ARITHMETIC PROPERTIES OF COEFFICIENTS OF HALF-INTEGRAL WEIGHT MAASS-POINCARÉ SERIES

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1 ARITHMETIC PROPERTIES OF COEFFICIENTS OF HALF-INTEGRAL WEIGHT MAASS-POINCARÉ SERIES KATHRIN BRINGMANN AND KEN ONO 1 Introdution and Statement of Results Let j(z be the usual modular funtion for SL (Z j(z = q q q +, where q = e πiz The values of modular funtions suh as j(z at imaginary quadrati arguments in H, the upper half of the omplex plane, are known as singular moduli Singular moduli are algebrai integers whih play many roles in number theory For example, they generate lass fields of imaginary quadrati fields, and they parameterize isomorphism lasses of ellipti urves with omplex multipliation We shall slightly abuse terminology by referring to the value of any modular invariant at an imaginary quadrati argument as a singular modulus In an important paper [3], Zagier gave a new proof of Borherds famous theorem on the infinite produt expansions of integer weight modular forms on SL (Z with Heegner divisor This proof, as well as all of the results of [3], are onneted to his beautiful observation that the generating funtions for traes of singular moduli are essentially weight 3/ weakly holomorphi modular forms Remark The observation that oeffiients of ertain automorphi forms an be expressed in terms of singular moduli is not entirely new Earlier works by Maass [17], and Katok and Sarnak [13] ontain suh results in different ontexts Zagier s results have inspired an astonishing number of reent works (for example, see [1, 3, 6, 7, 8, 9, 11, 1, 14, 15, 1, ] In view of the importane of his paper, ombined with the fat that he only provides skethes of proofs for some of the key theorems (eg Theorems 6, 9, 10, 11, here we systematially revisit his work from the ontext of Maass-Poinaré series Our uniform approah inludes these key theorems as speial ases of orollaries of a single theorem (ie Theorem 1, and, as an added bonus, gives exat formulas for traes of singular moduli Date: August 3, Mathematis Subjet Classifiation 11F30, 11F37 The authors thank the National Siene Foundation for their generous support The seond author is grateful for the support of a Pakard, a Romnes, and a Guggenheim Fellowship 1

2 KATHRIN BRINGMANN AND KEN ONO To make Zagier s results more preise, we first reall some definitions and fix notation For integers λ, let M! be the omplex vetor spae of weight λ + 1 weakly λ+ 1 holomorphi modular forms on Γ 0 (4 satisfying the Kohnen plus-spae ondition Reall that a meromorphi modular form is said to be weakly holomorphi if its poles (if there are any are supported at the usps, and it is said to satisfy Kohnen s plusspae ondition if its Fourier expansion has the form (11 a(nq n ( 1 λ n 0,1 (mod 4 Let M + (resp S + be the subspae of M! onsisting of those forms whih are λ+ 1 λ+ 1 λ+ 1 holomorphi modular forms (resp usp forms Throughout, let d 0, 3 (mod 4 be a positive integer (so that d is the disriminant of an order in an imaginary quadrati field, and let H(d be the Hurwitz- Kroneker lass number for the disriminant d Let Q d be the set of positive definite integral binary quadrati forms (note inluding imprimitive forms, if there are any Q(x, y = [a, b, ] = ax + bxy + y with disriminant d = b 4a For eah Q, let τ Q be the unique omplex number in the upper half-plane whih is a root of Q(x, 1 = 0 The singular modulus f(τ Q, for any modular invariant f(z, depends only on the equivalene lass of Q under the ation of Γ := PSL (Z If ω Q {1,, 3} is given by if Q Γ [a, 0, a], ω Q := 3 if Q Γ [a, a, a], 1 otherwise, then, for a modular invariant f(z, define the trae Tr(f; d by (1 Tr(f; d := f(τ Q ω Q Q Q d /Γ In this notation, Theorems 1 and 5 of [3] imply the following Theorem (Zagier If f(z Z[j(z] has a Fourier expansion with onstant term 0, then there is a finite prinipal part A f (z = n 0 a f(nq n for whih A f (z + 0<d 0,3 (mod 4 Tr(f; dq d M! 3 Remark For brevity, we do not give a preise desription of A f (z It is easily determined using Theorems 1 and 5 of [3]

