}} \left(\frac{c}{\delta}\right) \epsilon_{\delta}^{-1} \mathbf{e} \left(\frac{a - \overline{\delta}}{4c}\right)^\alpha \mathbf{e} \left(\frac{d-\delta}{4c}\right)^\beta, \end{align} $$

where $\overline {\delta }$ is the inverse of $\delta $ in $(\mathbb {Z}/4c\mathbb {Z})^\times $ . This immediately implies the proposition.

To confirm (4.8), let and ( $j = 0,1,2,3$ ). For simplicity, let $\rho "(\gamma )_{\alpha \beta }$ denote the right-hand side of (4.8) and show that $\rho "(\gamma )_{\alpha \beta } = \rho (\widetilde {\gamma })_{\alpha \beta }$ . If $\delta $ is odd, then we can easily check that

$$\begin{align*}\overline{\delta} = \begin{cases} a(1+b_j c) &\text{if } c \equiv 1\quad \pmod{2} \text{ and } a \equiv 1\quad \pmod{2},\\ (a+c)(1+(b_j+d_j)c) &\text{if } c \equiv 1\quad \pmod{2} \text{ and } a \equiv 0\quad \pmod{2},\\ a(1-ab_j cd_j) &\text{if } c \equiv 2\quad \pmod{4},\\ a(1-b_jc) &\text{if } c \equiv 0\quad \pmod{4}. \end{cases} \end{align*}$$

We prove the case where $c \equiv 1\ \pmod {2}$ and $a \equiv 1\ \pmod {2}$ , leaving the others to the reader. We have

$$ \begin{align*} \rho"(\gamma)_{\alpha \beta} &= \frac{1}{2} \sum_{\substack{0 \le j \le 3 \\ d_j \equiv 1\ (2)}} \left(\frac{c}{d_j}\right) \epsilon_{d_j}^{-1} \mathbf{e} \left(\frac{-ab_j}{4}\right)^\alpha \mathbf{e} \left(\frac{-j}{4}\right)^\beta = \frac{i^{-ab \alpha}}{2} \left(\frac{d}{c}\right) \sum_{\substack{0 \le j \le 3 \\ d_j \equiv 1\ (2)}} (-1)^{\frac{(c-1)(d_j-1)}{4}} \epsilon_{d_j}^{-1} i^{-j(\alpha+\beta)}. \end{align*} $$

Since the value of the sum depends only on $c, d\ \pmod {4}$ , a direct calculation yields

$$\begin{align*}\rho"(\gamma)_{\alpha \beta} = \frac{i^{-ab \alpha}}{2} \left(\frac{d}{c}\right) \frac{2\epsilon_c}{1+i} \times \begin{cases} 1 &\text{if } (\alpha, \beta) = (0,0),\\ i^{cd} &\text{if } (\alpha, \beta) = (0,1), (1,0),\\ (-1)^{d-1} &\text{if } (\alpha, \beta) = (1,1). \end{cases} \end{align*}$$

Combining simple calculations with Proposition 3.1, one obtains $\rho (\widetilde {\gamma })_{\alpha \beta }$ .

4.3 The Maass–Poincaré series

Using the two previous subsections, we now construct the Maass–Poincaré series. We let and . Assume that $k \in \frac {1}{2} \mathbb {Z}$ satisfies $2k \equiv 3\ \pmod {4}$ . For $\alpha \in \{0, 1\}$ and $m \equiv -\alpha\ \pmod {4}$ , we define the Maass–Poincaré series of weight k with respect to $\rho ^*$ by

(4.9)

This series converges absolutely and uniformly on compact subsets in $\operatorname {\mathrm {Re}}(s)> 1$ [6, p.29], and we note that is invariant under $|_{k,\rho ^*} (\gamma , \phi )$ for any $(\gamma , \phi ) \in \widetilde {\Gamma }_\infty $ as $2k \equiv 3\ \pmod {4}$ .

