aa r X i v : . [ m a t h . N T ] A p r NIEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI
STEFFEN L ¨OBRICH
Abstract.
We compute the Fourier coefficients of analogues of Kohnen and Zagier’s modularforms f k, ∆ of weight 2 and negative discriminant. These functions can also be written as twistedtraces of certain weight 2 Poincar´e series with evaluations of Niebur-Poincar´e series as Fouriercoefficients. This allows us to study twisted traces of singular moduli in an integral weight setting.In particular, we recover explicit series expressions for twisted traces of singular moduli and extendalgebraicity results by Bengoechea to the weight 2 case. We also compute regularized inner productsof these functions, which in the higher weight case have been related to evaluations of higher Green’sfunctions at CM-points. Introduction
For a positive discriminant ∆ and an integer k >
1, Zagier [24] introduced the weight 2 k cuspforms (in a different normalization)(1.1) f k, ∆ ( τ ) := ∆ k − π X Q ∈ Q ∆ Q ( τ, − k , where Q ∆ denotes the set of binary integral quadratic forms of discriminant ∆. The functions f k, ∆ were extensively studied by Kohnen and Zagier and have several applications. For example,they used these functions to construct the kernel function for the Shimura and Shintani lifts andto prove the non-negativity of twisted central L -values [21]. Furthermore, the even periods Z ∞ f k, ∆ ( it ) t n dt, (0 ≤ n ≤ k − f k, ∆ are rational [22]. Bengoechea [2] introduced analogous functions for negative discrim-inants and showed that their Fourier coefficients are algebraic for small k . These functions areno longer holomorphic, but have poles at the CM-points of discriminant ∆. They were realizedas regularized theta lifts by Bringmann, Kane, and von Pippich [7] and Zemel [26]. Moreover,Bringmann, Kane, and von Pippich related regularized inner products of the f k, ∆ to evaluations ofhigher Green’s functions at CM-points.The right-hand side of (1.1) does not converge for k = 1. However, one can use Hecke’s trick toobtain weight 2 analogues of the f k, ∆ . These were introduced by Zagier [24] and further studiedby Kohnen [20]. The aim of this paper is to analyze these weight 2 analogues for negative discrimi-nants. Here we deal with generalizations f ∗ d,D,N for a level N , a discriminant d , and a fundamentaldiscriminant D of opposite sign (see Definition 3.1). The f ∗ d,D,N transform like modular forms ofweight 2 for Γ ( N ) and have simple poles at the Heegner points of discriminant dD and level N .Let Q dD,N denote the set of quadratic forms [ a, b, c ] of discriminant dD with a > N | a , χ D the generalized genus character associated to D , and H ( d, D, N ) the twisted Hurwitz class numberof discriminants d , D and level N (see Subsection 2.2 for precise definitions). Then we obtain thefollowing Fourier expansion ( v := Im( τ ) throughout). Theorem 1.1.
For v > √ | dD | , we have f ∗ d,D,N ( τ ) = − H ( d, D, N ) π [SL ( Z ) : Γ ( N )] v − X n ≥ X a> N | a S d,D ( a, n ) sinh πn p | dD | a ! e ( nτ ) , where e ( w ) := e πiw for all w ∈ C and S d,D ( a, n ) := X b (mod 2 a ) b ≡ dD (mod 4 a ) χ D (cid:18)(cid:20) a, b, b − dD a (cid:21)(cid:19) e (cid:18) nb a (cid:19) . Remark.
The exponential sums S d,D also occur for example in [13] and [20].Note that we obtain a non-holomorphic term in the Fourier expansion of f ∗ d,D,N , just like inthe case of the non-holomorphic weight 2 Eisenstein series E ∗ (see Subsection 2.1). Therefore, incontrast to the higher weight case, the f ∗ d,D,N are in general no longer meromorphic modular forms,but polar harmonic Maass forms . This class of functions is defined and studied in Subsection 2.3.We also use a different approach to compute the coefficients of the f ∗ d,D,N , writing them as tracesof certain Poincar´e series denoted by H ∗ N ( z, · ) (see Proposition 2.5). The H ∗ N ( z, · ) are weight 2analogues of Petersson’s Poincar´e series and were introduced by Bringmann and Kane [5] to obtainan explicit version of the Riemann-Roch Theorem in weight 0. We obtain the following differentFourier expansion of the f ∗ d,D,N , realizing their coefficients as twisted traces of the Niebur-Poincar´eseries j N,n (see Definition 2.1 and Theorem 2.4).
Theorem 1.2.
For v > max (cid:26) √ | dD | , (cid:27) , we have f ∗ d,D,N ( τ ) = − H ( d, D, N ) π [SL ( Z ) : Γ ( N )] v − X n> tr d,D,N ( j N,n ) e ( nτ ) . An interesting phenomenon occurs when Γ ( N ) has genus 0. Subgroups of this type and theirHauptmoduln play a fundamental role in Monstrous Moonshine (see for example [11] for a classicaland [14] for a more modern treatment). When we apply the suitably normalized n -th Hecke operator T n to the Hauptmodul J N for Γ ( N ), then the Niebur-Poincar´e series j N,n coincides with T n J N , upto an additive constant. Zagier [25] showed that, for discriminants d < D >
0, the functions q d + X D> tr d,D,N ( T n J N ) q D and q − D + B n ( D,
0) + X d> tr d,D,N ( T n J N ) q d are weakly holomorphic modular forms for Γ (4 N ) of weight resp. in the Kohnen plus-space. Now summing over n instead of D or d , Theorem 1.2 states that the twisted Hecketraces { tr d,D,N ( T n J N ) } n> give rise to Fourier coefficients of the meromorphic modular forms f ∗ d,D,N − H ( d,D,N )[SL ( Z ):Γ ( N )] E ∗ . IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 3
We give three applications of Theorem 1.2. First, comparing Theorems 1.1 and 1.2, we obtainexplicit series expressions for traces of Niebur-Poincar´e series.
