Traces of reciprocal singular moduli

We show that the generating series of traces of reciprocal singular moduli is a mixed mock modular form of weight $3/2$ whose shadow is given by a linear combination of products of unary and binary theta functions. To prove these results, we extend the Kudla-Millson theta lift of Bruinier and Funke to meromorphic modular functions.


Introduction and statement of results
The special values of the modular j-invariant j(z) = q −1 + 744 + 196884q + 21493760q 2 + 864299970q 3 + . . . at imaginary quadratic points in the upper half-plane are called singular moduli. By the theory of complex multiplication they are algebraic integers in the ray class fields of certain orders in imaginary quadratic fields. In particular, their traces are known to be rational integers. Here Q + D denotes the set of positive definite quadratic forms of discriminant D < 0 on which Γ = SL 2 (Z) acts with finitely many orbits. Further, Γ Q is the stabilizer of Q in Γ = PSL 2 (Z) and z Q ∈ H is the CM point associated to Q.
The starting point for the present article was the question whether the generating series of traces of reciprocal singular moduli has similar modular transformation properties. Notice that 1/j has a third order pole at ρ = e πi/3 , so the CM value of 1/j at this point is not defined. However, if we replace 1/j(ρ) by the constant term in the elliptic expansion of 1/j around ρ (see (2.11)), then tr 1/j (D) is defined for every D < 0. By the theory of complex multiplication the traces tr 1/j (D) are rational numbers, but they are usually not integers.
In order to obtain a convenient modularity statement we have to add a constant term, which is given by the regularized average value of 1/j over Γ\H, .
We refer to Section 2.5 for the definition of the regularized average value and to Corollary 4.3 for the evaluation of tr 1/j (0). To describe the shadow of the generating series of traces of reciprocal singular moduli (that is, the image of its modular completion under the lowering operator L 3/2,τ = −2iv 2 ∂ ∂τ ) we require the theta functions with τ = u+iv ∈ H and the Hermite polynomial H 3 (x) = 8x 3 −12x. They are components of vector valued theta functions associated to positive definite one-and two-dimensional even lattices and certain polynomials as explained in [Bor98]. In particular, using Theorem 4.1 in [Bor98] one can show that they transform like modular forms of weight 7/2 and of weight 4 for Γ(24), respectively. We obtain the following modularity statement.
In order to prove the theorem, we extend the Kudla-Millson theta lift of Bruinier and Funke [BF06] to modular functions which are allowed to have poles in H. For simplicity, we restrict our attention to those meromorphic modular functions which decay like cusp forms towards ∞. We let S 0 be the space of all such meromorphic cusp forms of weight 0. The theta lift of f ∈ S 0 is defined by the regularized inner product (see Section 2.4) where Θ KM (z, τ ) is the Kudla-Millson theta function (see Section 2.2). The theta function transforms like a modular form of weight 3/2 for Γ 0 (4) in τ , and thus the same holds for the theta lift Φ KM (f, τ ). The technical heart of this work is the explicit computation of the Fourier expansion of Φ KM (f, τ ) (see Theorem 3.1). We show that the "holomorphic part" of Φ KM (f, τ ) is given by the generating series of traces of CM values of f . On the other hand, the "non-holomorphic part" of Φ KM (f, τ ) can be viewed as a termwise preimage of an indefinite theta function under the lowering operator (see Corollary 3.2). In particular, we obtain an explicit formula for the image of Φ KM (f, τ ) under the lowering operator. Choosing f = 1/j and doing some simplicifications (see Section 4) then yields Theorem 1.1.
Remark 1.2. Theorem 1.1 could also be proved using results of Bringmann, Ehlen, and the second author, namely by taking the constant term in the elliptic expansion at z = ρ of the function 1 j ′ (z) A * (z, τ ) defined in Theorem 1.1 of [BES18]. However, the Kudla-Millson theta lift considered in the present work yields the modularity of traces of CM values of arbitrary meromorphic cusp forms of weight 0, of which Theorem 1.1 is just an illustrative example.