The Fourier expansions of the functions involve the Bessel functions (see [17, Ch. 3] and [30, Ch. 17])

Proposition 4.4. For $\operatorname {\mathrm {Re}}(s)> 1$ , we have

where

$$ \begin{align*} b_{m,k}^{(\beta)}(n, s) &= 2\pi i^{-k} \sum_{c> 0} \left(1+\left(\frac{4}{c}\right)\right) \frac{K_{3/2}(m,n,4c)}{4c}\\ &\qquad \qquad \times \begin{cases} \displaystyle{|mn|^{\frac{1-k}{2}} J_{2s-1} \left(\frac{\pi \sqrt{|mn|}}{c}\right)} &\text{if } mn > 0,\\ \displaystyle{|mn|^{\frac{1-k}{2}} I_{2s-1} \left(\frac{\pi \sqrt{|mn|}}{c}\right)} &\text{if } mn < 0,\\ \displaystyle{2^{k-1} \pi^{s+k/2-1} |m+n|^{s-k/2} (4c)^{1-2s}} &\text{if } mn=0, m+n \neq 0,\\ \displaystyle{2^{2k-2} \pi^{k-1} \Gamma(2s) (8c)^{1-2s}} &\text{if } m = n = 0. \end{cases} \end{align*} $$

Proof. Dividing the sum of the Poincaré series into the identity class and the remaining part, we have

Let $H_{k, \rho ^*}^{(\alpha , m)}(\tau , s)$ denote the sum of the second term. By following the exact same argument as in the proof of Theorem 1.9 in Bruinier’s book [6], we obtain the Fourier expansion,

$$\begin{align*}H_{k,\rho^*}^{(\alpha, m)}(\tau,s) = \sum_{\beta \in \{0,1\}} \sum_{n \in \mathbb{Z}} \left( \sum_{c> 0} \sum_{d \in (\mathbb{Z}/c\mathbb{Z})^\times} \rho(\widetilde{\gamma})_{\alpha \beta} \mathbf{e}\left(\frac{ma+nd}{4c}\right) I_m(n) \right) \mathbf{e} \left(\frac{nu}{4}\right) \mathfrak{e}_\beta, \end{align*}$$

where $I_m(n)$ is given by

By combining this with Proposition 4.3, we obtain Proposition 4.4.

4.4 Traces of singular moduli

The coefficients of these functions are related to traces of singular moduli, as shown in several previous works (for example, see [5, 8, 12]). To make this precise, we consider weight 3/2 modular forms h on $\Gamma _0(4)$ satisfying

(4.10)

We define for $i \in \{0, 1\}$ , and then we have that

(4.11)

is a weight 3/2 vector-valued modular form with respect to $\rho ^*$ (see [13, Section 5] and [4, Ch. 2]).

We relate the $g_m(\tau )$ in (2.6) and to the Maass–Poincaré expressions

(4.12)

where $\alpha \equiv n^2\ \pmod {4}$ for each n. To be precise, we have the following theorem.

Theorem 4.5. If m is a nonnegative integer, then we have

$$\begin{align*}\lim_{s \to 3/4} G_m(\tau, s) = \begin{pmatrix}g_{m,0}(\tau) \\ g_{m,1}(\tau)\end{pmatrix}. \end{align*}$$

Remark. We note that the case of $m=0$ was stated by Williams [31, Example 5.1].

Sketch of the Proof. This result is standard, and so we sketch the proof. We first recall facts about Niebur–Poincaré series $F_m(\tau , s)$ (see [23] or [12, Section 4]), which are defined for $\operatorname {\mathrm {Re}}(s)> 1,$ and give alternative expressions for the $j_m(\tau )$ . Specifically, as described in [12, (4.10)], it is known that

$$\begin{align*}\mathop{\mathrm{Res}}_{s=1} F_0(\tau, s) = \frac{3}{\pi} \end{align*}$$

and

(4.13) $$ \begin{align} \lim_{s \to 1} F_{-m}(\tau, s) = j_m(\tau) + 24 \sigma_1(m) \quad (m> 0). \end{align} $$

