Traces of Reciprocal Singular Moduli, We Require the Theta Functions

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of traces of singular moduli is a mock modular form of weight 3/2 for Γ0(4) whose shadow n2 is a multiple of the Jacobi theta function θ(τ) = n Z q . These results have been generalized in various directions, for example to generating∈ series of traces of CM values of weakly holomorphic modular functions for congruenceP subgroups by Bruinier and Funke [BF06], using the so-called Kudla-Millson theta lift. The starting point for the present article was the question whether the generating series of traces 1/j(zQ) tr1/j(D)= ΓQ Q + /Γ ∈QXD | | of reciprocal singular moduli has similar modular transformation properties. Notice that 1/j has a third order pole at ρ = eπi/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 (2.13)), then tr1/j(D) is defined for every D < 0. By the theory of complex multiplication the traces tr1/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, \ 1 reg dxdy 1 1 tr1/j(0) = 1/j(z) 2 = 11 4 = . −4π Γ H y −2 3 −165888 Z \ · We refer to Section 2.6 for the definition of the regularized average value and to Corol- lary 4.4 for the evaluation of tr1/j(0). To describe the shadow of the generating series of traces of reciprocal singular moduli, we require the theta functions

3/2 a a2/3 θ7/2,h(τ)= v− H3 2√πv q , √3 a Z a hX(mod∈ 3) ≡ 3 2 2 θ (τ)= b i√3c qb /3+c , 4,h − b,c Z b cX(mod∈ 2) b≡h (mod 3) ≡ with τ = u + iv H and the Hermite polynomial H (x)=8x3 12x. Note that θ is ∈ 3 − 7/2,h (up to a non-zero constant multiple) the image under the Maass raising operator R3/2 = ∂ 3 1 a2/3 2i ∂τ + 2 v− of the weight 3/2 unary theta function θ3/2,h = a h (mod 3) aq , and θ4,h is a binary theta function associated to a harmonic polynomial of degre≡ e 3. In particular, they transform like modular forms of weight 7/2 and of weight 4P for Γ(24), respectively. We obtain the following modularity statement. Theorem 1.1. The generating series

D 1 23 3 1 4 1 7 1 8 tr (D)q− = + q + q q + q + ... 1/j −165888 331776 3456 − 3375 8000 D 0 X≤ 3

of traces of reciprocal singular moduli converges absolutely and locally uniformly, and de- ﬁnes a mixed mock modular form of weight 3/2 and depth 2 for Γ0(4) (in the sense of [DMZ19, GMN19], compare Section 2.1). Its shadow is a non-zero multiple of

7/2 v θ7/2,h(τ)θ4,h(τ). h (modX 3) 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 S0 be the space of all such meromorphic modular forms of weight 0. The theta lift∞ of f S is defined by the regularized inner product (see Section 2.5) ∈ 0 reg (1.1) Φ (f, τ)= f, Θ ( , τ) , KM KM · D E where ΘKM(z, τ) is the Kudla-Millson theta function (see Section 2.3). The theta function transforms like a modular form of weight 3/2 for Γ0(4) in τ, and thus the same holds for the theta lift ΦKM(f, τ). The technical heart of this work is the explicit computation of the Fourier expansion of ΦKM(f, τ) (see Theorem 3.1). We show that the “holomorphic part” of ΦKM(f, τ) is given by the generating series of traces of CM values of f. On the other hand, the “non-holomorphic part” of ΦKM(f, τ) can be viewed as a termwise preimage of an indefinite theta function under the lowering operator (see Corollary 3.3). In particular, we obtain an explicit formula for the image of ΦKM(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 1 of the function j′(z) A∗(z, τ) defined in Theorem 1.1 of [BES18]. 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 [ANBS19]. It was shown there that the generating series of traces of cycle integrals of meromorphic modular 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. 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 the 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. Remark 1.5. Using a vector-valued setup as in [BF06] one can generalize the results of the present work to arbitrary congruence subgroups. For the convenience of the reader we state the Fourier expansion of the Kudla-Millson theta lift of meromorphic modular functions for congruence subgroups in Section 5. 4 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

The work is organized as follows. In Section 2 we recall the necessary facts about mixed mock modular forms, (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 modular forms. In Section 3 we study the Kudla-Millson theta lift ΦKM(f, τ) of meromorphic modular forms f S0. We compute its Fourier expansion and determine its image under the lowering operator.∈ In Section 4 we show how Theorem 1.1 can be deduced using the Kudla-Millson theta lift of 1/j. Finally, in Section 5 we state the Fourier expansion of the Kudla-Millson theta lift of meromorphic modular forms for congruence subgroups.