Remark 1.3. Similar theta lifts of meromorphic modular forms were studied by Bruinier, Imamoglu, Funke, and Li in [BFIL18] and by Bringmann and the authors of the present work in [ANBS18]. It was shown there that the generating series of traces of cycle integrals of meromorphic cusp forms of positive even weight can be completed to real-analytic modular forms of half-integral weight whose images under the lowering operator are given by certain indefinite theta functions. Using a vector-valued setup as in [BF06] one can generalize the results of the present work to arbitrary congruence subgroups.
Remark 1.4. In [Zag02] Zagier also showed the weight 1/2 modularity of twisted traces of J = j − 744. Using the so-called Millson theta function (compare [ANS18]) one can investigate the modularity of generating series of twisted traces of 1/j. This is the topic of an ongoing Master's thesis under the supervision of the first author.
The work is organized as follows. In Section 2 we recall the necessary facts about (singular) theta functions, regularized inner products of meromorphic and real-analytic modular forms, as well as traces of CM values and average values of meromorphic cusp forms. In Section 3 we study the Kudla-Millson theta lift Φ KM (f, τ ) of meromorphic cusp forms f ∈ S 0 . We compute its Fourier expansion and determine its image under the lowering operator. Finally, in Section 4 we show how Theorem 1.1 can be deduced using the Kudla-Millson theta lift of 1/j.

Preliminaries
2.1. Quadratic forms. We let Q D be the set of all integral binary quadratic forms Q = [a, b, c] of discriminant D = b 2 − 4ac, and for D < 0 we let Q + D be the subset of positive definite forms. The group Γ = SL 2 (Z) acts from the right on Q D and Q + D , with finitely many orbits if D = 0.
For Q = [a, b, c] ∈ Q D and z = x + iy ∈ H we define the quantities They are related by The function Θ KM (z, τ ) is real-analytic in both variables and transforms like a modular form of weight 0 in z for Γ and weight 3/2 in τ for Γ 0 (4) (see [KM86], [BF04]). Moreover, as a function of z it decays square exponentially towards the cusp ∞ (see [BF06]). We also require the theta function It is a multiple of the derivative in z of the Siegel theta function Θ S (z, τ ), which is the theta function associated to ϕ S (Q, z, v) = v exp(−4πv|Q(z, 1)| 2 /y 2 ). In particular, it transforms like a modular form of weight 2 in z for Γ and weight −1/2 in τ for Γ 0 (4) (see [Bor98]).

Singular theta functions.
For Q = 0 we consider the function It is the derivative of Kudla's Green function ξ KM (see [Kud97]), which was also used in [BF06] to compute the Fourier expansion of the Kudla-Millson theta lift of harmonic Maass forms. For D ≥ 0 the function η KM is real-analytic in z on all of H. For D < 0 it is only real-analytic for z ∈ H \ {z Q }, but then the difference extends to a real-analytic function around z Q which vanishes at z Q . This can easily be proved using the first relation in (2.1).
For z ∈ H with Q(z, 1) = 0 the function η KM is related to ϕ KM and ϕ * KM via the lowering operator L κ = −2iy 2 ∂ ∂z by which can be checked by a direct calculation using (2.1).