3 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 3 Example If η(z := q 1 4 n=1 (1 qn is Dedekind s eta-funtion, then for f(z = j(z 744 we have (13 η(z ( n=1 v n v3 q 4n η(zη(4z 6 = q <d 0,3 (mod 4 Tr(f; dq d = q q q 4 + M! 3 Remark Using Poinaré series onstruted in [7], fats about non-holomorphi modular forms due to Niebur [19], and fats about half-integral weight Kloosterman-Salié sums, Duke [8] and Jenkins [11] have provided new proofs of Theorems 1 and 5 of [3] Remark Kim [14, 15] has established the modularity for traes of singular moduli on ertain genus zero ongruene subgroups Using theta lifts, Bruinier and Funke [6] have reently proven a more general theorem whih holds for modular funtions on modular urves of arbitrary genus In all of these ases, the orresponding generating funtions are weight 3/ weakly holomorphi modular forms In [3], Zagier inludes several generalizations of these results Here we highlight two of these; the first onerns twisted traes of singular moduli For fundamental disriminants D 1, let χ D1 denote the assoiated genus harater for positive definite binary quadrati forms whose disriminants are multiples of D 1 (see (3 If λ is an integer and D is a non-zero integer for whih ( 1 λ D 0, 1 (mod 4 and ( 1 λ D 1 D < 0, then define the twisted trae of a modular invariant f(z, say Tr D1 (f; D, by (14 Tr D1 (f; D := Q Q D1 D /Γ χ D1 (Qf(τ Q ω Q Zagier proved that these traes are also given by oeffiients of weight 3/ modular forms (see Theorem 6 of [3] The seond generalization is disussed in the last setion of [3] There Zagier onsiders Tr(f; d for speial non-holomorphi modular funtions f(z In these ases, the orresponding generating funtions have weight λ+ 1, where λ { 6, 4, 3,, 1} (see Theorems 10 and 11 of [3] We expliitly represent the oeffiients of ertain half-integral weight Maass-Poinaré series as traes of singular moduli This result (see Theorem 1 inludes the results of Zagier desribed above, and, as an added bonus, gives exat formulas for these traes Apart from the alulations required to ompute the Fourier expansions of these series, the proofs require little more than lassial fats about Kloosterman and Salié sums To prove all of his results, Zagier instead depends on the existene of speial bases These good bases our in pairs, and they satisfy a striking duality A for M! λ+ 1 prominent example of this phenomenon pertains to the spaes M! 1 and M! 3 There is

4 4 KATHRIN BRINGMANN AND KEN ONO a natural infinite basis {F 1 ( 1; z, F 1 ( 4; z, F 1 ( 5; z, } for M! 3, and the first few oeffiients of these series are (15 F 1 ( 1; z = q 1 +48q 3 49q q 7, F 1 ( 4; z = q 4 675q q q 7, F 1 ( 5; z = q q q q 7 Similarly, there is a basis {F 0 (0; z, F 0 ( 3; z, F 0 ( 4; z, } for M! 1, and the first few oeffiients of these forms are (16 F 0 (0; z = 1 +q +q 4 +0q 5 +, F 0 ( 3; z = q 3 48q +675q q 5 +, F 0 ( 4; z = q 4 +49q q q 5 +, F 0 ( 7; z = q q q q 5 + A brief inspetion reveals a striking pattern relating the oeffiients of the F 1 ( m; z, grouped by olumn, and the oeffiients of the individual forms F 0 ( n; z Zagier proved (see Theorem 4 of [3], for non-negative integers n, that the nth oeffiient of a form F 1 ( m; z is the negative of the mth oeffiient of F 0 ( n; z Zagier s examples turn out to be speial ases of generi duality whih holds for the half-integral weight Maass-Poinaré series onsidered here (see Theorem 11 To onstrut these series, let k := λ + 1, where λ is an arbitrary integer, and let M ν, µ (z be the usual M-Whittaker funtion (for example, see Chapter 4 of [] whih is a solution to the differential equation u z + ( 14 + νz µ z u = 0 Following Bruinier [4] (note see [4] and [10] for bakground on Poinaré series and [7], for s C and y R {0} we define M s (y := y k M k sgn(y, s 1 ( y Suppose that m 1 is an integer with ( 1 λ+1 m 0, 1 (mod 4 Define ϕ m,s (z by ϕ m,s (z := M s ( 4πmye( mx, where z = x + iy, and e(w := e πiw Furthermore, let { ( } 1 n Γ := ± : n Z 0 1 denote the translations within SL (Z Using this notation, define the Poinaré series (17 F λ ( m, s; z := (ϕ m,s k A(z A Γ \Γ 0 (4 for Re(s > 1 (see Setion for the definition of k

5 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 5 Remark The Fourier expansions of these series were previously omputed in [7] for integers λ 1 If pr λ is Kohnen s projetion operator (see page 50 of [16] to the weight λ + 1 plus-spae for Γ 0 (4, then for λ {0, 1} define F λ ( m; z by 3 F ( λ m, k ; z pr λ if λ, (18 F λ ( m; z := 3 F ( (1 kγ(1 k λ m, 1 k ; z pr λ if λ 1 Remark Stritly speaking, Kohnen defined pr λ for holomorphi half-integral weight forms It is easy to see that its definition applies for weakly holomorphi forms Remark For λ 1, the normalization above is hosen so that (19 and (110 below are valid If λ = 0 or 1, then we also have Poinaré series F λ ( m; z, but the onstrution requires more are (see Setion If λ 6 with λ 5, then F λ ( m; z M! λ+ 1 For suh λ, it turns out F λ ( m; z has an expansion of the form (19 F λ ( m; z = q m + Remark If λ = 1 and m = 1, then n 0 ( 1 λ n 0,1 (mod 4 b λ ( m; nq n M! λ+ 1 F 1 ( 1; z = q q q 4 + M! 3 is the modular form in (13 Furthermore, when λ {0, 1}, these modular forms oinide with those given in (15 and (16 If λ = 5 or λ 7, then F λ ( m; z is a weak Maass form of weight λ + 1, and has a Fourier expansion of the form (110 F λ ( m; z = B λ ( m; z + q m + b λ ( m; nq n, n 0 ( 1 λ n 0,1 (mod 4 where B λ ( m; z is the non-holomorphi part of F λ ( m; z desribed in Setion The duality illustrated by (15 and (16 is a speial ase of the following general result whih holds for all of the F λ ( m; z Theorem 11 Assume the notation above Suppose that λ 1, and that m is a positive integer for whih ( 1 λ+1 m 0, 1 (mod 4 For every positive integer n with ( 1 λ n 0, 1 (mod 4 we have b λ ( m; n = b 1 λ ( n; m Remark Theorem 11 for λ = 1 (resp λ {, 3, 4, 5, 7} is Theorem 4 of [3] (resp Theorem 9 of [3]