For nonnegative integers m, the trace functions

have a direct connection to the coefficients of the earlier Maass–Poincaré series. Indeed, by combining the result of Duke, Imamoḡlu and Tóth in [12, Proposition 4] with our Proposition 4.4, for $\operatorname {\mathrm {Re}}(s)> 1$ , $m \ge 0$ , and $d> 0$ with $d \equiv 0, 3\ \pmod {4}$ , we obtain that

$$\begin{align*}\mathrm{Tr}_d(F_{-m}(\cdot, s)) = \begin{cases} \displaystyle{- d^{1/2} \sum_{n|m} n b_{-n^2, 3/2}^{(\beta)} \left(d, \frac{s}{2} + \frac{1}{4}\right)} &\text{if } m> 0,\\ \displaystyle{-2^{s-2} \pi^{-s/2-1} d^{1/2} \zeta(s) b_{0, 3/2}^{(\beta)} \left(d, \frac{s}{2} + \frac{1}{4}\right)} &\text{if } m = 0. \end{cases} \end{align*}$$

Therefore, (4.13) implies that

(4.14) $$ \begin{align} \mathbf{t}_m(d) = \lim_{s \to 3/4} \begin{cases} \displaystyle{- d^{1/2} \sum_{n|m} n b_{-n^2, 3/2}^{(\beta)} (d, s) + \frac{4d^{1/2}}{\sqrt{\pi}} \sigma_1(m) b_{0, 3/2}^{(\beta)}(d,s)} &\text{if } m> 0,\\ \displaystyle{- \frac{d^{1/2}}{6 \sqrt{\pi}} b_{0, 3/2}^{(\beta)} (d, s)} &\text{if } m = 0. \end{cases} \end{align} $$

By applying (4.3) and (4.4), we thereby conclude the proof of the theorem.

Remark. We note that subtle technicalities arise in the proof of Theorem 4.5, which have been addressed in the aforementioned works but deserve commentary. The $G_m(\tau , s)$ are defined for $\operatorname {\mathrm {Re}}(s)> 1$ , where they enjoy the Fourier series expansion in Proposition 4.4. As we can only be analytically continued up to $\operatorname {\mathrm {Re}}(s)> 3/4$ , care is required when letting $s \to 3/4$ . In fact, the Fourier coefficients $b_{-m^2, 3/2}^{(\beta )}(-n^2, s)$ have a simple pole at $s = 3/4$ , which cancels out with a zero from , (for example, see [12, Lemma 3]). This issue is addressed by examining the growth of the Fourier coefficients of $G_m(\tau , s)$ , including $\mathrm {Tr}_d(F_{-m}(\cdot , s))$ , as $d \to \infty $ and the behavior as $s \to 3/4$ . We refer the reader to [8, 11, 12] for these details.

5 Proof of Theorem 1.4

We have constructed the Poincaré series $G_m(\tau , s)$ whose Fourier coefficients give the traces of singular moduli. We turn to the problem of providing the Hecke decomposition of $\mathcal {G}_{m,\nu }(\tau )$ . Specifically, we compute the Petersson inner product $\langle \mathcal {G}_{m,\nu }, f \rangle $ with a normalized Hecke eigenform f of $S_{2\nu +2}$ . We first recall useful facts about Jacobi forms to relate the Rankin–Cohen brackets to these Poincaré series.

5.1 Jacobi forms and the modified heat operator

For a function $\varphi : \mathbb {H} \times \mathbb {C} \to \mathbb {C}$ , $\gamma \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ , and positive integers $k, m \in \mathbb {Z}_{>0}$ , we define the slash operator

and the weighted heat operator

where . Then, we have

(5.1) $$ \begin{align} L_{k,m} (\varphi|_{k,m} \gamma) = (L_{k,m} \varphi)|_{k+2,m} \gamma \end{align} $$

for any $\gamma \in \operatorname {\mathrm {SL}}_2(\mathbb {Z})$ (see [13, (11) in Section 3]). For simplicity, we put .