The "theta function" formed by summing η KM (Q, z, v)e −2πiDτ over D ∈ Z and Q ∈ Q D \ {0} would behave very badly as a function of z since it would have a singularity at every CM point, that is, on a dense subset of H. However, for a fixed ̺ ∈ H we define the "singular" theta function which is a real-analytic function of τ . By a slight abuse of notation, for m ∈ N 0 we also set for every m ∈ N 0 . By construction, R m 2,z Θ * KM (̺, τ ) can be viewed as a termwise L 3/2,τpreimage of R m 2,z Θ * KM (̺, τ ). In particular, it is no surprise that these singular theta functions do not transform like modular forms in τ . As we will see later, they appear as the "non-holomorphic parts" of real-analytic modular forms of weight 3/2. 2.4. Regularized inner products. We now describe the regularized inner product in (1.1). Let f ∈ S 0 . We denote by [̺ 1 ], . . . , [̺ r ] ∈ Γ\H the equivalence classes of the poles of f on H and we choose a fundamental domain F * for Γ\H containing ̺ 1 , . . . , ̺ r such that each ̺ ℓ lies in the interior of Γ ̺ ℓ F * . For any ̺ ∈ H and ε > 0 we let be the ε-ball around ̺. Let g : H → C be a real-analytic and Γ-invariant function, and assume that it is of moderate growth at ∞. We define the regularized Petersson inner product of f and g by It was shown in Proposition 3.2 of [ANBS18] that this regularized inner product exists under the present assumptions on f and g. In particular, the theta lift defined in (1.1) converges due to the rapid decay of the Kudla-Millson theta function at ∞.
Recall that f has an elliptic expansion of the shape around every ̺ ∈ H (see Proposition 17 in Zagier's part of [BvdGHZ08]). In order to evaluate regularized inner products as in (2.10) we will typically apply Stokes' Theorem, which yields integrals over the boundaries of the balls B ε ℓ (̺ ℓ ). To compute such boundary integrals the following formula is useful.
One can view tr f (0) as the regularized average value of f on Γ\H. It can be evaluated in terms of special values of the real-analytic weight 2 Eisenstein series as follows.
where c f,̺ (−n) are the coefficients of the elliptic expansion (2.11) of f .
Proof. The Eisenstein series satisfies L 2 E * 2 (z) = 3 π , so we can write Now using Stokes' Theorem (in the form given in Lemma 2.1 of [BKV13]) and Lemma 2.1 easily gives the stated formula.

The Kudla-Millson theta lift
Let f ∈ S 0 be a meromorphic cusp form of weight 0 and let [̺ 1 ], . . . , [̺ r ] ∈ Γ\H be the classes of poles of f mod Γ. We define the Kudla-Millson theta lift of f by Since Θ KM (z, τ ) is real-analytic in z and decays square exponentially as y goes to ∞, it follows from Proposition 3.2 of [ANBS18] that the theta lift converges for every τ ∈ H. In particular, it transforms like a modular form of weight 3/2 for Γ 0 (4). We now compute the Fourier expansion of the Kudla-Millson theta lift.
Theorem 3.1. The Kudla-Millson theta lift of f ∈ S 0 is a real-analytic modular form of weight 3/2 for Γ 0 (4) whose Fourier expansion is given by where c f,̺ (−n) are the coefficients of the elliptic expansion (2.11) of f and R n−1 2,z Θ * KM (̺, τ ) is the singular theta function defined in (2.8).
Proof. We plug in the definition of the Kudla-Millson theta function (2.3) and obtain the Fourier expansion We now compute the coefficients c(D, v) for fixed D ∈ Z and v > 0.
The coefficients of index D > 0. In this case the function η KM (Q, z, v) defined in (2.5) is real-analytic in z on all of H. We use the differential equation (2.6) and apply Stokes' Theorem (in the form given in Lemma 2.1 of [BKV13]) to obtain Here we also used that f decays like a cusp form towards ∞ and that all other boundary integrals cancel out in Γ-equivalent pairs due to the modularity of the integrand. Using the disjoint union we see that integrating over the full boundary ∂B ε ℓ (̺ ℓ ) on the right-hand side of (3.2) gives an additional factor 1/|Γ ̺ ℓ |. For D > 0 the function Q∈Q D η KM (Q, z, v) is real-analytic in z on H, so using Lemma 2.1 we find This finishes the computation in the case D > 0.
The coefficients of index D = 0. We split off the summand for Q = 0 in (3.1), which yields The remaining part with Q = 0 can be computed as in the case D > 0.