2. Preliminaries 2.1. Mixed mock modular forms. We brieﬂy recall the notion of mixed mock modular forms of higher depth from [DMZ19], Section 7.3, and [GMN19], Section 3.2. First, a mixed 1 Z H mock modular form h of weight k 2 and depth 1 is a holomorphic function on which is of polynomial growth at the cusps,∈ and for which there exist ﬁnitely many holomorphic 1 modular forms fj Mℓ and gj M2 k+ℓ for some ℓj Z such that the completion ∈ j ∈ − j ∈ 2

(2.1) h(τ)= h(τ)+ fj(τ)gj∗(τ) j X is a real-analytic modular formb of weight k. Here

i 1 ℓ ∞ ℓ 2 g∗(τ)=( 2i) − gj( z)(z + τ) − dz − τ − Z− denotes the non-holomorphic Eichler integral of a modular form g Mℓ. In this case the function ∈

ℓj ξkh(τ)= v fj(τ)gj(τ) j X b is called the shadow of h. Here ξk is the antilinear differential operator ∂ ξ =2ivk . k ∂τ A mixed mock modular form h of weight k and depth n 2 is defined to be the holomorphic ≥ part of a real-analytic modular form h of weight k whose image under the ξ-operator is a finite linear combination of the form

b ℓj ξkh(τ)= v fj(τ)gj(τ), j X b where gj M2 k+ℓ , and fj is the modular completion of a mixed mock modular form of ∈ − j weight ℓj and depth n 1. Here, the meaning of the holomorphic part of a real-analytic modular form is deliberately− kept vague in order to include many natural examples. 5

Example 2.1. We show that the unary theta function θ7/2,h from the introduction is the completion of a mixed mock modular form of weight 7/2 and depth 1. To this end, we write it as 3/2 1/2 64π 2 24π 2 θ (τ)= a3qa /3 aqa /3. 7/2,h √ − √ 3 3 a Z 3v a Z a hX(mod∈ 3) a hX(mod∈ 3) ≡ ≡ The ﬁrst sum is holomorphic on H and at the cusps, and is the mixed mock modular part 1 of θ7/2,h(τ). The second summand is a product of v− (which is the Eichler integral of a constant) and a multiple of the holomorphic unary theta series θ3/2,h. Furthermore, we see 3/2 that the shadow of θ7/2,h, that is, its image under ξ7/2, is a multiple of v θ3/2,h(τ).

2.2. Quadratic forms. We let D be the set of all integral binary quadratic forms Q = 2 Q + [a, b, c] of discriminant D = b 4ac, and for D < 0 we let D be the subset of positive − Q + definite forms. The group Γ = SL2(Z) acts from the right on D and D, with finitely many orbits if D = 0. Q Q For Q = [a, b, c]6 and z = x + iy H we define the quantities ∈ QD ∈ 1 Q(z, 1) = az2 + bz + c, Q = (a z 2 + bx + c). z y | | They are related by

2 2 2 ∂ i (2.2) Q = y− Q(z, 1) D, Q = Q(z, 1). z | | − ∂z z −2y2

For D < 0 the CM point zQ H associated to Q D is deﬁned as the unique root of Q(z, 1) = 0 in H. We can factor∈ Q(z, 1) as ∈ Q D (2.3) Q(z, 1) = | | (z zQ)(z zQ). 2p Im(zQ) − − 2.3. Theta functions. The Kudla-Millson theta function is deﬁned for z = x + iy H and τ = u + iv H as ∈ ∈ 2πiDτ (2.4) ΘKM(z, τ)= ϕKM(Q, z, v)e− , D Z Q X∈ X∈QD where we set |Q(z,1)|2 2 1 4πv ϕ (Q, z, v)= 4vQ e− y2 . KM z − 2π The function ΘKM(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 Γ0(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 ∞ 2πiDτ (2.5) ΘKM∗ (z, τ)= ϕKM∗ (Q, z, v)e− , D Z Q X∈ X∈QD 6 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

where

2 |Q(z,1)|2 v 4πv 2 ϕ∗ (Q, z, v)= Q(z, 1)Q e− y . KM −2y2 z

It is a multiple of the derivative in z of the Siegel theta function ΘS(z, τ), which is the theta 2 2 function associated to ϕS(Q, z, v)= v exp( 4πv Q(z, 1) /y ). In particular, it transforms like a modular form of weight 2 in z for Γ− and weight| |1/2 in τ for Γ (4) (see [Bor98]). − 0 2.4. Singular theta functions. For Q = 0 we consider the function 6 2 Qz 4πv |Q(z,1)| (2.6) η (Q, z, v)= e− y2 . KM 2πQ(z, 1)

It is the derivative of Kudla’s Green function ξKM (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 ηKM is real-analytic in z on all of H. For D < 0 it is only real-analytic for≥ z H z , but then the diﬀerence ∈ \{ Q}

sgn(Qz) D ηKM(Q, z, 1) | | − 2πQ(z,p1)

extends to a real-analytic function around zQ which vanishes at zQ. This can easily be proved using the ﬁrst relation in (2.2). For z H with Q(z, 1) = 0 the function ηKM is related to ϕKM and ϕKM∗ via the lowering operator∈L = 2iy2 ∂ by6 κ − ∂z (2.7) L2,zηKM(Q, z, v)= ϕKM(Q, z, v) and

(2.8) L3/2,τ ηKM(Q, z, v)= ϕKM∗ (Q, z, v), which can be checked by a direct calculation using (2.2). 2πiDτ The “theta function” formed by summing η (Q, z, v)e− over D Z and Q KM ∈ ∈ D 0 would behave very badly as a function of z since it would have a singularity at everyQ \{ CM} point, that is, on a dense subset of H. However, for ﬁxed D Z and ̺ H we deﬁne the function ∈ ∈

(2.9) ΘKM∗ ,D(̺, v)= ηKM(Q, ̺, v).

Q D 0 ∈QzX=\{̺ } e Q6

We can imagine it as the ( D)-th coeﬃcient of the “singular theta function” ΘKM∗ (̺, τ) − 2πiDτ that one would obtain by multiplying (2.9) with e− and summing up over all D.