Lemma 5.1. For a Poincaré series defined by

$$\begin{align*}G(\tau) = \sum_{(\gamma, \phi) \in \widetilde{\Gamma}_\infty \backslash \operatorname{\mathrm{Mp}}_2(\mathbb{Z})} \begin{pmatrix}\psi_0(\tau) \\ \psi_1(\tau)\end{pmatrix} \bigg|_{3/2, \rho^*} (\gamma, \phi), \end{align*}$$

with test functions $\psi _0, \psi _1: \mathbb {H} \to \mathbb {C}$ , we have

$$\begin{align*}{}^t \Theta(\tau, z) G(\tau) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} (\theta_0(\tau, z) \psi_0(\tau) + \theta_1(\tau, z) \psi_1(\tau)) |_{2,1} \gamma. \end{align*}$$

Proof. By a direct calculation with (3.4), we have

$$ \begin{align*} {}^t \Theta(\tau, z) G(\tau) = \sum_{(\gamma, \phi) \in \widetilde{\Gamma}_\infty \backslash \operatorname{\mathrm{Mp}}_2(\mathbb{Z})} \phi(\tau)^{-4} \mathbf{e} \left(\frac{-cz^2}{c\tau+d}\right) {}^t \Theta\left(\gamma \tau, \frac{z}{c\tau+d}\right) {}^t \rho((\gamma, \phi))^{-1} \rho^*((\gamma, \phi))^{-1} \begin{pmatrix}\psi_0(\gamma\tau) \\ \psi_1(\gamma \tau)\end{pmatrix}. \end{align*} $$

Since ${}^t \rho ((\gamma , \phi ))^{-1} \rho ^*((\gamma , \phi ))^{-1} = I$ and $\phi (\tau )^{-4} = (c\tau +d)^{-2}$ , we obtain the result.

We require the following proposition for the $p_k(r,n)$ in the Eichler–Selberg trace formula.

Proposition 5.2. For $\nu , l \in \mathbb {Z}_{\ge 0}$ and $r \in \mathbb {Z}$ , we define the differential operator by

(5.2)

Then, for a function $: \mathbb {H} \to \mathbb {C}$ , we have the Taylor expansion

$$\begin{align*}L_{2\nu} \circ \cdots \circ L_2 f(\tau)(\zeta^r + \zeta^{-r}) = 2 \sum_{l=0}^\infty p_{2\nu+2}(r,D,l) f(\tau) \frac{(2\pi irz)^{2l}}{(2l)!}. \end{align*}$$

In particular, letting $p_k(r,n)$ as in (1.4), we have that $p_{2\nu +2}(r,D,0) = p_{2\nu +2}(r,D)$ and

$$\begin{align*}\lim_{z \to 0} L_{2\nu} \circ \cdots \circ L_2 f(\tau)(\zeta^r + \zeta^{-r}) = 2 p_{2\nu+2}(r,D) f(\tau). \end{align*}$$

Proof. We check that the Taylor coefficients of $L_{2\nu } \circ \cdots \circ L_2 f(\tau ) (\zeta ^r + \zeta ^{-r})$ and the sequence (5.2) satisfy the same recursion. The claim is clear for $\nu = 0$ . For $\nu> 0$ , let

Then $S_{\nu , l, j}$ satisfies the recursion

$$\begin{align*}S_{\nu, l, j} = -D S_{\nu-1, l, j-1} + \frac{r^2}{4} \left(1 + \frac{4\nu-1}{2l+1}\right) S_{\nu-1, l+1, j}, \end{align*}$$

for $\nu \ge 1$ and $0 \le j \le \nu $ with $S_{\nu , l, -1} = 0$ , which implies that

$$\begin{align*}p_{2\nu+2}(r,D,l) = -D p_{2\nu}(r,D,l) + \frac{r^2}{4} \left(1 + \frac{4\nu-1}{2l+1}\right) p_{2\nu} (r,D,l+1). \end{align*}$$

One can check that the Taylor coefficients also satisfy this recursion.

We use this proposition to understand the combinatorial properties of the Rankin–Cohen bracket operators, which is a slight generalization of [13, Theorem 5.5].