Corollary 1.3.
We have tr d,D,N ( j N,n ) = 2 X a> N | a S d,D ( a, n ) sinh πn p | dD | a ! . Corollary 1.3 was obtained by Duke for N = D = 1 ([12], Proposition 4) and Jenkins for N = 1and D > ( p ) + for p prime, later generalized by Kang and Kim [19] to Γ ( N ) + for arbitrary N .Next we examine algebraicity properties of the Fourier coefficients of f ∗ d,D,N . Bengoechea [2]showed that for ∆ < k ∈ { , , , , } (so that S k = { } ), the Fourier coefficients of f k, ∆ liein the Hilbert class field of Q ( √ ∆). We have the following extension to the weight 2 case. Theorem 1.4. If Γ ( N ) has genus , then the Fourier coefficients of the meromorphic part of f ∗ d,D,N are real algebraic integers in the field Q (cid:16) √ D (cid:17) . Eventually, we compute regularized inner products of meromorphic analogues of the f ∗ d,D,N . Forthis we restrict to the case D = N = 1 and consider the meromorphic modular forms f d ( τ ) := f ∗ d, , ( τ ) − H ( d, , E ∗ ( τ ) . The usual inner product h f d , f δ i for negative discriminants d, δ does not converge, so we need touse a regularization by Bringmann, Kane, and von Pippich. Moreover, since the f d do not decaylike cusp forms towards i ∞ , we also have to apply Borcherds’s regularization near the cusp i ∞ (seeSection 4 for a precise definition). We obtain the following evaluations, where J ( z ) := j , ( z ) − normalized modular j -invariant . Theorem 1.5.
Let d be a negative discriminant and Q d := Q d, . (i) If δ < d is another negative discriminant such that δd is not a square, then h f d , f δ i = 12 π X Q ∈ Q d/ SL2( Z ) Q∈ Q δ/ SL2( Z ) w Q w Q log | J ( z Q ) − J ( z Q ) | . (ii) If neither − d nor − d is a square, then h f d , f d i = 12 π X Q ∈ Q d / SL ( Z ) log (cid:12)(cid:12)(cid:12)(cid:12)p | d | J ′ ( z Q ) Q (1 , (cid:12)(cid:12)(cid:12)(cid:12) + 12 π X Q, Q∈ Q d/ SL2( Z ) Q = Q log | J ( z Q ) − J ( z Q ) | . (iii) We have h f − , f − i = 118 π log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ J ′′′ i √ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and h f − , f − i = 18 π log (cid:12)(cid:12)(cid:12) J ′′ ( i ) (cid:12)(cid:12)(cid:12) . STEFFEN L ¨OBRICH
Note that log | J ( z ) − J ( z ) | is a Green’s function on the modular curve X (1). The double tracesover CM-values of Green’s functions occurring in Theorem 1.5 have been related to heights of Heeg-ner points on modular curves by Gross and Zagier [17]. Since the f d are modular forms of weight2, it would be enlightening to find a geometric interpretation of their inner products and see howthey relate to height functions. In higher weight, Bringmann, Kane, and von Pippich [7] wroteregularized inner products of the functions f k, ∆ for ∆ < higher Green’s functions , so we can see Theorem 1.5 as an extension of their result to the weight2 case.The paper is organized as follows: In Section 2, we introduce the necessary notation and defini-tions. In Section 3, we prove Theorems 1.1, 1.2, and 1.4. Eventually, in Section 4, we compute theregularized inner products, proving Theorem 1.5. Acknowledgements
We thank Kathrin Bringmann, Stephan Ehlen, and Markus Schwagenscheidt for valuable ad-vice on writing the paper. We are also grateful to John Duncan, Michael Mertens, Ken Ono,Dillon Reihill, Larry Rolen, Tonghai Yang, Shaul Zemel, and the referee for helpful comments andconversations. 2.
Definitions and Preliminaries
General notation.
Throughout this paper, we denote variables in the complex upper half-plane H by τ , z , and ̺ with v := Im( τ ) , y := Im( z ) , η := Im( ̺ ) and for w ∈ C we write e ( w ) := e πiw .For a matrix M = (cid:0) a bc d (cid:1) ∈ Γ ( N ) and τ ∈ H , we set M τ := aτ + bcτ + d and j ( M, τ ) := cτ + d. For each point ̺ ∈ H we let Γ N,̺ denote the stabilizer of ̺ in Γ ( N ) and set w N,̺ := N,̺ . Notethat if ρ := i √ denotes the sixth order root of unity in H , then we have w ̺ := w ,̺ = , if ̺ ∈ SL ( Z ) ρ ,2 , if ̺ ∈ SL ( Z ) i ,1 , otherwise.Furthermore, we define the divisor sum function σ ( m ) := P d | m d and the weight 2 Eisensteinseries E ( τ ) := 1 − X m ≥ σ ( m ) e ( mτ ) , as well as its non-holomorphic completion E ∗ ( τ ) := − πv + E ( τ ) , which transforms like a weight 2 modular form for SL ( Z ). In general, we will use a star to denotenon-holomorphic modular forms (cf. Proposition 2.5 and Definition 3.1). IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 5
Quadratic forms and traces of singular moduli.