However, it is in not clear (and seems to be diﬃcult to prove) that ΘKM∗ (̺, τ)e converges for every ̺. Hence, for simplicity, we will not work with the full function ΘKM∗ (̺, τ). By a e e 7

slight abuse of notation, for m N we also set ∈ 0 m m (2.10) R Θ∗ (̺, v)= R η (Q, z, v) , 2,z KM,D 2,z KM |z=̺ Q D 0 ∈QzX=\{̺ } e Q6 m 0 where R2 = Rm Rm 2 R2 with R2 = id is an iterated version of the raising operator ∂ ◦1 − ◦···◦ Rκ =2i ∂z + κy− . Using (2.8) we obtain m m (2.11) L3/2,τ R2,zΘKM∗ ,D(̺, v)= R2,zΘKM∗ ,D(̺, v) m 2πiDτ for every m N0 and D Z, where R2,zΘKM∗ ,D(̺, v) denotes the coeﬃcient at e− of m ∈ ∈ e R2,zΘKM∗ (̺, τ) 2.5. Regularized inner products. We now describe the regularized inner product in (1.1). Let f S . We denote by [̺ ],..., [̺ ] Γ H the equivalence classes of the poles of ∈ 0 1 r ∈ \ f on H and we choose a fundamental domain ∗ for Γ H containing ̺ ,...,̺ such that F \ 1 r each ̺ lies in the interior of Γ ∗. For any ̺ H and ε> 0 we let ℓ ̺ℓ F ∈ z ̺ B (̺)= z H : X (z) <ε , X (z)= − , ε { ∈ | ̺ | } ̺ z ̺ − 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 ∞

reg dxdy (2.12) f,g = lim f(z)g(z) 2 . ε1,...,εr 0 ∗ r h i S Bε (̺ ) y → ZF \ ℓ=1 ℓ ℓ It was shown in Proposition 3.2 of [ANBS19] 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 ∞

n (2.13) f(z)= cf,̺(n)X̺ (z) n ≫−∞X around every ̺ H (see Proposition 17 in Zagier’s part of [BvdGHZ08]). In order to evaluate regularized∈ inner products as in (2.12) we will typically apply Stokes’ Theorem,

which yields integrals over the boundaries of the balls Bεℓ (̺ℓ). To compute such boundary integrals the following formula is useful.

Lemma 2.2 (Lemma 4.1 in [ANBS19]). Let ̺ H, let f S0 be meromorphic near ̺, and let g : H C be real-analytic near ̺. Then∈ we have the∈ formula → n Im(̺) n 1 lim f(z)g(z)dz = 4π cf,̺( n)R2 − g(̺), ε 0 − (n 1)! − → ∂Bε(̺) n 1 Z X≥ − where c ( n) are the coeﬃcients of the elliptic expansion (2.13) of f. f,̺ − 8 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

2.6. Traces of CM values and average values of meromorphic modular forms. For D< 0 we deﬁne the D-th trace of f S by ∈ 0 cf,zQ (0) trf (D)= , ΓQ Q + /Γ ∈QXD | | where cf,zQ (0) is the constant coeﬃcient in the elliptic expansion (2.13) of f around the

CM point zQ. If f is holomorphic at zQ then we have cf,zQ (0) = f(zQ). We deﬁne the trace of index 0 of f S with poles at [̺ ],..., [̺ ] Γ H by ∈ 0 1 r ∈ \ 1 reg 1 dxdy trf (0) = f, 1 = lim f(z) 2 . ε1,...,εr 0 ∗ r −4π h i −4π S Bε (̺ ) y → ZF \ ℓ=1 ℓ ℓ One can view trf (0) as the regularized average value of f on Γ H. Similarly as in [Bor98], Theorem 9.2, or [BF06], Remark 4.9, it can be evaluated in terms\ of special values of the real-analytic weight 2 Eisenstein series

∞ 3 2πinz E∗(z)= +1 24 σ (n)e 2 −πy − 1 n=1 X as follows. Lemma 2.3. For f S with poles at [̺ ],..., [̺ ] Γ H we have ∈ 0 1 r ∈ \ r n π 1 Im(̺ℓ) n 1 tr (0) = c ( n)R − E∗(̺ ), f 3 (n 1)! f,̺ℓ − 2 2 ℓ ℓ=1 Γ̺ℓ n 1 X | | X≥ − where c ( n) are the coeﬃcients of the elliptic expansion (2.13) of f. f,̺ − 3 Proof. The Eisenstein series satisﬁes L2E2∗(z)= π , so we can write 1 dxdy ∗ trf (0) = lim f(z)L2E2 (z) 2 . ε1,...,εr 0 ∗ r −12 S Bε (̺ ) y → ZF \ ℓ=1 ℓ ℓ Now using Stokes’ Theorem (in the form given in Lemma 2.1 of [BKV13]) and Lemma 2.2 easily gives the stated formula.