6 6 KATHRIN BRINGMANN AND KEN ONO Remark As an alternative approah, one an derive Theorem 11, for λ, by applying Proposition 116 of [4], and by interpreting the relevant forms as vetor valued forms on Mp (Z Motivated by Zagier s results, we now desribe the oeffiients of the F λ ( m; z in terms of singular moduli To state these results, we require the following Poinaré series defined by Niebur [19] Throughout, let I s (x denote the usual I-Bessel funtion, the so-alled modified Bessel funtion of the first kind If λ > 1, then let (111 F λ (z := π A Γ \SL (Z Im(Az 1 Iλ 1(πIm(Aze( Re(Az Remark For λ = 1, Niebur s [19] definition requires a areful argument involving analyti ontinuation It turns out that F 1 (z = 1 (j(z 744, and this is the ontinuation, as s 1 from the right, of the expansion 1 + π Im(Az 1 Is 1(πIm(Aze( Re(Az A Γ \SL (Z The oeffiients of F λ ( m; z are traes and twisted traes of the singular moduli for F λ (z In view of Theorem 11 on duality, to state this result we may without loss of generality assume that λ 1 Theorem 1 If λ, m 1 are integers for whih ( 1 λ+1 m is a fundamental disriminant (note whih inludes 1, then for eah positive integer n with ( 1 λ n 0, 1 (mod 4 we have b λ ( m; n = ( 1[(λ+1/] n λ 1 m λ Tr ( 1 λ+1 m (F λ ; n Three remarks 1 For λ = 1, Theorem 1 relates b 1 ( m; n to traes and twisted traes of F 1 (z = 1(j(z 744 Therefore, if λ = 1 and m = 1, then this is Theorem 1 of [3] (ie example (13, and for general m is Theorem 6 of [3] For λ > 1, the oeffiients b λ ( m; n are traes of singular moduli of non-harmoni Maass forms This relates to Theorem 11 of [3] where these non-harmoni Maass forms are images of weakly holomorphi modular forms under differential operators whih have the additional property that their singular moduli are algebrai This phenomenon is explained in more detail in a reent preprint of Miller and Pixton [18] 3 Although we do not inlude the details here for brevity, we note that Theorem 1 has a straightforward generalization involving the ation of the half-integral weight Heke operators on F λ ( m; z For λ = 1, this generalization is Theorem 5 of [3] To prove Theorem 1, we first reall and derive the Fourier expansions of the half-integral weight series onsidered here This is done in Setion These Fourier expansions are desribed in terms of the half-integral weight Kloosterman sums In

7 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 7 Setion 3, we give standard fats about Kloosterman sums and Salié sums In partiular, we reall how Kloosterman sums are related to ertain Salié sums, and how suh sums may be reformulated as Poinaré-type series over orbits of CM points In Setion 3 we prove Theorem 11 using the formulas obtained in Setion In Setion 4 we desribe Niebur s Poinaré series F λ (z, and explain their relation to weakly holomorphi modular forms In the last setion we use the results of Setions and 3 to diretly prove Theorem 1 Remark Although we have not arried out the details, it is not diffiult to generalize Theorems 11 and 1 to arbitrary ongruene subgroups Γ 0 (4N, where N is odd and square-free Aknowledgements The authors thank Sott Ahlgren, Matt Boylan, Jan Bruinier, Winfried Kohnen, Alison Miller, Aaron Pixton and Jeremy Rouse for their omments on an earlier version of this paper Half-integral weight Maass-Poinaré series Here we reall and derive the Fourier expansions of the series F λ ( m; z The ommon feature of these series is that their Fourier oeffiients for positive exponents are given as expliit infinite sums in half-integral weight Kloosterman sums weighted by Bessel funtions To define these Kloosterman sums, for odd δ let { 1 if δ 1 (mod 4, (1 ɛ δ := i if δ 3 (mod 4 If λ is an integer, then we define the λ + 1 weight Kloosterman sum K λ(m, n, by K λ (m, n, := ( ( m v + nv ( ɛ λ+1 v e v v (mod In the sum, v runs through the primitive residue lasses modulo, and v denotes the multipliative inverse of v modulo Here we give the Fourier expansions of the Maass-Poinaré series F λ ( m; z For λ 1, these expansions are omputed in [7] For ompleteness, we give the following result giving formulas for the oeffiients b λ ( m; n for all λ For onveniene, we define δ,m {0, 1} by { 1 if m is a square, (3 δ,m := 0 otherwise,