Proposition 5.3. Let $\nu \ge 0$ be a nonnegative integer. For a modular form h of weight $3/2$ on $\Gamma _0(4)$ of the form (4.10), we have

Proof. By definition, we have

A direct calculation implies that

and

$$ \begin{align*} \sum_{\substack{r, s \ge 0\\ r+s = \nu}} (-1)^r \frac{\Gamma(3/2+\nu) \Gamma(1/2+\nu)}{s! \Gamma(3/2+r) r! \Gamma(1/2+s)} m^{2s} (4D-m^2)^r = \binom{2\nu}{\nu} p_{2\nu+2}(m,D). \end{align*} $$

The last equation immediately follows from Proposition 5.2.

For each $n \ge 0$ and $\nu \ge 0$ , we define

(5.3)

Combining Theorem 4.5 and Lemma 5.3, for $m\geq 1,$ we obtain the following key expressions:

(5.4) $$ \begin{align} \mathcal{G}_{m,\nu}(\tau) = -\frac{1}{2\binom{2\nu}{\nu}} \cdot [g_m, \theta]_\nu |U_4 = -\frac{1}{2} \lim_{s \to 3/4} \left( -\frac{\sqrt{\pi}}{2} \sum_{n \mid m} n \Phi_{n, \nu}(\tau, s) + 2\sigma_1(m) \Phi_{0,\nu}(\tau, s)\right). \end{align} $$

The order of limits of s and z is interchanged, which is justified by the Remark at the end of Section 4.4.

5.2 The Selberg–Poincaré series

To prove Theorem 1.4 using (5.4), we must calculate $\Phi _{n,\nu }(\tau , s)$ and $\langle \Phi _{n, \nu }(\cdot , s), f\rangle $ at $s = 3/4$ for Hecke eigenforms f. To this end, we use Selberg’s generalization [27] of the Poincaré series in (1.7). For integers $k\ge 2$ and $m\in \mathbb {Z},$ they are defined by

(5.5)

This series converges absolutely and uniformly on compact subsets for $\operatorname {\mathrm {Re}}(s)> 1-k/2$ and admits meromorphic continuation. In particular, it is known that $P_{k,m}(\tau , s)$ is holomorphic at $s = 1-k/2$ . This fact follows from comparing it with the Maass–Poincaré series defined by

Indeed, from (4.5), we have

Thus, for $\operatorname {\mathrm {Re}}(s)> -k/2$ , the poles of these two types of Poincaré series agree. However, the Fourier expansion of the Maass–Poincaré series (see [15, Theorem 3.2]) and the Weil bound for the Kloosterman sums imply its holomorphy at $s = 1-k/2$ .

The next lemma describes the Petersson inner product of cusp forms with these series.

Lemma 5.4. For $f \in S_k$ and $m> 0$ , we have

Proof. It follows from the classical unfolding argument (see [4, Ch. 10.1], for instance).

5.3 The case of $n=0$

Here, we calculate $\langle \Phi _{0, \nu }(\cdot , s), f\rangle $ at $s = 3/4$ for a normalized Hecke eigenform f. To this end, we decompose $\Phi _{0,\nu }(\tau ,s)$ in terms of the Selberg–Poincaré series.

Proposition 5.5. We have that

$$\begin{align*}\Phi_{0,\nu}(\tau, s) = 4^{-s+3/4} \sum_{0 \le l \le \nu} \frac{(s-3/4)^{\underline{l}}}{(4\pi)^l} \binom{2\nu+1}{2l+1} \sum_{r \in \mathbb{Z}} r^{2\nu-2l} P_{2\nu+2, r^2} \left(\tau, s-\frac{3}{4}-l\right). \end{align*}$$

Proof. By applying (5.1), Lemma 5.1 and Proposition 5.2,

The summand is calculated as

Then the claim follows from the Leibniz rule, where , and the fact that

$$\begin{align*}\sum_{l \le j \le \nu} (-1)^j 2^{2(\nu-j)} \binom{2\nu-j}{j} \binom{j}{l} = (-1)^l \binom{2\nu+1}{2l+1}. \end{align*}$$

The next result provides a formula for the Petersson norm of a cusp form f.