Finally, for notational convenience, we set trf (D)=0 for D> 0. 3. The Kudla-Millson theta lift Let f S be a meromorphic modular form of weight 0 which decays like a cusp form ∈ 0 towards 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 ∈ \ reg dxdy (3.1) ΦKM(f, τ)= f, ΘKM( , τ) = lim f(z)ΘKM(z, τ) 2 . ε1,...,εr 0 ∗ r · S Bε (̺ ) y D E → ZF \ ℓ=1 ℓ ℓ Since ΘKM(z, τ) is real-analytic in z and decays square exponentially as y goes to , it follows from Proposition 3.2 of [ANBS19] that the theta lift converges for every τ H∞. In ∈ particular, it transforms like a modular form of weight 3/2 for Γ0(4). 9

We now compute the Fourier expansion of the Kudla-Millson theta lift. Theorem 3.1. The Fourier expansion of the Kudla-Millson theta lift of f S is given by ∈ 0 r n 1 Im(̺ℓ) n 1 D Φ (f, τ)= 2tr (D) 4π c ( n)R − Θ∗ (̺ , v) q− , KM f − (n 1)! f,̺ℓ − 2,z KM,D ℓ D Z ℓ=1 Γ̺ℓ n 1 ! X∈ X | | X≥ − e n 1 where c ( n) are the coefficients of the elliptic expansion (2.13) of f and R − Θ∗ (̺, v) f,̺ − 2,z KM,D is defined in (2.10). Recall that we set trf (D)=0 for D> 0. e Remark 3.2. We state the Fourier expansion of the Kudla-Millson theta lift of meromorphic modular forms for congruence subgroups in Section 5. Proof of Theorem 3.1. We plug in the definition of the Kudla-Millson theta function (2.4) and obtain the Fourier expansion

2πiDτ ΦKM(f, τ)= c(D, v)e− D Z X∈ with coefficients dxdy (3.2) c(D, v)= lim f(z) ϕKM(Q, z, v) 2 . ε1,...,εr 0 ∗ r y → Sℓ=1 Bεℓ (̺ℓ) Q ! ZF \ X∈QD 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 ηKM(Q, z, v) defined in (2.6) is real-analytic in z on all of H. We use the differential equation (2.7) and apply Stokes’ Theorem (in the form given in Lemma 2.1 of [BKV13]) to obtain r

(3.3) c(D, v)= lim f(z) ηKM(Q, z, v)dz. εℓ 0 ∗ ℓ=1 → ∂(Bεℓ (̺ℓ) ) Q X Z ∩F X∈QD 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

B (̺ )= γ(B (̺ ) ∗) εℓ ℓ εℓ ℓ ∩F γ Γ̺ ∈[ ℓ

we see that integrating over the full boundary ∂Bεℓ (̺ℓ) on the right-hand side of (3.3) gives an additional factor 1/ Γ . For D > 0 the function η (Q, z, v) is real-analytic ̺ℓ Q D KM in z on H, so using Lemma| | 2.2 we ﬁnd ∈Q P r n 1 Im(̺ℓ) n 1 c(D, v)= 4π cf,̺ℓ ( n)R2,z− ηKM(Q, z, v) . − Γ̺ (n 1)! − z=̺ ℓ=1 ℓ n 1 Q D ! ℓ X | | X≥ − X∈Q

This ﬁnishes the computation in the case D> 0. 10 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

The coefficients of index D =0. We split off the summand for Q = 0 in (3.2), which yields 1 dxdy lim f(z) 2 = 2trf (0). ε1,...,εr 0 ∗ r −2π S Bε (̺ ) y → ZF \ ℓ=1 ℓ ℓ The remaining part with Q = 0 can be computed as in the case D> 0. 6 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 [z1],..., [zs] Γ H be the Γ-classes of CM points of discriminant D, and let Q ,...,Q + be the∈ corresponding\ quadratic forms. We can 1 s ∈ QD assume that z ,...,z ∗ and that every z lies in the interior of Γ ∗. We cut out 1 s ∈ F ℓ zℓ F a ball Bδℓ (zℓ) around every CM point zℓ and then apply Stokes’ Theorem as in the case D> 0 to obtain r 1 c(D, v)= lim f(z) ηKM(Q, z, v)dz εℓ 0 Γ̺ℓ → ∂Bε (̺ℓ) ℓ=1 Z ℓ Q D (3.4) X | | X∈Q s 1 + lim f(z) ηKM(Q, z, v)dz. Γ δℓ 0 ∂B (z ) ℓ=1 zℓ → δℓ ℓ Q X | | Z X∈QD The first line can be evaluated as in the case D > 0. In the second line, for fixed ℓ the summands with Q = Q are real-analytic near z , so their integrals vanish as δ 0. 6 ± ℓ ℓ ℓ → Replacing Qℓ by Qℓ gives a factor 2. Further, since ηKM(Q, z, v) sgn(Qz) D /2πQ(z, 1) is real analytic− near the CM point z , the second line in (3.4) is− given by | | Q p s 1 sgn((Q ) ) D 2 lim f(z) ℓ z | |dz. Γz δℓ 0 ∂B (z ) 2πQℓ(z, 1) ℓ=1 ℓ → δℓ ℓ p X | | Z Note that sgn((Qℓ)z)=1 for z close to zℓ. If we now plug in the elliptic expansion (2.13) of f and the expression (2.3) for Q(z, 1), we arrive at

s n 1 1 Im(zℓ) (z zℓ) − (3.5) 2 cf,zℓ (n) lim − n+1 dz. Γ π δℓ 0 ∂B (z ) (z zℓ) ℓ=1 zℓ n 0 → δℓ ℓ X | | X≥ Z − By the residue theorem, the last integral vanishes unless n = 0, in which case it equals π/ Im(zℓ). Hence the second line in (3.4) is given by s cf,zℓ (0) 2 = 2trf (D), Γzℓ Xℓ=1 | | 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 = z1 is a pole of f which is also a CM point. In this case we do not need to cut out an additional δ1-ball around z1 before applying Stokes’ Theorem since we already cut out an ε1-ball around ̺1. 11