8 8 KATHRIN BRINGMANN AND KEN ONO and we let (4 δ odd (v := { 1 if v is odd, 0 otherwise Assuming the notation of (19 and (110, we have the following theorem Theorem 1 Suppose that λ is an integer, and suppose that m is a positive integer for whih ( 1 λ+1 m 0, 1 (mod 4 Furthermore, suppose that n is a non-negative integer for whih ( 1 λ n 0, 1 (mod 4 (1 If λ, then b λ ( m; 0 = 0, and for positive n we have b λ ( m; n = ( 1 [(λ+1/] π (n/m λ 1 4 (1 ( 1 λ i (1 + δ odd (/4 K λ( m, n, ( If λ 1, then >0 0 (mod 4 I λ 1 ( 4π mn b λ ( m; 0 = ( 1 [(λ+1/] π 3 λ 1 λ m 1 λ (1 ( 1 λ i 1 ( 1 λγ(1 λ (1 + δ odd (/4 K λ( m, 0,, 3 λ and for positive n we have >0 0 (mod 4 >0 0 (mod 4 b λ ( m; n = ( 1 [(λ+1/] π (n/m λ 1 4 (1 ( 1 λ i (1 + δ odd (/4 K λ( m, n, ( 4π mn I1 λ (3 If λ = 1, then b 1 ( m; 0 = δ,m, and for positive n we have b 1 ( m; n = 4δ,m H(n π (n/m 1 4 (1 + i >0 0 (mod 4 (1 + δ odd (/4 K 1( m, n, (4 If λ = 0, then b 0 ( m; 0 = 0, and for positive n we have b 0 ( m; n = 4δ,n H(m + π (m/n 1 4 (1 i (1 + δ odd (/4 K 0( m, n, >0 0 (mod 4 I1 I1 ( 4π mn ( 4π mn

9 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 9 Remark For positive integers m and n, the formulas for b λ ( m; n are nearly uniform in λ In partiular, the only differene between Theorem 1 (1 and ( ours in the I-Bessel funtion fator For λ we have I λ 1, and for λ 1 we have I1 λ instead Before we prove this result, we first reall the onstrution ( of these forms Suppose α β that λ is an integer, and that k := λ + 1 For eah A = Γ γ δ 0 (4, let ( γ j(a, z := ɛ 1 δ (γz + δ 1 δ be the usual fator of automorphy for half-integral weight modular forms If f : H C is a funtion, then for A Γ 0 (4 we let (5 (f k A (z := j(a, z λ 1 f(az As usual, let z = x + iy be the standard variable on H For s C and y R {0}, we let (6 M s (y := y k M k sgn(y, s 1 ( y, where M ν,µ (z is the standard M-Whittaker funtion whih is a solution to the differential equation u ( z νz µ u = 0 z If m is a positive integer, then define ϕ m,s (z by and onsider the Poinaré series (7 F λ ( m, s; z := ϕ m,s (z := M s ( 4πmye( mx, A Γ \Γ 0 (4 (ϕ m,s k A(z It is easy to verify that ϕ m,s (z is an eigenfuntion, with eigenvalue (8 s(1 s + (k k/4, of the weight k hyperboli Laplaian k := y ( x + y + iky ( x + i y ( Sine ϕ m,s (z = O y Re(s k as y 0, it follows that F λ ( m, s; z onverges absolutely for Re(s > 1, is a Γ 0 (4-invariant eigenfuntion of the Laplaian, and is real analyti These series provide examples of weak Maass forms of half-integral weight Following Bruinier and Funke [5], we make the following definition

10 10 KATHRIN BRINGMANN AND KEN ONO Definition A weak Maass form of weight k for the group Γ 0 (4 is a smooth funtion f : H C satisfying the following: (1 For all A Γ 0 (4 we have (f k A(z = f(z ( We have k f = 0 (3 The funtion f(z has at most linear exponential growth at all the usps Remark If a weak Maass form f(z is holomorphi on H, then it is a weakly holomorphi modular form In view of (8, it follows that the speial s-values at k/ and 1 k/ of F λ ( m, s; z are weak Maass forms of weight k = λ + 1 when the defining series is absolutely onvergent If λ {0, 1} and m 1 is an integer for whih ( 1 λ+1 m 0, 1 (mod 4, then we reall the definition 3 F ( λ m, k ; z pr λ if λ, (9 F λ ( m; z := 3 F ( (1 kγ(1 k λ m, 1 k ; z pr λ if λ 1 By the disussion above, it follows that F λ ( m; z is a weak Maass form of weight k = λ + 1 on Γ 0(4 If λ = 1 and m is a positive integer for whih m 0, 1 (mod 4, then define F 1 ( m; z by (10 F 1 ( m; z := 3 F 1 ( m, 34 ; z pr 1 + 4δ,m G(z The funtion G(z is given by the Fourier expansion G(z := where H(0 = 1/1 and H(nq n π y n=0 β(s := 1 n= t 3 e st dt β(4πn yq n, Proposition 36 of [7] proves that eah F 1 ( m; z is in M! 3 These series form the basis given in (15 Remark Note that the integral β(s is easily reformulated in terms of the inomplete Gamma-funtion We make this observation sine the non-holomorphi parts of the F λ ( m; z, for λ 7 and λ = 5, will be desribed in suh terms