Theorem 5.6. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , we have

$$\begin{align*}\lim_{s \to 3/4} \langle \Phi_{0,\nu} (\cdot, s), f \rangle = 24 \|f\|^2. \end{align*}$$

Proof. First, we note that the Fourier coefficients of a normalized Hecke eigenform are real. By Lemma 5.4 and Proposition 5.5, we find that

$$ \begin{align*} \langle \Phi_{0,\nu}(\cdot, s), f \rangle &= 4^{-s+3/4} \sum_{0 \le l \le \nu} \frac{(s-3/4)^{\underline{l}}}{(4\pi)^l} \frac{\Gamma(2\nu+s+1/4-l)}{(4\pi)^{2\nu+s+1/4-l}} \binom{2\nu+1}{2l+1} \cdot 2 \sum_{r =1}^\infty \frac{c_f(r^2)}{r^{2s+2\nu+1/2}}. \end{align*} $$

As in [10, Lemma 11.12.6], let

for $f \in S_{2\nu +2}$ . Then, it is known that $B(f,s)$ admits the meromorphic continuation to the whole $\mathbb {C}$ -plane, and $L(\mathrm {Sym}^2(f), s)$ has no poles (see [10, Remark 11.12.8]). In particular, $B(f, 2s+2\nu +1/2)$ has no pole at $s = 3/4$ . Therefore, by [10, Corollary 11.12.7], we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{0,\nu}(\cdot, s), f\rangle &= \frac{\Gamma(2\nu+1)}{(4\pi)^{2\nu+1}} (2\nu+1) 2B(f, 2\nu+2)\\ &= \frac{2(2\nu+1)!}{(4\pi)^{2\nu+1}} \frac{6}{\pi^2} \frac{\pi}{2} \frac{(4\pi)^{2\nu+2}}{(2\nu+1)!} \langle f, f \rangle\\ &=24 \|f\|^2.\\[-47pt] \end{align*} $$

5.4 The cases of $n> 0$

We turn to the case of positive $n.$ Again, we first decompose $\Phi _{n,\nu }(\tau ,s).$

Proposition 5.7. For $n> 0$ , we have

$$ \begin{align*} \Phi_{n, \nu}(\tau, s) &= \frac{1}{\Gamma(2s)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2)}} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} Q_{\nu, i_1+i_2} (n,r) \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}}\widetilde{P}_{n,r}^{i_1, i_2} (\tau, s), \end{align*} $$

where we let

Proof. Arguing as above, by applying (5.1), Lemma 5.1 and Proposition 5.2, we obtain

The summand is calculated as

Similar to the case of $n=0$ , direct calculation utilizing yields

For the second term, by Lemma 4.1, we find that

The claim follows by combining these results.

We split the sum defining $\Phi _{n, \nu }(\tau , s)$ into $\Phi _{n, \nu }^+(\tau , s)$ and $\Phi _{n, \nu }^-(\tau , s),$ based on the inequalities $r^2> n^2$ or $r^2 \le n^2.$ We consider them as $s \to 3/4$ . By (4.5), the summand of the Poincaré series $\widetilde {P}_{n,r}^{i_1, i_2} (\tau , s)$ satisfies

as . Therefore, for $\operatorname {\mathrm {Re}}(s)> -\nu +i_1 + 3/4$ , the Poincaré series is holomorphic (in s). In particular, $\widetilde {P}_{n,r}^{i_1, i_2} (\tau , s)$ is holomorphic at $s = 3/4$ for $0 \le i_1 < \nu $ . Regarding the case of $i_1 = \nu $ , by a similar argument as in Section 5.2 – that is, by comparing it with the Selberg–Poincaré series or the Maass–Poincaré series – we see that it is also holomorphic at $s=3/4$ . Therefore, we have

$$ \begin{align*} \lim_{s \to 3/4} \Phi_{n, \nu}^-(\tau, s) &= \frac{1}{\Gamma(3/2)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2)\\r^2 \le n^2}} Q_{\nu,0} (n,r) \widetilde{P}_{n,r}^{0,0}(\tau, 3/4). \end{align*} $$