In particular, the summand for z1 in the second line of (3.4) has to be omitted, and the summand for ̺1 in the ﬁrst line of (3.4) has to be computed as follows. We write

lim f(z) ηKM(Q, z, v)dz ε1 0 → ∂Bε1 (̺1) Q Z X∈QD D (3.6) = 2 lim f(z) | | dz ε1 0 ∂B (̺ ) 2πQ1(z, 1) → Z ε1 1 p sgn(Qz) D + lim f(z) ηKM(Q, z, v) δ̺1=zQ | | dz. ε1 0 ∂Bε (̺1) − 2πQ(z, 1) → 1 Q D p ! Z X∈Q The factor 2 in the second line comes from the fact that Q and Q have the same CM 1 − 1 point z1, and the sign is gone since sgn((Q1)z)=1 for z close to z1. The sum over Q D in the third line of (3.6) is real-analytic near ̺1, hence the expression in the third∈ lineQ can be computed as in the case D > 0 to

n Im(̺1) n 1 sgn(Qz) D 4π cf,̺1 ( n)R2,z− ηKM(Q, z, v) δ̺1=zQ | | . − (n 1)! − − 2πQ(z, 1) z=̺ n 1 Q D p !! 1 X≥ − X∈Q

For ̺1 = zQ we have

n 1 sgn(Qz) D R − η (Q, z, v) δ | | =0 2,z KM − ̺1=zQ 2πQ(z, 1) p ! z=̺1

for all n 1, which follows from the fact that the diﬀerence in the brackets can be written ≥ as a real-analytic multiple of Q(z, 1). Therefore we can just omit the summands for ̺1 = zQ. In the second line in (3.6), we plug in the elliptic expansion of f and obtain n 1 D Im(̺1) (z ̺1) − 2 lim f(z) | | dz =2 cf,̺1 (n) lim − n+1 dz. ε1 0 ∂B (̺ ) 2πQ1(z, 1) π ε1 0 ∂B (̺ ) (z ̺ ) → ε1 1 p n → ε1 1 1 Z ≫−∞X Z − Note that, in comparison to the expression in (3.5), the sum now has ﬁnitely many terms with negative n. However, using the residue theorem one can check that it is still true that the last integral vanishes unless n = 0, in which case it equals π/ Im(̺1). Hence the last

displayed formula becomes 2cf,̺1 (0), which contributes to the trace of index D. We can proceed in the same way for every pole of f which is also a CM point of discriminant D. The proof is ﬁnished. From the Fourier expansion of the Kudla-Millson theta lift and (2.11) we immediately obtain its image under the lowering operator.

Corollary 3.3. The image under the lowering operator L3/2,τ of the Kudla-Millson theta lift of f S is given by ∈ 0 r n 1 Im(̺ℓ) n 1 L Φ (f, τ)= 4π c ( n)R − Θ∗ (̺ , τ). 3/2,τ KM − (n 1)! f,̺ℓ − 2,z KM ℓ ℓ=1 Γ̺ℓ n 1 X | | X≥ − 12 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

4. The proof of Theorem 1.1 We now prove Theorem 1.1. Let ρ = eπi/3. By Theorem 3.1 the Fourier expansion of the Kudla-Millson theta lift of 1/j is given by

n 1 Im(ρ) n 1 D Φ (1/j, τ)= 2tr (D) 4π c ( n)R − Θ∗ (ρ, v) q− . KM 1/j − (n 1)! 1/j,ρ − 2,z KM,D D Z Γρ n 1 ! X∈ | | X≥ − We ﬁrst consider the growth of the traces of reciprocal singular moduli.e

Proposition 4.1. The traces tr1/j(D) are of polynomial growth in D . In particular, the generating series of traces of reciprocal singular moduli converges| absolutely| and locally uniformly. Proof. We start with the simple estimate

tr1/j(D) H(D) max c1/j,zQ (0) , | | ≤ Q + /Γ | | ∈QD

where H(D)= Q + /Γ 1/ ΓQ is the D-th Hurwitz class number. It follows from Dirich- ∈QD | | let’s class number formula that H(D) is of polynomial growth in D . Since the value P | | c1/j,ρ(0) does not grow with D , it remains to estimate c1/j,zQ (0) = 1/j(zQ) for CM | | + | | | | | | points zQ D with zQ = ρ in terms of D . We can assume that all the CM points zQ lie inside the∈ rectangle Q z 6 H :0 x 1,y| | 1/2 , which contains ρ in its interior but does not contain any of the{ ∈ other Γ-translates≤ ≤ ≥ of ρ. If} we cut out a small ε-ball around ρ from this rectangle, then 1/j is bounded on the remaining set since it has no other poles there and decays like a cusp form towards . In particular, it suﬃces to estimate 1/j(zQ) on ∞ | | n the ε-ball around ρ. If we look at the Taylor expansion 1/j(z)= n α1/j,̺(n)(ρ z) 1 ≫−∞ − of 1/j on this ε-ball, we see that it suﬃces to estimate ρ zQ − in terms of D . We can write | − | P | | 1 √3 b D ρ = + i , zQ = + i | |, 2 2 −2a p2a with a, b, c Z. Then we have ∈ 1 4a2 2 = . ρ zQ (a + b)2 +(√3a D )2 | − | − | | Now there are two cases. If (a + b)2 = 0, then it is a leastp 1, and the whole denominator 6 is at least 1. Furthermore, the assumption that Im(zQ) 1/2 implies a D . Hence, in this case we get ≥ ≤ | | 1 p 4a2 4 D . ρ z 2 ≤ ≤ | | | − Q| If, on the other hand, we have (a + b)2 = 0, then 1 4a2 4a2(√3a + D )2 2 = = 2 | 2 | . ρ zQ (√3a D )2 (3a D ) | − | − | | −| p| p 13