11 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 11 Remark We may define the series F 0 ( m; z M! 1 using an argument analogous to Proposition 36 of [7] Instead, we simply note that the existene of the basis (16 of M! 1, together with the duality of Theorem 4 [3] and an elementary property of Kloosterman sums (see Proposition 31, gives a diret realization of the Fourier expansions of F 0 ( m; z in terms of the expansions of the F 1 ( n; z desribed above To ompute the Fourier expansions of these weak Maass forms, we require some further preliminaries For s C and y R {0}, we let (11 W s (y := y k W k sgn(y, s 1 ( y, where W ν,µ denotes the usual W-Whittaker funtion For y > 0, we shall require the following relations: (1 M k( y = e y, (13 W 1 k(y = W k(y = e y, and (14 W 1 k( y = W k( y = e y Γ (1 k, y, where Γ(a, x := x e t t adt t is the inomplete Gamma funtion For z C, the funtions M ν,µ (z and M ν, µ (z are related by the identity W ν,µ (z = From these fats, we easily find, for y > 0, that Γ( µ Γ(µ Γ( 1 µ νm ν,µ(z + Γ( 1 + µ νm ν, µ(z (15 M 1 k( y = (k 1e y y Γ(1 k, y + (1 kγ(1 ke Proof of Theorem 1 Although the onlusions of Theorem 1 (1 and (3 were obtained previously by Bruinier, Jenkins and the seond author in [7], for ompleteness we onsider the alulation for general λ {0, 1} In partiular, for suh λ, suppose that m is a positive integer for whih ( 1 λ+1 m 0, 1 (mod 4 Computing the Fourier expansion requires alulating the integral z k M s ( 4πm y z e ( mx z nx dx This integral is omputed on page 357 of [10] (see also page 33 of [4], and it implies that F λ ( m, s; z has a Fourier expansion of the form F λ ( m, s; z = M s ( 4πmye( mx + n Z (n, y, se(nx,

12 1 KATHRIN BRINGMANN AND KEN ONO where the oeffiients (n, y, s are given by ( πi k Γ(s n λ Γ(s k 1 4 K λ ( m, n, J s 1 m πi k Γ(s Γ(s + k (n/m λ >0 0 (mod >0 0 (mod 4 K λ ( m, n, λ π 3 4 +s λ i k m s λ 1 4y 3 4 s λ Γ(s 1 Γ(s + k Γ(s k >0 0 (mod 4 4π mn W s (4πny, n < 0, ( 4π mn I s 1 W s (4πny, n > 0, K λ ( m, 0, s, n = 0 Here J s (x denotes the usual J-Bessel funtion, the so-alled Bessel funtion of the first kind The Fourier expansion defines an analyti ontinuation of F λ ( m, s; z to Re(s > 3/4 For λ, we then find that F λ ( m, k ; z is a weak Maass form of weight λ + 1 Thanks to the Γ-fator, the Fourier oeffiients (n, y, s vanish for negative n, and so eah F λ ( m, k ; z is a weakly holomorphi modular form on Γ 0(4 Applying Kohnen s projetion operator (see page 50 of [16] to these series gives Theorem 1 (1 The situation is a little more ompliated if λ 1 Arguing as above, by letting s = 1 k/ we obtain a weak Maass form F λ ( m, 1 k; z of weight k = λ + 1 on Γ 0 (4 Using (14 and (15, we find that its Fourier expansion has the form ( F λ m, 1 k ; z (16 = (k 1 (Γ(1 k, 4πmy Γ(1 k q m + (n, ye(nz, n Z where the oeffiients (n, y, for n < 0, are given by πi k (1 k n m λ 1 4 Γ(1 k, 4π n y >0 0 (mod 4 K λ ( m, n, J1 λ ( 4π mn For n 0, (13 allows us to onlude that the (n, y are given by ( πi k Γ( k(n/m λ K 1 λ ( m, n, 4π 4 I1 λ λ 3 π λ i k m 1 λ >0 0 (mod 4 >0 0 (mod 4 mn, n > 0, K λ ( m, 0, 3 λ n = 0

13 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 13 The numbers b λ ( m; n are the Fourier oeffiients of the images of the forms above under Kohnen s projetion operator pr λ One readily heks that this returns the desired formulas Remark If λ { 6, 4, 3,, 1}, then the funtions F λ ( m; z are in M!, and λ+ 1 their q-expansions are of the form (17 F λ ( m; z = q m + b λ ( m; nq n n 0 ( 1 λ n 0,1 (mod 4 This laim is equivalent to the vanishing of the non-holomorphi terms appearing in the proof of Theorem 1 for these λ To justify this vanishing, reall that there is an anti-linear differential operator ξ k that takes weak Maass forms of weight k to weakly holomorphi modular forms of weight k (see Proposition 3 of [5] It is defined by (18 ξ k (f(z := iy k z f(z This operator has the property that ker(ξ k is the subset of weight k weak Maass forms whih are weakly holomorphi ( modular forms To prove our laim, apply ξ k to the Fourier expansion of F λ m, 1 k ; z given by (16 Sine ξ k (f = 0 for holomorphi f, and sine ξ k is anti-linear, the non-trivial ontributions an only arise from ξ k (Γ (1 k, 4π n y, where n is a negative integer For negative n it is easy to show that ξ k (Γ(1 k, 4π n y = (4π n 1 k e 4πny Therefore ( ( ξ k F λ m, 1 k ; z = (nq n, n>0 where (n, for n m, is πi (λ+1 ( 1 λ (n/m λ 1 4 (4πn 1 λ and for n = m is ( ( 1 1 λ Γ λ (4πm 1 λ πi (λ+1 ( 1 λ (4πm 1 λ >0 0 (mod 4 >0 0 (mod 4 K λ ( m, n, K λ ( m, m, J1 λ ( 4π mn, J1 λ ( 4π m ( ( From these alulations it follows that ξ k Fλ m, 1 k ; z is a weight k usp form on Γ 0 (4 Sine S k! is trivial for these λ, it follows that (n = 0 for all n