Since $Q_{\nu ,0}(n,r) = p_{2\nu +2}(r, (r^2-n^2)/4)$ and $\widetilde {P}_{n,r}^{0,0}(\tau , 3/4) = P_{2\nu +2, \frac {r^2-n^2}{4}}(\tau )$ , by (2.8), we have

(5.6) $$ \begin{align} \lim_{s \to 3/4} \Phi_{n, \nu}^-(\tau, s) = \frac{4}{n \sqrt{\pi}} \sum_{0 < r \le n} r^{2\nu+1} P_{2\nu+2, -r(n-r)}(\tau). \end{align} $$

As a counterpart to Theorem 5.6, the Petersson inner product of $\Phi _{n, \nu }^+(\tau , s)$ with a Hecke eigenform is expressed in terms of the symmetrized shifted convolution L-functions.

Theorem 5.8. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f \rangle = \frac{4}{n \sqrt{\pi}} \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \widehat{L}(f, n/d; 2\nu+1). \end{align*} $$

Proof. By Proposition 5.7, we have

$$ \begin{align*} &\langle \Phi_{n, \nu}^+(\cdot, s), f \rangle\\ &= \frac{1}{\Gamma(2s)} \sum_{\substack{r \in \mathbb{Z} \\ r \equiv n\ (2) \\ r^2> n^2}} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} Q_{\nu, i_1+i_2} (n,r) \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}} \langle \widetilde{P}_{n,r}^{i_1, i_2} (\cdot, s), f \rangle. \end{align*} $$

The unfolding argument, combined with Lemma 4.2, gives

By changing variables $r = 2m+n$ for $r> n$ and $r = -2m-n$ for $r < -n$ , we have

$$ \begin{align*} &\langle \Phi_{n,\nu}^+(\cdot, s), f\rangle\\ &= \frac{2}{\Gamma(2s)} \sum_{\substack{i_1, i_2 \ge 0 \\ i_1 + i_2 \le \nu}} \frac{(-1)^{i_1}}{i_1! i_2!} \left(\frac{n^2}{4} \right)^{i_1+i_2} \frac{(s-3/4)^{\underline{i_1}} (s-3/4)^{\overline{i_2}}}{(2s)^{\overline{i_2}}} (\pi n^2)^{s -\frac{3}{4}-i_1} \Gamma \left(s + 2\nu + \frac{1}{4} - i_1 \right)\\ &\times \sum_{m=1}^\infty \frac{Q_{\nu, i_1+i_2} (n,2m+n) c_f (m(m+n))}{(\pi (2m+n)^2)^{s + 2\nu + \frac{1}{4} - i_1}} {}_2 F_1 \left(s+\frac{3}{4}, s + 2\nu + \frac{1}{4} - i_1; 2s+i_2; \frac{n^2}{(2m+n)^2}\right), \end{align*} $$

where we note that $Q_{\nu ,i}(n,-r) = Q_{\nu ,i}(n,r)$ holds. For a normalized Hecke eigenform $f \in S_{2\nu +2}$ , since

$$\begin{align*}c_f(m(m+n)) = \sum_{d \mid (m, m+n)} \mu(d) d^{2\nu+1} c_f\left(\frac{m}{d}\right) c_f\left(\frac{m+n}{d}\right), \end{align*}$$

the last sum becomes

$$ \begin{align*} &\sum_{d \mid n} \mu(d) d^{2\nu+1} \sum_{m=1}^\infty \frac{Q_{\nu, i_1+i_2} (n,2dm+n) c_f(m) c_f(m+n/d)}{(\pi (2dm+n)^2)^{s + 2\nu + \frac{1}{4} - i_1}}\\ &\quad \times {}_2 F_1 \left(s+\frac{3}{4}, s + 2\nu + \frac{1}{4} - i_1; 2s+i_2; \frac{n^2}{(2dm+n)^2}\right).\end{align*} $$