Since the denominator is not 0 (as we assumed z = ρ), it is at least 1, so we ﬁnd Q 6 1 4a2(√3a + D )2 4(√3+1)2 D 2. ρ z 2 ≤ | | ≤ | | | − Q| 1 p In any case, we see that ρ zQ − is bounded by a polynomial in D , which ﬁnishes the proof. | − | | |

We now compute the coeﬃcients c1/j,ρ( n) for n 1 in the elliptic expansion of 1/j at ρ. They can be explicitly described in terms− of the Chowla-Selberg≥ period

3 1 2 1 Γ( 3 ) (4.1) Ω = ΩQ(√ 3) = 2 0.6409273802 − √6π Γ( ) ≈ 3 of Q(√ 3) (compare Section 6.3 in Zagier’s part of [BvdGHZ08]). − Proposition 4.2. The function 1/j has an elliptic expansion at ρ of the form 3 6 π− Ω− 3 23 X− (z)+ + O(X (z)). − 212 33 ρ 212 33 ρ · · Proof. The j-function has an elliptic expansion at ρ of the form

∞ Im(ρ)n j(z)= Rnj(ρ) Xn(z), 0 n! ρ n=3 X compare Proposition 17 in Zagier’s part of [BvdGHZ08]. Using the theory of complex mul- n n 2n tiplication one can show that the values R0 j(ρ) are algebraic multiples of π Ω (compare Proposition 26 in Zagier’s part of [BvdGHZ08]). Explicitly, we have R3j(ρ)= 216 32 √3 π3Ω6, 0 − · · · 4 5 R0j(ρ)= R0j(ρ)=0, R6j(ρ)= 222 32 5 23 π6Ω12. 0 − · · · · 4 5 Note that R0j(ρ) = R0j(ρ) = 0 follows from the simple fact that every function which transforms like a modular form of weight k 0 (mod6) for Γ vanishes at ρ. The other values can be computed, for example, by writing6≡ the almost holomorphic modular forms 3 6 R0j(z)∆(z) and R0j(z)∆(z) in terms of the Eisenstein series E2∗, E4 and E6 and using their values at ρ given in the table after Proposition 27 in Zagier’s part of [BvdGHZ08]. We also checked the above evaluations numerically. By inverting the elliptic expansion of j we obtain the elliptic expansion of 1/j at ρ. Corollary 4.3. We have 23 23 tr ( 3) = = . 1/j − 212 34 331776 · Proof. By deﬁnition tr ( 3) = c1/j,ρ(0) , so the result follows from Proposition 4.2. 1/j − 3 14 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

Corollary 4.4. The trace of index 0 of 1/j is given by 1 1 tr (0) = = . 1/j −211 34 −165888 · Proof. We have the special values

2 32 2 6 E∗(ρ)= R2E∗(ρ)=0, R E∗(ρ)= π Ω . 2 2 2 2 √3 2 They can be obtained, for example, by writing E2∗, R2E2∗ and R2E2∗ in terms of the Eisen- stein series E2∗, E4 and E6 (using Proposition 15 in Zagier’s part of [BvdGHZ08]) and then plugging in their values at ρ from the table after Proposition 27 in loc. cit. Combining the above values with Lemma 2.3 and Proposition 4.2 we obtain the value of tr1/j(0).

Next, we simplify the shadow of ΦKM(1/j, τ). By Corollary 3.3 and the elliptic expansion of 1/j from Proposition 4.2 the image of ΦKM(1/j, τ) under the lowering operator is given by 2π 3 2 Im(ρ) c1/j,ρ( 3)R Θ∗ (ρ, τ). − Γ − 2,z KM | ρ| Computing the action of the raising operators we get 3 |Q(ρ,1)|2 2 Q(ρ, 1) 3 2 4 3 4πv 2 2πiDτ R Θ∗ (ρ, τ)= 12πv Q 32π v Q e− Im(ρ) e− . 2,z KM Im(ρ)6 ρ − ρ D Z Q X∈ X∈QD 1 √3 If we write Q = [A,B,C] and ρ = 2 + i 2 then we have the evaluations 1 1 √3 Qρ = (2A + B +2C) , Q(ρ, 1) = ( A + B +2C) i (A + B). √3 2 − − 2 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 Z3 deﬁned by the conditions a b (mod∈ 3) and b c (mod 2). Thus we obtain ≡ ≡ 3 2 8 3 2 4 3 2πia2τ/3 2πi(b2/3+c2)τ R Θ∗ (ρ, τ)= 36πv a 32π v a b i√3c e e− . 2,z KM √ − − 81 3 a,b,c Z a bX(mod∈ 3) b≡c (mod 2) ≡ Splitting the sum over a into arithmetic progressions mod 3 and using the relation ξ3/2 = 1/2 v− L3/2 we easily obtain the shadow given in Theorem 1.1. Although we did not rigorously deﬁne what the holomorphic part of a real-analytic modular form should be, it seems reasonable from the Fourier expansion of ΦKM(1/j, τ) to view the generating series of traces of reciprocal singular moduli as its holomorphic part. Since the image of ΦKM(1/j, τ) under the ξ-operator is a linear combination of products 7/2 of the functions v θ7/2,h(τ)θ4,h(τ), and θ7/2,h is the completion of a mixed mock modular form of weight 7/2 and depth 1, see Example 2.1, we may view the generating series of 15

traces of reciprocal singular moduli as a mixed mock modular form of weight 3/2 and depth 2 by the deﬁnition given in Section 2.1. This ﬁnishes the proof of Theorem 1.1.