We denote an integral binary quadraticform Q ( X, Y ) = aX + bXY + cY ∈ Z [ X, Y ] by Q = [ a, b, c ]. The group SL ( Z ) acts on the set ofbinary quadratic forms via(2.1) (cid:16) Q ◦ (cid:16) α βγ δ (cid:17)(cid:17) ( X, Y ) := Q ( αX + βY, γX + δY ) , leaving the discriminant ∆ = b − ac invariant. For a positive integer N and a discriminant ∆, wewrite Q ∆ ,N for the set of all binary integral quadratic forms Q = [ a, b, c ] of discriminant ∆ with a > N | a . Then the group Γ ( N ) acts on Q ∆ ,N . For ∆ < Q ∈ Q ∆ ,N , we denote by z Q the Heegner point of Q , which is the unique zero of Q ( τ,
1) in H .For ∆ <
0, we consider a splitting ∆ = d · D into a discriminant d and a fundamental disriminant D that are both congruent to squares modulo 4 N (meaning that d, D ≡ D is not a proper square multiple of an integer congruent to 0 or 1 (mod 4)) and denote by χ D the generalized genus character corresponding to the decomposition ∆ = d · D as defined in [16]. Definition 2.1.
For a Γ ( N )-invariant function g : H → C , we define the twisted trace of singularmoduli of discriminants d and D of g astr d,D,N ( g ) := X Q ∈ Q dD,N / Γ ( N ) χ D ( Q ) w N,Q g ( z Q ) , where w N,Q := w N,z Q . Moreover, we call H ( d, D, N ) := tr d,D,N (1) = X Q ∈ Q dD,N / Γ ( N ) χ D ( Q ) w N,Q the
Hurwitz class number of discriminants d and D and level N .2.3. Polar Harmonic Maass Forms.
Now we define polar harmonic Maass forms and studytheir elliptic expansions, which we will need to compute the regularized inner products in Section4. See [4], Section 13.3 for an introduction to polar harmonic Maass forms and their applications.
Definition 2.2.
For k ∈ Z , a polar harmonic Maass form of weight k for Γ ( N ) is a continuousfunction F : H → C ∪ {∞} which is real-analytic outside a discrete set of points and satisfies thefollowing conditions:i) For every M ∈ Γ ( N ) and τ ∈ H , we have F ( M τ ) = j ( M, τ ) k F ( τ ) . ii) The function F is annihilated by the weight k hyperbolic Laplacian ∆ k := − v (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + ikv (cid:18) ∂∂u + i ∂∂v (cid:19) . iii) For every z ∈ H , there exists an n ∈ N such that ( τ − z ) n F ( τ ) is bounded in someneighborhood of z .iv) The function F grows at most linearly exponentially at the cusps of Γ ( N ).We denote by H k ( N ) the space of weight k polar harmonic Maass forms for Γ ( N ). STEFFEN L ¨OBRICH
Polar harmonic Maass forms have elliptic expansions around every point ̺ ∈ H , which convergeif(2.2) X ̺ ( τ ) := τ − ̺τ − ̺ is sufficiently small. These can be seen as counterparts to the more common q -series expansions atthe cusps and also break into two pieces. Proposition 2.3 (Proposition 2.2 of [5], see also Subsection 2.3 of [7]) . A polar harmonic Maassform F of weight k ≤ has an expansion around each point ̺ ∈ H of the form F = F + ̺ + F − ̺ ,where the meromorphic part F + ̺ is given by (2.3) F + ̺ ( τ ) := ( τ − ̺ ) − k X n ≫−∞ A + F,̺ ( n ) X n̺ ( τ ) and the non-meromorphic part F − ̺ by (2.4) F − ̺ ( τ ) := ( τ − ̺ ) − k X n ≪∞ A − F,̺ ( n ) β (cid:0) − | X ̺ ( τ ) | ; 1 − k, − n (cid:1) X n̺ ( τ ) . These expressions converge for | X ̺ ( τ ) | ≪ . Here, we have that β ( w ; a, b ) := β ( w ; a, b ) − C a,b with C a,b := X ≤ j ≤ a − j = − b (cid:18) a − j (cid:19) ( − j j + b , where the incomplete β -function is defined by β ( w ; a, b ) := R w t a − (1 − t ) b − dt. We refer to the terms in (2.3) and (2.4) which grow as τ → ̺ as the principal part of F around ̺ . Remark.
The hyperbolic Laplacian splits as(2.5) ∆ k = − ξ − k ◦ ξ k , where ξ k := 2 iv k ∂∂τ . If F satisfies weight k modularity, then ξ k ( F ) is modular of weight 2 − k . Moreover, ξ k annihilatesthe meromorphic part of a polar harmonic Maass form, so that it maps weight k polar harmonicMaass forms to weight 2 − k meromorphic modular forms and its kernel is given by the space ofweight k meromorphic modular forms.2.4. Niebur-Poincar´e series.