14 14 KATHRIN BRINGMANN AND KEN ONO Consequently, the oeffiients for n < 0, with the exeption of n = m must vanish For n = m, the normalization given by (18 now onfirms (17 3 Kloosterman and Salié sums and the proof of Theorem 11 Here we give lassial fats onerning half-integral weight Kloosterman sums and Salié sums We reall how suh sums are related, and give a reformulation of ertain Salié sums as Poinaré-type series over CM points However, we begin by using the formulas from Theorem 1 to prove Theorem Proof of Theorem 11 Here we prove Theorem 11 using the expliit formulas ontained in Theorem 1 Thanks to these formulas, duality follows from the following elementary proposition Proposition 31 Suppose that > 0 is a multiple of 4 If λ, m, and n are integers, then K λ ( m, n, = ( 1 λ i K 1 λ ( n, m, Proof For v oprime to, we have that ( ( v = v, and v v (mod 4, and so K λ ( m, n, = ( ( mv + nv ɛ λ+1 v e = v v (mod (mod v ( v ɛ λ+1 v e ( nv + mv Sine ɛ λ+1 v = ( 1 λ iɛv (1 λ+1, it follows that K λ ( m, n, = ( 1 λ i K 1 λ ( n, m, Proof of Theorem 11 In Theorem 1, it is lear that the I-Bessel funtion fators exatly orrespond for λ and 1 λ Furthermore, when λ = 1 and 1 λ = 0, Theorem 1 (3 and (4 shows that the lass number summands obey the alleged duality To omplete the proof one simply observes that we may transform the formula for b λ ( m; n into the formula for b 1 λ ( n; m Thanks to Proposition 31, this is ahieved by replaing, for eah, the Kloosterman sum K λ ( m, n, by ( 1 λ i K 1 λ ( n, m, 3 Fats about Kloosterman sums and Salié sums The results in this paper are derived from the lassial fat that the half-integral weight Kloosterman sums are easily desribed in terms of simpler sums, the Salié sums Suppose that D 1 0, 1 (mod 4 is a fundamental disriminant Reall that 1 is onsidered to be a fundamental disriminant If λ is an integer, D 0 is an integer for whih ( 1 λ D 0, 1 (mod 4,

15 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 15 and N is a positive multiple of 4, then define the generalized Salié sum S λ (D 1, D, N by ( ( N (31 S λ (D 1, D, N := χ D1 4, x, x ( 1 λ D 1 D x e, N N x (mod N x ( 1 λ D 1 D (mod N where χ D1 (a, b,, for a binary quadrati form Q = [a, b, ], is given by (3 { 0 if (a, b,, D 1 > 1, χ D1 (a, b, := ( D1 r if (a, b,, D 1 = 1 and Q represents r with (r, D 1 = 1 Remark If D 1 = 1, then χ D1 is trivial Therefore, if ( 1 λ D 0, 1 (mod 4, then ( x S λ (1, D, N = e N x (mod N x ( 1 λ D (mod N Half-integral weight Kloosterman sums are essentially equal to suh Salié sums Relations of this type are well known, and they play fundamental roles throughout the theory of half-integral weight modular forms (for example, in the work of Iwanie, and later work of Duke, bounding oeffiients of half-integral weight usp forms Here we reall a speial ase of suh relations (for example, see Proposition 5 of [16] Proposition 3 Suppose that N is a positive multiple of 4, and that D 1 is a fundamental disriminant If λ is an integer, and D is a non-zero integer for whih ( 1 λ D 0, 1 (mod 4, then N 1 (1 ( 1 λ i(1 + δ odd (N/4 K λ (( 1 λ D 1, D, N = S λ (D 1, D, N Using Proposition 3 and Theorem 1, we may rewrite the oeffiients of the halfintegral weight Maass-Poinaré series in terms of the simpler looking Salié sums This simple reformulation, ombined with the next proposition, underlies and explains the general phenomenon in whih oeffiients of half-integral weight modular forms are desribed as traes of singular moduli The following proposition, well known to experts, desribes these Salié sums themselves as Poinaré series over CM points Proposition 33 Suppose that λ is an integer, and that D 1 is a fundamental disriminant If D is a non-zero integer for whih ( 1 λ D 0, 1 (mod 4 and ( 1 λ D 1 D < 0, then for every positive integer a we have S λ (D 1, D, 4a = Q Q D1 D /Γ χ D1 (Q ω Q A Γ \SL (Z D1 D Im(Aτ Q= a e ( Re (Aτ Q