The coefficients of index D < 0. Let us first suppose that f does not have a pole at any CM point of discriminant D. Let [z 1 ], . . . , [z s ] ∈ Γ\H be the Γ-classes of CM points of discriminant D, and let Q 1 , . . . , Q s ∈ Q + D be the corresponding quadratic forms. We can assume that z 1 , . . . , z s ∈ F * and that every z ℓ lies in the interior of Γ z ℓ F * . We cut out a ball B δ ℓ (z ℓ ) around every CM point z ℓ and then apply Stokes' Theorem as in the case D > 0 to obtain (3. 3) The first line can be evaluated as in the case D > 0. In the second line, for fixed ℓ the summands with Q = ±Q ℓ are real-analytic near z ℓ , so their integrals vanish as δ ℓ → 0. Replacing −Q ℓ by Q ℓ gives a factor 2. Further, since η KM (Q, z, v)−sgn(Q z ) |D|/2πQ(z, 1) is real analytic near the CM point z Q , the second line in (3.3) is given by Note that sgn((Q ℓ ) z ) = 1 for z close to z ℓ . If we now plug in the elliptic expansion (2.11) of f and the expression (2.2) for Q(z, 1), we arrive at By the residue theorem, the last integral vanishes unless n = 0, in which case it equals π/ Im(z ℓ ). Hence the second line in (3.3) is given by which finishes the computation for D < 0 if f does not have a pole at any CM point of discriminant D.
We now indicate the changes in the computation if f has poles at some CM points of discriminant D. For notational convenience, let us assume that ̺ 1 = z 1 is a pole of f which is also a CM point. In this case we do not need to cut out an additional δ 1 -ball around z 1 before applying Stokes' Theorem since we already cut out an ε 1 -ball around ̺ 1 . In particular, the summand for z 1 in the second line of (3.3) has to be omitted, and the summand for ̺ 1 in the first line of (3.3) has to be computed as follows. We write (3.5) The factor 2 in the second line comes from the fact that Q 1 and −Q 1 have the same CM point z 1 , and the sign is gone since sgn((Q 1 ) z ) = 1 for z close to z 1 . The sum over Q ∈ Q D in the third line of (3.5) is real-analytic near ̺ 1 , hence the expression in the third line can be computed as in the case D > 0 to Proof. The j-function has an elliptic expansion at ρ of the form compare Proposition 17 in Zagier's part of [BvdGHZ08]. Using the theory of complex multiplication one can show that the values R n 0 j(ρ) are algebraic multiples of π n Ω 2n (compare Proposition 26 in Zagier's part of [BvdGHZ08]). Explicitly, we have R 4 0 j(ρ) = R 5 0 j(ρ) = 0, R 6 0 j(ρ) = −2 22 · 3 2 · 5 · 23 · π 6 Ω 12 . Note that R 4 0 j(ρ) = R 5 0 j(ρ) = 0 follows from the simple fact that every function which transforms like a modular form of weight k ≡ 0 (mod 6) for Γ vanishes at ρ. The other values can be computed, for example, by writing the almost holomorphic modular forms R 3 0 j(z)∆(z) and R 6 0 j(z)∆(z) in terms of the Eisenstein series E * 2 , E 4 and E 6 and using their values at ρ given in the table on p. 87 in [BvdGHZ08]. We also checked the above evaluations numerically.
Combining this with Lemma 2.2 and Proposition 4.1 we obtain the value of tr 1/j (0).
Computing the action of the raising operators we get We set a = 2A + B + 2C, b = −A + B + 2C, c = A + B. It is then not hard to see that with A, B, C ∈ Z the pairs (a, b, c) run through the sublattice of Z 3 defined by the conditions a ≡ b (mod 3) and b ≡ c (mod 2). Thus we obtain Splitting the sum over a into arithmetic progressions mod 3 we easily obtain the shadow given in Theorem 1.1, which finishes the proof.