Here we introduce
Niebur-Poincar´e series and give their explicitFourier expansion. Following [23], we define for n > F N, − n,s ( z ) := 2 π √ n X M ∈ Γ ∞ \ Γ ( N ) e ( − n Re(
M z )) Im(
M z ) I s − (2 πn Im(
M z )) , where I s − denotes the I -Bessel function and Γ ∞ := {± ( n ) : n ∈ Z } . This series converges abso-lutely and locally uniformly for Re( s ) >
1. The function F N, − n,s is a Γ ( N )-invariant eigenfunctionof the hyperbolic Laplacian with eigenvalue s (1 − s ). Niebur showed that F N, − n,s is analytic in s and has an analytic continuation to s = 1. IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 7
Proposition 2.4 (Theorem 1 of [23]) . The function F N, − n,s has an analytic continuation j N,n to s = 1 , and j N,n ∈ H ( N ) . It has a Fourier expansion of the form j N,n ( z ) = e ( − nz ) − e ( − nz ) + c N ( n,
0) + X m ≥ ( c N ( n, m ) e ( mz ) + c N ( n, − m ) e ( − mz )) . The coefficients are given by c N ( n, m ) := 2 π √ n X c ≥ N | c K ( m, − n ; c ) c × √ m I (cid:16) π √ mnc (cid:17) , if m > , π √ nc , if m = 0 , √ | m | J (cid:18) π √ | m | nc (cid:19) , if m < ,where K ( m, n ; c ) := X a,d (mod c ) ad ≡ c ) e (cid:18) md + nac (cid:19) denotes the Kloosterman sum and I , J are the first order I - and J -Bessel functions , respectively. The constants c N ( n,
0) can be explicitly evaluated. For example, for N = 1 we obtain c ( n,
0) = 24 σ ( n )and for p prime we have(2.6) c p ( n,
0) = − p − (cid:18) σ ( n ) − p σ (cid:18) np (cid:19)(cid:19) , where σ ( ℓ ) := 0 if ℓ / ∈ Z (see [10] for a similar calculation).2.5. Petersson’s Poincar´e series.
For w, s ∈ C , w = 0, we let(2.7) φ s ( w ) := w − | w | − s and define(2.8) H N,s ( z, τ ) := − v s π X M ∈ Γ ( N ) φ s (cid:18) j ( M, τ ) ( M τ − z )( M τ − z ) y (cid:19) = − v s π X M ∈ Γ ( N ) φ s (cid:18) ( τ − M z )( τ − M z )Im(
M z ) (cid:19) . These sums converge locally uniformly for Re( s ) > s . Theysatisfy modularity of weight 0 in z and of weight 2 in τ . Bringmann and Kane showed that theyhave an analytic continuation H ∗ N to s = 0, which are a polar harmonic Maass forms of weight 2with simple poles at Γ ( N )-equivalent points to z . For this, they used a splitting of the sum due toPetersson and obtained an analytic continuation of the Fourier expansion of every part by Poissonsummation and locally uniform estimates. STEFFEN L ¨OBRICH
Proposition 2.5 (Lemma 4.4 of [5]) . The function H N,s has an analytic continuation H ∗ N to s = 0 .We have z H ∗ N ( z, τ ) ∈ H ( N ) and τ H ∗ N ( z, τ ) ∈ H ( N ) . Furthermore, the function τ H N ( z, τ ) := H ∗ N ( z, τ ) + 1[SL ( Z ) : Γ ( N )] E ∗ ( τ ) is a meromorphic modular form of weight for Γ ( N ) with only simple poles at points that are Γ ( N ) -equivalent to z .Remark. Note that H ∗ N ( z, · ) has principal part − w N,z πi τ − z at τ = z .The Fourier coefficients of H ∗ N ( z, · ) were computed in [6], where it was shown that they are givenby the Niebur-Poincar´e series j N,n of Proposition 2.4, evaluated at z . Proposition 2.6 (Theorem 1.1 of [6]) . For v > max n y, y o , we have H ∗ N ( z, τ ) = 3 π [SL ( Z ) : Γ ( N )] v + X n> j N,n ( z ) e ( nτ ) . Weight Modular Forms Associated to Imaginary Quadratic Fields
Now we define and study the weight 2 analogues of the functions f k, ∆ . Definition 3.1.
For N ∈ N , discriminants d , D that are congruent to squares modulo 4 N with D fundamental and dD negative, and s ∈ C with Re( s ) >
0, we let f d,D,N,s ( τ ) := ( | dD | ) s v s s π X Q ∈ Q dD,N χ D ( Q ) φ s ( Q ( τ, φ s as in (2.7) and define f ∗ d,D,N to be the analytic continuation of f d,D,N,s to s = 0. Remark.
The existence of the analytic continuation is established by combining Lemma 3.2 withthe analytic continuation of H N,s stated in Proposition 2.5.With the trace operation from Definition 2.1, we obtain the following relation.
Lemma 3.2.
We have f d,D,N,s ( τ ) = − tr d,D,N ( H N,s ( · , τ )) . Proof.
For M ∈ Γ ( N ), Q ∈ Q dD,N , and the group action defined in (2.1), we have z Q ◦ M = M − z Q and Q ( τ,
1) = p | dD | τ − z Q )( τ − z Q )Im( z Q ) , since Im( z Q ) = √ | dD | a for Q = [ a, b, c ]. Thus it follows H N,s ( z Q , τ ) = − v s π X M ∈ Γ ( N ) φ s (cid:18) ( τ − M z Q )( τ − M z Q )Im( M z Q ) (cid:19) IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 9 = − v s π X M ∈ Γ ( N ) φ s (cid:18) ( τ − z Q ◦ M )( τ − z Q ◦ M )Im( z Q ◦ M ) (cid:19) = − v s π X M ∈ Γ ( N ) φ s p | dD | ( Q ◦ M )( τ, ! = − v s ( | dD | ) s s π X M ∈ Γ ( N ) φ s (( Q ◦ M )( τ, d,D,N ( H N,s ( · , τ )) = − v s ( | dD | ) s s π X Q ∈ Q dD,N / Γ ( N ) χ D ( Q ) w N,Q X M ∈ Γ ( N ) φ s (( Q ◦ M )( τ, − v s ( | dD | ) s s π X Q ∈ Q dD,N χ D ( Q ) w N,Q · w N,Q φ s ( Q ( τ, − f d,D,N,s ( τ ) (cid:3) Theorem 1.2 now follows directly from Proposition 2.6 and taking the analytic continuation to s = 0 in Lemma 3.2.We now move on to compute the Fourier expansion of f ∗ d,D,N directly and prove Theorem 1.1. Proof of Theorem 1.1.