16 16 KATHRIN BRINGMANN AND KEN ONO Proof For every integral binary quadrati form Q(x, y = ax + bxy + y of disriminant ( 1 λ D 1 D, there is a unique point τ Q in the upper half of the omplex plane that is a root of Q(x, 1 = 0 Clearly τ Q is equal to τ Q = b + i D 1 D (33, a and the oeffiient b of Q solves the ongruene (34 b ( 1 λ D 1 D (mod 4a Conversely, every solution of (34 orresponds to a quadrati form with an assoiated CM point as above There is a one-to-one orrespondene between the solutions of b 4a = ( 1 λ D 1 D (a, b, Z, a, > 0 and the points of the orbits { } AτQ : A SL (Z/Γ τq, Q where Γ τq denotes the isotropy subgroup of τ Q in SL (Z, and where Q varies over the representatives of Q D1 D /Γ The group Γ preserves the imaginary part of suh a CM point τ Q, and preserves (34 However, it does not preserve the middle oeffiient b of the orresponding quadrati forms modulo 4a It identifies the ongruene lasses b, b +a (mod 4a appearing in the definition of S λ (D 1, D, 4a Sine χ D1 (Q is fixed under the ation of Γ, the orresponding summands for suh pairs of ongruene lasses are equal Proposition 33 follows easily sine #Γ τq = ω Q, and sine both Γ τq and Γ ontain the negative identity matrix 4 Modular invariant Poinaré series Here we reall the properties of the Poinaré series F λ (z whih are the modular invariants whose singular values determine the oeffiients of the Maass-Poinaré series F λ ( m; z These series were defined and investigated by Niebur in [19] Reall that they are defined by (41 π A Γ \SL (Z I Im(Az1 λ 1(πIm(Aze( Re(Az if λ > 1, F λ (z := 1 + π A Γ \SL (Z Im(Az1 I1(πIm(Aze( Re(Az if λ = 1 Stritly speaking, this definition is not well defined when λ = 1 sine the defining series is not absolutely onvergent A signifiant portion of Niebur s paper is devoted to defining F 1 (z via analyti ontinuation, and it turns out that F 1 (z = 1(j(z 744

17 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 17 For the remainder of this setion suppose that λ > 1 is an integer Sine we have y 1/ I λ 1/ (y = O ( y λ (y 0, it follows that the defining series for F λ (z is absolutely uniformly onvergent Sine the funtion f λ (z := πim(z 1 Iλ 1(πIm(ze( Re(z satisfies 0 (f λ (z = λ(1 λf λ (z, where ( 0 := y x +, y we have that 0 (F λ (z = (1 λλf λ (z Therefore, F λ (z is an eigenfuntion of the weight 0 hyperboli Laplaian 0, and has eigenvalue λ(1 λ Consequently, F λ (z is a weight 0 Maass form However, it is not a weak Maass form sine it is not harmoni Niebur [19] omputed the Fourier expansion of F λ (z, and he found that where F λ (z = πi λ 1(πye πy y 1 q 1 + b λ (n, yq n, n Z y 1 λ π λ+1 (λ 1(λ 1! >0 (4 b λ (n, y := πy 1/ e πny K λ 1(π n y >0 πy 1/ e πny K λ 1(π n y >0 S( 1, 0; λ n = 0, ( S(n, 1; 4π n I λ 1 ( S(n, 1; 4π n J λ 1 Here S(n, m; denotes the integer weight Kloosterman sum S(n, m; := ( nν + mν ν (mod e n > 0, n < 0 Remark The reader is warned that there are typographial errors in Niebur s formulas In Theorems 10 and 11 of [3], Zagier proved that the oeffiients of several halfintegral weight modular forms are traes of non-holomorphi modular forms Moreover, he notes that these forms are images of weakly holomorphi modular forms under standard differential operators Theorem 1 inludes these results thanks to Niebur s

18 18 KATHRIN BRINGMANN AND KEN ONO formulas (4 For these results, it turns out that the relevant weakly holomorphi modular forms (see [19] g λ (z of weight λ on SL (Z have Fourier expansions of the form (43 g λ (z = q 1 4( 1 λ+1 π λ+1 + Γ(λ 1 (λ 1(λ 1! >0 + π( 1 λ+1 n λ+1 n 1 >0 S(n, 1; S( 1, 0; λ I λ 1 ( 4π n q n A brief inspetion reveals that the same Kloosterman sum and I-Bessel fators appear in both (4 and (43 In a reent preprint [18], Miller and Pixton elaborate on these observations to obtain general theorems about traes of algebrai values of suitable non-holomorphi modular forms as oeffiients of Maass-Poinaré series These nonholomorphi modular forms, as in Theorem 11 of [3], are obtained by applying suitable differential operators to weakly holomorphi modular forms Example For ompleteness, we note that the modular forms g λ (z are simple to realize in terms of the lassial modular forms on SL (Z For example, we have that g 1 (z = j(z 70 = q q +, g (z = E 4(zE 6 (z (z = q q, g 3 (z = E 4(z (z = q q + Note that if λ = 1, then we have F 1 (z + 4 = g 1 (z If λ =, then F (z = 1 ( 1 πi z + 1 g (z πim(z 5 Proof of Theorem 1 Here we prove Theorem 1 using the definitions of the series F λ ( m; z and F λ (z, and Propositions 3 and 33 Proof of Theorem 1 Here we prove the ases where λ The argument when λ = 1 is idential For λ, Theorem 1 (1 implies that b λ ( m; n = ( 1 [(λ+1/] π (n/m λ 1 4 (1 ( 1 λ i (1 + δ odd (/4 K λ( m, n, >0 0 (mod 4 I λ 1 ( 4π mn