Then, this Dirichlet series is holomorphic at $s = 3/4$ . Indeed, since $Q_{\nu , i_1+i_2} (n,2dm+n)$ has degree $2(\nu -i_1-i_2)$ in m, it suffices to show that

$$\begin{align*}\sum_{m=1}^\infty \frac{c_f(m) c_f(m+n/d)}{m^{2(s+\nu+1/4+i_2)}} \end{align*}$$

(conditionally) converges at $s = 3/4$ . This can be seen by partial summation using the estimate

$$\begin{align*}\sum_{1 \le m \le x} c_f(m) c_f(m+n/d) \ll x^{2\nu+2-\delta}, \end{align*}$$

with some $\delta> 0$ , (see [2, Corollary 1.4]). Therefore, all terms corresponding to nonzero $(i_1, i_2)$ vanish as $s \to 3/4$ , and we obtain

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f\rangle &= \frac{4}{\sqrt{\pi}} \Gamma(2\nu+1) \sum_{d \mid n} \mu(d) d^{2\nu+1} \\ &\quad \times \sum_{m=1}^\infty \frac{Q_{\nu, 0} (n,2dm+n) c_f(m) c_f(m+n/d)}{(\pi (2dm+n)^2)^{2\nu + 1}} {}_2 F_1 \left(\frac{3}{2}, 2\nu + 1; \frac{3}{2}; \frac{n^2}{(2dm+n)^2}\right). \end{align*} $$

Since we have

$$\begin{align*}\frac{1}{r^{2(2\nu+1)}} {}_2 F_1 \left(\frac{3}{2}, 2\nu + 1; \frac{3}{2}; \frac{n^2}{r^2}\right) = \frac{1}{(r^2-n^2)^{2\nu+1}} \end{align*}$$

and $Q_{\nu ,0}(n,r) = p_{2\nu +2}(r, (r^2-n^2)/4)$ with (2.8) again, the proof is complete as

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi_{n,\nu}^+(\cdot, s), f\rangle &= \frac{4}{n \sqrt{\pi}} \frac{\Gamma(2\nu+1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \sum_{m=1}^\infty c_f(m) c_f(m+n/d) \left(\frac{1}{m^{2\nu+1}} - \frac{1}{(m+n/d)^{2\nu+1}}\right). \end{align*} $$

5.5 Proof of Theorem 1.4

We apply the results from the two previous subsections. For $m> 0$ and $\nu \ge 0$ , let

As stated in (5.4), we have

$$\begin{align*}\lim_{s \to 3/4} \Phi(\tau, s) = \mathcal{G}_{m,\nu}(\tau). \end{align*}$$

However, from (5.6), the minus part

For the plus part, by Theorem 5.6 and Theorem 5.8 and the Möbius inversion formula, we have

$$ \begin{align*} \lim_{s \to 3/4} \langle \Phi^+(\cdot, s), f \rangle &= \sum_{n \mid m} \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \sum_{d \mid n} \mu(d) \widehat{L}(f, n/d; 2\nu+1) - 24\sigma_1(m) \|f\|^2\\ &= \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \widehat{L}(f, m; 2\nu+1) - 24 \sigma_1(m) \|f\|^2. \end{align*} $$

Combining these facts, we are pleased to obtain the conclusion of the theorem

$$ \begin{align*} \lim_{s \to 3/4} \Phi(\tau, s) &= \sum_{n \mid m} \sum_{0 < r \le n} r^{2\nu+1} P_{2\nu+2, -r(n-r)}(\tau) - \sum_{j=1}^{d_{2\nu+2}} \left(24 \sigma_1(m) - \frac{\Gamma (2\nu + 1)}{(4\pi)^{2\nu + 1}} \frac{\widehat{L}(f, m; 2\nu+1)}{\|f_j\|^2} \right) f_j. \end{align*} $$