We follow the approach of Appendix 2 of [24]. For v > √ | dD | , we obtain byPoisson summation f d,D,N,s ( τ ) = ( | dD | ) s v s s π X a> N | a X b ∈ Z b ≡ dD (mod 4 a ) χ D (cid:18)(cid:20) a, b, b − dD a (cid:21)(cid:19) φ s (cid:18) aτ + bτ + b − dD a (cid:19) = ( | dD | ) s v s s π X a> N | a X n ∈ Z X b (mod 2 a ) b ≡ dD (mod 4 a ) χ D (cid:18)(cid:20) a, b, b − dD a (cid:21)(cid:19) × Z R φ s (cid:18) a ( τ + t ) + b ( τ + t ) + b − dD a (cid:19) e ( − nt ) dt. Here we used that aτ + ( b + 2 an ) τ + ( b + 2 an ) − dD a = a ( τ + n ) + b ( τ + n ) + b − dD a and that χ D is invariant under translation. Together with Z R φ s (cid:18) a ( τ + t ) + b ( τ + t ) + b − dD a (cid:19) e ( − nt ) dt = a − − s e ( nτ ) Z R + iv φ s (cid:18) t − dD a (cid:19) e ( − nt ) dt we obtain f d,D,N,s ( τ ) = ( | dD | ) s v s s π X a> N | a X n ∈ Z S d,D ( a, n ) a − − s e ( nτ ) Z R + iv φ s (cid:18) t − dD a (cid:19) e ( − nt ) dt. First we consider terms with n = 0. We have to show locally uniform convergence in s of thedouble sum. For this we will bound the integral locally uniformly for σ := Re( s ) > − ε for some ε > a and n . First we write Z R + iv φ s (cid:18) t − dD a (cid:19) e ( − nt ) dt = Z R (cid:18) t − v − dD a (cid:19) + 4 v t ! − s e ( − nt ) dt ( t + iv ) − dD a . Note that the integrand is holomorphic in t in the region Im( t ) > √ | dD | − v . Thus for n <
0, wemay shift the path of integration to R + i ∞ and the integral vanishes.For n >
0, we may fix α ∈ (cid:18) , v − √ | dD | (cid:19) and shift the path of integration to R − iα . Thisyields (cid:12)(cid:12)(cid:12)(cid:12)Z R − iα φ s (cid:18) ( t + iv ) − dD a (cid:19) e ( − nt ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ e − πnα Z ∞ (cid:18) t − ( v − α ) − dD a (cid:19) + 4( v − α ) t ! − σ dt. Now we apply the estimates (cid:18) t − ( v − α ) − dD a (cid:19) + 4( v − α ) t ≥ ((cid:0) ( v − α ) + dD (cid:1) , for every t , (cid:0) t + ( v − α ) (cid:1) , for t > v − α ,to obtain Z ∞ (cid:18) t − ( v − α ) − dD a (cid:19) + 4( v − α ) t ! − σ dt ≤ Z v − α (cid:18) ( v − α ) + dD (cid:19) − − σ dt + Z ∞ v − α (cid:0) t + ( v − α ) (cid:1) − − σ dt. The last bound is locally uniform for σ > − and independent of a and n . Thus the overall sumis uniformly bounded by ≪ X n ≥ X a> N | a S d,D ( a, n ) a − − σ e − πnα . We define the half-integral weight Kloosterman sum as K ∗ ( m, n, c ) := X a,d (mod c ) ∗ ad ≡ c ) (cid:16) cd (cid:17) (cid:18) − d (cid:19) / e (cid:18) na + mdc (cid:19) , IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 11 where ( ·· ) denotes the Kronecker symbol. Plugging c a into Proposition 3 of [13] and notingthat the definition of S given there differs from ours by a factor 2, we obtain S d,D ( a, n ) = 1 − i X r | ( a,n ) (cid:18) Dr (cid:19) r ra (cid:18) (cid:18) a/r (cid:19)(cid:19) K ∗ (cid:18) d, n Dr , ar (cid:19) , and hence X a> N | a S d,D ( a, n ) a − − σ = 1 − i X r | n r − − σ (cid:18) Dr (cid:19) X a> N | a (cid:18) (cid:18) a (cid:19)(cid:19) K ∗ (cid:16) d, n Dr , a (cid:17) a + σ . It has been observed in the remark following Theorem 2.1 of [15] that the Selberg-Kloostermanzeta function S m,n ( s ) := X a> K ∗ ( m, n, a ) a s has an analytic continuation to s = for mn <
0. Since S m,n has only finitely many poles in [1 , ε > S d, n Dr (cid:0) + σ (cid:1) has an analytic continuation to σ > − ε .This gives a locally uniform bound for σ > − ε and we obtain the analytic continuation for the sumover the positive n by just plugging in s = 0.Now we have, for v > √ | dD | and n > Z R + iv (cid:18) t − dD a (cid:19) − e ( − nt ) dt = − πa p | dD | sinh πn p | dD | a ! , since t sinh( κt ) is the inverse Laplace transform of s κs − κ (see for example (29.3.17) of [1]).So all in all we obtain that for n >
0, the n -th Fourier coefficient of f ∗ d,D,N equals − X a> N | a S d,D ( a, n ) sinh πn p | dD | a ! . Finally, it follows from Proposition 2.6 and Lemma 3.2 that the remaining part of the Fourierexpansion, i.e. the n = 0 term, equals − tr d,D,N (cid:18) π [SL ( Z ) : Γ ( N )] v (cid:19) = − H ( d, D, N ) π [SL ( Z ) : Γ ( N )] v . (cid:3) Proof of Theorem 1.4.