19 COEFFICIENTS OF HALF-INTEGRAL WEIGHT POINCARÉ SERIES 19 Using Proposition 3, where D 1 = ( 1 λ+1 m and D = n, for integers N = whih are positive multiples of 4, we have 1 (1 ( 1 λ i(1 + δ odd (/4 K λ ( m, n, = S λ (( 1 λ+1 m, n, These identities, ombined with the hange of variable = 4a, give b λ ( m; n = ( 1[(λ+1/] π (n/m λ S 1 λ (( 1 λ+1 m, n, 4a 4 I a λ 1 Using Proposition 33, this beomes a=1 b λ ( m; n = ( 1[(λ+1/] π (n/m λ 1 4 a=1 A Γ \SL (Z Im(Aτ Q = mn a Q Q nm/γ χ ( 1 λ+1 m(q ω Q ( π mn I λ 1(πIm(Aτ Q e( Re(Aτ Q a The definition of F λ (z in (111, ombined with the obvious hange of variable relating 1/ a to Im(Aτ Q 1, gives b λ ( m; n = ( 1[(λ+1/] n λ 1 m λ A Γ \SL (Z π Q Q nm/γ χ ( 1 λ+1 m(q ω Q Im(Aτ Q 1 Iλ 1(πIm(Aτ Q e( Re(Aτ Q a = ( 1[(λ+1/] n λ 1 m λ Tr ( 1 λ+1 m(f λ ; n Referenes [1] S Ahlgren and K Ono, Arithmeti of singular moduli and lass polynomials, Compositio Math 141 (005, pages [] G E Andrews, R Askey and R Roy, Speial funtions, Cambridge Univ Press, Cambridge, 1999 [3] M Boylan, -adi properties of Heke traes of singular moduli, Math Res Letters 1 (005, pages [4] J H Bruinier, Borherds produts on O(, l and Chern lasses of Heegner divisors, Springer Let Notes 1780, Springer-Verlag, Berlin, 00 [5] J H Bruinier and J Funke, On two geometri theta lifts, Duke Math J 15 (004, pages [6] J H Bruinier and J Funke, Traes of CM-values of modular funtions, J Reine Angew Math 594 (006, pages 1-33

20 0 KATHRIN BRINGMANN AND KEN ONO [7] J H Bruinier, P Jenkins, and K Ono, Hilbert lass polynomials and traes of singular moduli, Math Annalen 334 (006, pages [8] W Duke, Modular funtions and the uniform distribution of CM points, Math Annalen 334 (006, pages 41-5 [9] B Edixhoven, On the p-adi geometry of traes of singular moduli, Int J of Number Th 1, No 4 (005, pages [10] D A Hejhal, The Selberg trae formula for PSL (R, Springer Let Notes in Math 1001, Springer-Verlag, 1983 [11] P Jenkins, Kloosterman sums and traes of singular moduli, J Number Th, aepted for publiation [1] P Jenkins, p-adi properties for traes of singular moduli, Int J of Number Th, 1, No 1 (005, pages [13] S Katok and P Sarnak, Heegner points, yles and Maass forms, Israel J Math, 84, no 1- (1993, pages 19-7 [14] C H Kim, Borherds produts assoiated to ertain Thompson series, Compositio Math 140 (004, pages [15] C H Kim, Traes of singular values and Borherds produts, preprint [16] W Kohnen, Fourier oeffiients of modular forms of half-integral weight, Math Ann 71 (1985, pages [17] H Maass, Über die räumlihe Verteilung der Punkte in Gittern mit indefiniter Metrik, Math Ann 138 (1959, pages [18] A Miller and A Pixton, Arithmeti traes of non-holomorphi modular invariants, preprint [19] D Niebur, A lass of nonanalyti automorphi funtions, Nagoya Math J 5 (1973, pages [0] K Ono, The web of modularity: Arithmeti of the oeffiients of modular forms and q-series, CBMS Regional Conferene, 10, Amer Math So, Providene, R I, 004 [1] R Osburn, Congruenes for traes of singular moduli, Ramanujan J, aepted for publiation [] J Rouse, Zagier duality for the exponents of Borherds produts for Hilbert modular forms, JLondon Math So 73 (006, pages [3] D Zagier, Traes of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998 (00, Int Press Let Ser, 3, I, Int Press, Somerville, MA, pages Department of Mathematis, University of Wisonsin, Madison, Wisonsin address: bringman@mathwisedu address: ono@mathwisedu