Remark 4.5. The above arguments work more generally for meromorphic modular forms f S0 having poles only at CM points in H. In particular, one can generalize Theorem 1.1 to∈ such f, that is, the generating series

D trf (D)q− D 0 X≤ of traces of CM values of f converges absolutely and locally uniformly, and deﬁnes a mixed mock modular form of weight 3/2 and higher depth whose shadow is a linear combination of products of unary and binary theta functions. Indeed, the proof of Proposition 4.1 works in the same way if we replace ρ by any other CM point, which shows that trf (D) 2 is of polynomial growth in D . Furthermore, the above splitting of R2,zΘKM∗ (ρ, τ) into a sum of products of unary and| | binary theta functions is a special case of a more general n 1 principle. In particular, a similar splitting exists for R2,z− ΘKM∗ (z0, τ) for any CM point n 1 z0 H and n 1. First, one can show by induction that R2,z− ΘKM∗ (z, τ) is a non-zero constant∈ multiple≥ of

n 2 n |Q(z,1)| +1 Q(z, 1) 4πv 2 2πiDτ v 2 H 2√πvQ e− y e− , y2n n z D Z Q X∈ X∈QD where Hn(x) denotes the n-th Hermite polynomial. If we plug in a CM point z0 H, n 1 ∈ the function R2,z− ΘKM∗ (z0, τ) will split into a linear combination of products of complex conjugates of unary theta functions of weight n+1/2 (associated to the Hermite polynomial Hn(x)) and binary theta functions of weight n + 1 (associated to harmonic homogeneous polynomials of degree n). n 1 Finally, we remark that the splitting of R2,z− ΘKM∗ (z0, τ) at a general CM point z0 H can be described most conveniently in the vector-valued setup alluded to in Section∈ 5 below. For an instance of the general principle, we refer the reader to [Ehl17], where an analogous splitting of the Siegel theta function at an arbitrary CM point is worked out in the vector-valued setting.

5. The Fourier expansion of the Kudla-Millson theta lift of meromorphic modular forms for congruence subgroups As mentioned in Remark 1.5, the results of this work can easily be generalized to arbitrary congruence subgroups by using a vector-valued setup as in [BF06]. For the convenience of the reader we state the Fourier expansion of the Kudla-Millson theta lift of a meromorphic modular form in the general case. However, the necessary computations are analogous to the ones given in the proof of Theorem 3.1, so we leave the details to the reader. 16 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

Let V be the real quadratic space of signature (1, 2) given by the set b/2 c V = Q = : a, b, c R , a b/2 ∈ − − equipped with the quadratic form q(Q) = det(Q) and the associated bilinear form (Q1, Q2)= 2 tr(Q1Q2). We can also think of the elements of V as binary quadratic forms ax + bxy + cy− 2, with q being 1 times the discriminant. The group SL (R) acts on V as isometries − 4 2 by conjugation, and this action is compatible with the usual action of SL2(R) on binary quadratic forms. Let L V be an even lattice with dual lattice L′, and let C[L′/L] be its group ring with ⊂ standard basis (eh)h L′/L. We let Λ be a congruence subgroup of SL2(Z) which acts on L ∈ and fixes the classes of L′/L. For simplicity we assume that 1 Λ. For h L′/L and − ∈ ∈ m Q we let Lm,h be the set of all X L + h with q(X) = m, on which Λ acts with ∈ ∈ b/2 c finitely many orbits if m = 0. For z = x + iy H and Q = a b/2 V we define the 6 ∈ − − ∈ quantities 1 Q(z)= az2 + bz + c, Q = (a z 2 + bx + c). z y | | For Q L with m > 0 we let z H denote the unique root of Q(z) in H. Given ∈ m,h Q ∈ h L′/L, m Q>0, and a meromorphic modular form f S0(Λ) of weight 0 for Λ which decays∈ like a∈ cusp form at all cusps, we define the trace function∈

1 f(zQ) trf (m, h)= , 2 Λ Q Λ L Q ∈X\ m,h | | where f(zQ) is again defined as the constant term in the elliptic expansion of f(z) at z = z if z is a pole of f. Further, for m = 0 and h = 0 we set tr (m, h) = 0, and we Q Q 6 f define trf (0, 0) as the regularized average value of f analogously as in Section 2.6, but with the fundamental domain ∗ for Γ= SL (Z) replaced with a suitable fundamental domain F 2 ∗(Λ) for Λ. Finally, we set tr (m, h)=0 for m< 0. F f The C[L′/L]-valued Kudla-Millson theta function is defined by

|Q(z)|2 2 1 πv 2πiq(Q)τ Θ (z, τ)= vQ e− y2 e e , KM z − 2π h h L′/L Q L+h ∈X X∈ and the Kudla-Millson theta lift of a meromorphic modular form f S (Λ) of weight 0 ∈ 0 for Λ is defined analogously as in (3.1), but with the fundamental domain ∗ replaced by F ∗(Λ). The Kudla-Millson theta lift transforms like a modular form of weight 3/2 for the F Weil representation ρ . Furthermore, for fixed ̺ H and h L′/L, m Q, we define L ∈ ∈ ∈ |Q(̺)|2 Q̺ πv 2 Θ∗ (̺, τ)= e− Im(̺) , KM,m,h 2πQ(̺) Q Lm,h 0 ∈ zX=̺\{ } e Q6 and we define its raised version similarly as in (2.10). In this setting, the Fourier expansion of the Kudla-Millson theta lift is given follows. 17