By Theorem 1.2, the n -th Fourier coefficient of f ∗ d,D,N is − tr d,D,N ( j N,n ). IfΓ ( N ) has genus 0, then j N,n is weakly holomorphic on the modular curve X ( N ). Lemma 5.1 (v)of [8] states that the twisted Heegner divisor Z d,D,N := X Q ∈ Q dD,N / Γ ( N ) χ D ( Q ) w N,Q z Q is defined over Q ( √ D ). This means that h Z d,D,N , j N,n i := X Q ∈ Q dD,N / Γ ( N ) χ D ( Q ) w N,Q j N,n ( z Q ) = tr d,D,N ( j N,n ) ∈ Q ( √ D ) . By Theorem I of [9], j N, ( z Q ) is an algebraic integer for every quadratic form Q ∈ Q dD,N . Now j N,n is a polynomial in j N, , so the twisted sum tr d,D,N ( j N,n ) is also an algebraic integer, whichimplies the statement. (cid:3) Regularized Inner Products
In this section we restrict to the full modular group and therefore drop the subscript N through-out. Let f , g be meromorphic modular forms of weight k which decay like cusp forms at i ∞ and have poles at z , . . . , z r ∈ SL ( Z ) \ H . We choose a fundamental domain F such that for ev-ery j ∈ { , . . . , r } , the representative of z j in F lies in the interior of Γ z j F . We identify the z , . . . , z r ∈ SL ( Z ) \ H with their representatives in F .For an analytic function A ( s ) in s = ( s , . . . , s r ), denote by CT s =0 A ( s ) the constant term of themeromorphic continuation of A ( s ) around s = · · · = s r = 0. Then the regularized inner productintroduced in [7] is given by(4.1) h f, g i := CT s =0 Z F f ( τ ) r Y ℓ =1 | X z ℓ ( τ ) | s ℓ g ( τ ) v k dudvv ! . Note that, as z → z ℓ , we have X z ℓ ( z ) →
0, so the integral in (4.1) converges if we have Re( s ℓ ) ≫ ≤ ℓ ≤ r . One can show that the regularization is independent of the choice of fundamentaldomain. Since the functions we integrate do not decay like cusp forms, we need to use anotherregularization by Borcherds [3]. Namely for holomorphic modular forms f, g of weight k , we define(4.2) h f, g i := CT s =0 (cid:18) lim T →∞ Z F T f ( τ ) g ( τ ) v k − s dudvv (cid:19) , whenever it exists. Here, for a fundamental domain F and T >
0, we set F T := { z ∈ F : Im( z ) ≤ T } . To compute the inner products in Theorem 1.5, we split the domain of integration into a partwhich contains all the poles of the integrands, where we apply the regularization of Bringmann,Kane, and von Pippich, and a part around the cusp i ∞ , where we apply Borcherds’s regularization.Therefore, the regularized integral will look like(4.3) h f, g i = CT s =0 Z F Y f ( τ ) r Y ℓ =1 | X z ℓ ( τ ) | s ℓ g ( τ ) v k dudvv ! + CT s =0 lim T →∞ Z F T \F Y f ( τ ) g ( τ ) v k − s dudvv ! , where f, g are meromorphic modular forms of weight k and Y > f and g lie in F Y . Here we can assume that F contains all the poles of f and g as well as IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 13 [0 ,
1] + i [ Y, ∞ ].To prepare the proof, we first look at elliptic expansions of the polar harmonic Maass forms H z ( τ ) := H ( z, τ ). For X ̺ ( τ ) ≪
1, we have(4.4) H z ( τ ) = − πi J ′ ( τ ) J ( τ ) − J ( z ) = 1( τ − ̺ ) − δ z,̺ w z yπ X ̺ ( τ ) − + X n ≥ a z,̺ ( n ) X ̺ ( τ ) n , where δ z,̺ is defined to be 1 if z ∈ SL ( Z ) ̺ and 0 otherwise (cf. the remark following Proposition2.5). Furthermore, let G z ( τ ) := − π log | J ( τ ) − J ( z ) | , so that ξ ( G z ) = H z with ξ as in (2.5). Lemma 4.1.
The function G z is a weight polar harmonic Maass form. For every ̺ ∈ H , it hasan elliptic expansion (4.5) G z ( τ ) = − δ z,̺ w z π log( | X ̺ ( τ ) | ) + X n ≥ A + z,̺ ( n ) X ̺ ( τ ) n + X n> A − z,̺ ( n ) X ̺ ( τ ) n , which converges for | X ̺ ( τ ) | ≪ . Moreover, we have A − z,̺ ( n ) = a z,̺ ( n − ηn with a z,̺ ( n ) as in (4.4) and (4.6) A + z,̺ (0) = − π × log | J ( ̺ ) − J ( z ) | , if ̺ / ∈ SL ( Z ) z , log | yJ ′ ( z ) | , if ̺ ∈ SL ( Z ) z and i, ρ / ∈ SL ( Z ) z , log | J ′′ ( i ) | , if ̺, z ∈ SL ( Z ) i , log (cid:12)(cid:12)(cid:12) √ J ′′′ ( ρ ) (cid:12)(cid:12)(cid:12) , if ̺, z ∈ SL ( Z ) ρ .Proof. One easily checks that G z is a polar harmonic Maass form of weight 0. By Proposition 2.3,it has for every ̺ ∈ H an elliptic expansion G z ( τ ) = X n ≫−∞ A + z,̺ ( n ) X ̺ ( τ ) n + X n ≪∞ A − z,̺ ( n ) β (cid:16) − | X ̺ ( τ ) | , , − n (cid:17) X ̺ ( τ ) n for X ̺ ( τ ) ≪
1. Noting that β (1 − r ; 1 , − n ) = Z − r (1 − t ) − n − dt + δ n =0 · n = ( r − n n , if n = 0, − log( r ) , if n = 0,we obtain an elliptic expansion of the shape G z ( τ ) = A − z,̺ (0) log( | X ̺ ( τ ) | ) + X n ≫−∞ A + z,̺ ( n ) X ̺ ( τ ) n + X n ≪∞ n =0 A − z,̺ ( n ) X ̺ ( τ ) − n Now using ξ (cid:16) X ̺ ( τ ) (cid:17) = 2 i∂ τ τ − ̺τ − ̺ = − η ( τ − ̺ )
24 STEFFEN L ¨OBRICH and ξ (cid:0) log( | X ̺ ( τ ) | ) (cid:1) = 2 i∂ τ log( | X ̺ ( τ ) | ) = 2 i ∂ τ | X ̺ ( τ ) | | X ̺ ( τ ) | = 2 iX ̺ ( τ ) ∂ τ X ̺ ( τ ) | X ̺ ( τ ) | = − η ( τ − ̺ ) X ̺ ( τ ) − gives ξ ( G z ( τ )) = − η ( τ − ̺ ) A − z,̺ (0) X ̺ ( τ ) − + X n ≪∞ n =0 nA − z,̺ ( n ) X ̺ ( τ ) − n − . We compare with (4.4) and obtain A − z,̺ ( n ) = 0 for n > A − z,̺ (0) = − δ z,̺ w z π , and − ηnA ̺,z ( n ) = a ̺,z ( − n − . for n <
0. We also have A + z,̺ ( n ) = 0 for n < G z at ̺ comes entirelyfrom the n = 0 term.For the evaluation of A + z,̺ (0), note that A + z,̺ (0) = − π lim τ → ̺ (log | J ( τ ) − J ( z ) | − δ z,̺ w z log( | X ̺ ( τ ) | )) , which equals G z ( ̺ ) = − πw z log | J ( τ ) − J ( z ) | if ̺ = z . If ̺ = z , note thatlim τ → z (log | J ( τ ) − J ( z ) | − w z log( | X ̺ ( τ ) | )) = lim τ → z (cid:18) log | J ( τ ) − J ( z ) | + log (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) τ − zτ − z (cid:12)(cid:12)(cid:12)(cid:12) w z (cid:19)(cid:19) = lim τ → z log (cid:12)(cid:12)(cid:12)(cid:12) J ( τ ) − J ( z )( τ − z ) w z (cid:12)(cid:12)(cid:12)(cid:12) + log ((2 y ) w z ) = log (cid:12)(cid:12)(cid:12)(cid:12) (2 y ) w z w z ! J ( w z ) ( z ) (cid:12)(cid:12)(cid:12)(cid:12) , which implies the statement. (cid:3) Lemma 4.2.
For every z, z ∈ H , we have h H z , H z i = − A + z ,z (0) with A + z ,z (0) as in (4.6) .Proof. Applying Stokes’s Theorem to the second summand of (4.3), we obtain for Re( s ) > Z F T \F Y H z ( τ ) H z ( τ ) v − s dudv = − Z F T \F Y H z ( τ ) ξ (cid:16) G z ( τ ) (cid:17) v − s dudv = − Z F T \F Y ξ (cid:16) H z ( τ ) G z ( τ ) v − s (cid:17) dudv − s Z F T \F Y H z ( τ ) G z ( τ ) v − s − dudv = − Z ∂ ( F T \F Y ) H z ( τ ) G z ( τ ) v − s dτ − s Z F T \F Y H z ( τ ) G z ( τ ) v − s − dudv = Z H z ( u + iT ) G z ( u + iT ) T − s du − Z H z ( u + iY ) G z ( u + iY ) Y − s du IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 15 + s Z TY Z H z ( u + iv ) G z ( u + iv ) v − s − dudv. Now we have G z ( τ ) = v + O (1) and H z ( τ ) = O (1) as v → ∞ . Hence, the first summand vanishesas T → ∞ , provided that Re( s ) is sufficiently large, and the limit as T → ∞ of the third integralhas a meromorphic continuation to C with the only pole at s = 1. Therefore the contributions ofthe first and third summand vanish in the analytic continuation to s = 0. The second summand isanalytic in s = 0, so in total we have(4.7) CT s =0 Z F\F Y H z ( τ ) H z ( τ ) v − s dτ = − Z H z ( u + iY ) G z ( u + iY ) du. To compute the first summand of (4.3), note that the functions H z and H z have simple polesonly at z and z . Thus the regularized inner product equals h H z , H z i = CT ( s ,s )=(0 , Z F H z ( τ ) | X z ( τ ) | s | X z ( τ ) | s H z ( τ ) dudv = − CT ( s ,s )=(0 , Z F H z ( τ ) | X z ( τ ) | s | X z ( τ ) | s ξ (cid:16) G z ( τ ) (cid:17) dudv, By Stokes’s Theorem, the integral equals(4.8) − Z F H z ( τ ) ξ ( | X z ( τ ) | s | X z ( τ ) | s ) G z ( τ ) dudv − Z ∂ F Y H z ( τ ) | X z ( τ ) | s | X z ( τ ) | s G z ( τ ) dτ. Since there are no poles on F Y , the analytic continuation of the second summand is given by justplugging in ( s , s ) = (0 , Z ∂ F Y H z ( τ ) G z ( τ ) dτ = − Z H z ( u + iY ) G z ( u + iY ) du, which cancels with (4.7). Therefore the contribution from the cusp i ∞ vanishes.We are left to compute the analytic continuation of the first summand of (4.8). For this, weclosely follow the proof of Theorem 6.1 in [7]. For δ > ̺ ∈ H , we let B δ ( ̺ ) denote theclosed disc of radius δ around ̺ and split the domain of integration into B δ ( z ) ∩ F , B δ ( z ) ∩ F ,and F \ ( B δ ( z ) ∪ B δ ( z )). The integral over F \ ( B δ ( z ) ∪ B δ ( z )), away from the poles, vanishes at( s , s ) = (0 , B δ ( z ) (resp. B δ ( z )) we can plug in s = 0 (resp. s = 0). By construction of F , the points z and z lie in the interior of Γ z F , resp. Γ z F . For ̺ ∈ { z, z } , we decompose B δ ( ̺ ) = [ M ∈ Γ ̺ M ( B δ ( ̺ ) ∩ F )to write − Z B δ ( ̺ ) ∩F H z ( τ ) ξ ( | X ̺ ( τ ) | s ) G z ( τ ) dudv = − w ̺ Z B δ ( ̺ ) H z ( τ ) ξ ( | X ̺ ( τ ) | s ) G z ( τ ) dudv. Now using ξ (cid:0) | X ̺ ( τ ) | s (cid:1) = − sη | X ̺ ( τ ) | s − X ̺ ( τ )( τ − ̺ ) , we have to compute(4.9) Res s =0 ηw ̺ Z B δ ( ̺ ) G z ( τ ) | X ̺ ( τ ) | s − X ̺ ( τ )( τ − ̺ ) H z ( τ ) dudv ! for ̺ ∈ { z, z } . Plugging in the elliptic expansions (4.4) and (4.5) around ̺ , we obtain4 ηw ̺ Z B δ ( ̺ ) H z ( τ ) | X ̺ ( τ ) | s − X ̺ ( τ )( τ − ̺ ) G z ( τ ) dudv = 4 ηw ̺ Z B δ ( ̺ ) − δ z,̺ w z ηπ X ̺ ( τ ) − + X n ≥ a z,̺ ( n ) X ̺ ( τ ) n | X ̺ ( τ ) | s − X ̺ ( τ )( τ − ̺ ) ( τ − ̺ ) × − w z π log( | X ̺ ( τ ) | ) + X n ≥ A + z ,̺ ( n ) X ̺ ( τ ) n + X n> A − z ,̺ ( n ) X ̺ ( τ ) n dudv We substitute X ̺ ( τ ) = Re ( θ ) and use η | τ − ̺ | dudv = 2 πR dθdR to obtain2 πηw ̺ Z δ Z − δ z,̺ w z ηπ + X n ≥ a z,̺ ( n ) R n +1 e (( n + 1) θ ) R s − × − δ z ,̺ w z π log( R ) + X n ≥ A + z ,̺ ( n ) R n e ( nθ ) + X n> A − z ,̺ ( n ) R n e ( − nθ ) dθdR = Z δ δ z,̺ δ z ,̺ w z w z πw ̺ log( R ) − δ z,̺ w z w ̺ A + z ,̺ (0) + 2 πηw ̺ X n ≥ A − z ,̺ ( n + 1) a z,̺ ( n ) R n +2 R s − dR = δ z,̺ δ z ,̺ w z π Z δ log( R ) R s − dR − δ z,̺ A + z ,̺ (0) δ s s + 2 πηw ̺ X n ≥ A − z ,̺ ( n + 1) a z,̺ ( n ) δ n + s +1) n + s + 1) . The last sum is analytic in s = 0 and we have Z δ log( R ) R s − dR = δ s log( δ )2 s − δ s s = − s + O (1) , so only the second term contributes to the residue in (4.9), yielding the statement. (cid:3) Proof of Theorem 1.5.
It follows from Lemmas 3.2 and 4.2 that h f d , f δ i = X Q ∈ Q d/ SL2( Z ) Q∈ Q δ/ SL2( Z ) w z Q w z Q (cid:10) H z Q , H z Q (cid:11) = − X Q ∈ Q d/ SL2( Z ) Q∈ Q δ/ SL2( Z ) w z Q w z Q A + z Q ,z Q (0) . (i) Note that if two quadratic forms Q , Q have the same CM-point, then one has to be aninteger multiple of the other. The factor has to be q δd . So conversely, if δd is not a square,then we have A + z Q ,z Q (0) = − π log (cid:18) | J ( z Q ) − J ( z Q ) | wQw Q (cid:19) IEBUR-POINCAR´E SERIES AND TRACES OF SINGULAR MODULI 17 for any Q ∈ Q d and Q ∈ Q δ by the first case of (4.6).(ii) By the same argument as above, if neither d nor d is a square, then neither ρ nor i is aCM-point of any quadratic form of discriminant d . Thus by the first two cases of (4.6), wehave for any Q = Q ∈ Q d A + z Q ,z Q (0) = − π log | J ( z Q ) − J ( z Q ) | and A + z Q ,z Q (0) = − π log (cid:12)(cid:12) z Q ) J ′ ( z Q ) (cid:12)(cid:12) = − π log (cid:12)(cid:12)(cid:12)(cid:12)p | d | J ′ ( z Q ) Q (1 , (cid:12)(cid:12)(cid:12)(cid:12) . (iii) This follows directly from the last two cases of A + z Q ,z Q (0) given in (4.6). (cid:3) References [1] M. Abramowitz and I. Stegun,
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