Theorem 5.1. The Fourier expansion of the Kudla-Millson theta lift of f S0(Λ) is given by ∈

ΦKM(f, τ)= 2trf (m, h) h L′/L m Q ∈X X∈ r n 1 Im(̺ℓ) n 1 m 4π cf,̺ ( n)R − Θ∗ (̺ℓ, v) q eh, − Λ (n 1)! ℓ − 2,z KM,m,h ℓ=1 ̺ℓ n 1 X | | X≥ − e where cf,̺ℓ ( n) are the coefficients of the elliptic expansions of f around its poles ̺1,...,̺r mod Λ. − Finally, consider the lattice L given by the set of all integral traceless 2 by 2 matrices, and choose Λ = SL2(Z). Then Lm,h can be identified with the set 4m of all integral binary quadratic forms of discriminant 4m. For m > 0 the point zQ− H associated to − Q ∈ Q Lm,h is precisely the CM point associated to the binary quadratic form corresponding to ∈Q. The discriminant group of L is isomorphic to Z/2Z. By the results of [EZ85], Section 5, we can identify vector-valued modular forms for the Weil representation ρL with scalar-valued modular forms satisfying the Kohnen plus space condition via the map f (τ) e +f (τ) e f (4τ)+ f (4τ). In this way Theorem 5.1 generalizes Theorem 3.1. 0 0 1 1 7→ 0 1 References [ANBS19] Claudia Alfes-Neumann, Kathrin Bringmann, and Markus Schwagenscheidt. On the rational- ity of cycle integrals of meromorphic modular forms. Math. Ann., accepted for publication, 2019. [ANS18] Claudia Alfes-Neumann and Markus Schwagenscheidt. On a theta lift related to the Shintani lift. Adv. Math., 328:858–889, 2018. [BES18] Kathrin Bringmann, Stephan Ehlen, and Markus Schwagenscheidt. On the modular completion of certain generating functions. preprint, arXiv: 1804.07589, 2018. [BF04] Jan H. Bruinier and Jens Funke. On two geometric theta lifts. Duke Math. J., 125(1):45–90, 2004. [BF06] Jan H. Bruinier and Jens Funke. Traces of CM values of modular functions. J. reine angew. Math., 594:1–33, 2006. [BFIL18] Jan H. Bruinier, Jens Funke, Ozlem¨ Imamoglu, and Yingkun Li. Modularity of generating series of winding numbers. Res. Math. Sci, 5, 2018. [BKV13] Kathrin Bringmann, Ben Kane, and Maryna Viazovska. Theta lifts and local Maass forms. Math. Res. Lett., 20(2):213–234, 2013. [Bor98] Richard E. Borcherds. Automorphic forms with singularities on Grassmannians. Invent. Math., 132(3):491–562, 1998. [BvdGHZ08] Jan H. Bruinier, Gerard van der Geer, Günter Harder, and Don Zagier. The 1-2-3 of modular forms. Universitext. Springer-Verlag, Berlin, 2008. Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004, Edited by Kristian Ranestad. [DMZ19] Atish Dabholkar, Sameer Murthy, and Don Zagier. Quantum Black Holes, Wall Crossing, and Mock Modular Forms. Cambridge Monographs in Mathematical Physics, to appear, 2019. [Ehl17] Stephan Ehlen. CM values of regularized theta lifts and harmonic Maass forms of weight one. Duke Math. J., 166(13):2447–2519, 2017. 18 CLAUDIA ALFES-NEUMANN AND MARKUS SCHWAGENSCHEIDT

[EZ85] Martin Eichler and Don Zagier. The theory of Jacobi forms, volume 55 of Progress in Math- ematics. Birkh¨auser Boston Inc., Boston, MA, 1985. [GMN19] Rajesh Kumar Gupta, Sameer Murthy, and Caner Nazaroglu. Squashed toric manifolds and higher depth mock modular forms. High Energ. Phys., 2019(64), 2019. [KM86] Stephen S. Kudla and John J. Millson. The theta correspondence and harmonic forms. I. Math. Ann., 274(3):353–378, 1986. [Kud97] Stephen S. Kudla. Central derivatives of Eisenstein series and height pairings. Ann. of Math., 146:545–646, 1997. [Zag75] Don Zagier. Nombres de classes et formes modulaires de poids 3/2. C.R. Acad. Sci. Paris (A), 281:883–886, 1975. [Zag02] Don Zagier. Traces of singular moduli. In Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volume 3 of Int. Press Lect. Ser., pages 211–244. Int. Press, Somerville, MA, 2002.

Mathematical Institute, Paderborn University, Warburger Str. 100, D-33098 Pader- born, Germany E-mail address: [email protected] Mathematical Institute, University of Cologne, Weyertal 86-90, D–50931 Cologne, Germany E-mail address: [email protected]