INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS W. DUKE AND PAUL JENKINS ABSTRACT. We define traces associated to a weakly holomorphic modular form f of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of half-integral weight. If the coefficients of f are integral, then these traces are integral as well. We obtain a negative weight analogue of the classical Shintani lift and give an application to a generalization of the Shimura lift.. INTRODUCTION Recently there has been a resurgence of interest in the classical theory of singular moduli, these being the values of the modular j-function at quadratic irrationalities. This resurgence is due largely to the influential papers of Borcherds [], [] and Zagier [6]. The present paper arose from a suggestion made at the end of [6] to extend some of the results given there on traces of singular moduli to higher weights. One such generalization has been given recently by Bringmann and Ono [3], who provide an identity for the traces associated to certain Maass forms in terms of the Fourier coefficients of half-integral weight Poincaré series. However, it does not seem to be known when these traces are integral or even rational. Here we will identify the traces associated to a weakly holomorphic form f of negative integral weight with the coefficients of certain weakly holomorphic forms of half-integral weight and show that these coefficients are integral when the coefficients of f are integral. We will use this identification to obtain a negative weight analogue of the classical Shintani lift. We also give an application to Borcherds generalization of the Shimura lift to weakly holomorphic modular forms. Recall that a weakly holomorphic modular form of weight k, where k Z, is a holomorphic function f on the upper half-plane H that satisfies f k γ τ = cτ + d k f aτ+b = fτ cτ+d for all γ = a c d b Γ = PSL, Z and that has a q-expansion fτ = n anqn with an = 0 for all but finitely many n < 0; here, as usual, q = eτ = e πiτ. Let M k! denote the vector space of all weakly holomorphic modular forms of weight k. Similarly, for k = s + / with s Z let M k! denote the space of holomorphic functions on H that transform like θ k under Γ 0 4, have at most poles in the cusps, and have a q-expansion supported on integers n with s n 0, mod 4. Here, as usual, θτ = n Z qn. For any k let M k M k! denote the subspace of holomorphic forms and S k M k the subspace of cusp forms. In this paper d is always an integer with d 0, mod 4 and D is always a fundamental discriminant possibly. Suppose that dd < 0 and that F is a Γ-invariant function on H. Date: September 5, 008. Supported by National Science Foundation grants DMS-0355564, DMS-06037.

W. DUKE AND PAUL JENKINS Define the twisted trace Tr d,d F = Q w Q χqf τ Q, where the sum is over a complete set of Γ-inequivalent positive definite integral quadratic forms Qx, y = ax + bxy + cy with discriminant dd = b 4ac, and τ Q = b + dd a is the associated CM point. Here w Q = unless Q ax + y or Q ax + xy + y, in which case w Q = or 3 respectively, and { χd r, if a, b, c, D = and Q represents r, where r, D = ; 3 χq = χa, b, c = 0, if a, b, c, D >, where χ D is the Kronecker symbol. It is known that χ is well defined on classes, that χ restricts to a real character a genus character on the group of primitive classes, and that all such characters arise this way. For the usual j-function j = E 3 4/ M! 0 with Fourier expansion H jτ = q + 744 + 96884q + 493760q +..., it is classical that the value jτ Q is an algebraic integer in an abelian extension of Q dd. Let j = j 744. Zagier showed in [6] that for a fundamental discriminant D we have q D + d>0 d Trd,D j q d M! / if D < 0, q D D Tr d,d j q d M 3/! if D > 0, d<0 and that both forms have integral Fourier coefficients. For instance, when D = 3 and D = 5 we have the two weakly holomorphic forms q 3 48q + 675q 4 85995q 5 + M! / and q 5 + 85995q 3 565760q 4 + 5756480q 7 + M! 3/, and Tr 5, 3 j = Tr 3,5 j = j + 5 j + 5 = 85995 5. 4 In this paper we will give such a result when j is replaced by a function f of negative weight. To state it, first define the Maass raising operator k in τ = x + iy: 4 k = D k 4πy, where D = πi d dτ = q d dq. Now k f k γ = k f k+ γ for any γ PSL, R. Thus, if f M! s for s Z +, the function s f is Γ-invariant, where 5 s s 4 4 s s. After Maass we know that s f is an eigenfunction of the Laplacian = y + x y

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 3 with eigenvalue s s, so s f is a weak Maass form see e.g. [6, p.6] for a precise definition. As was shown in special cases in [6] by a method that readily generalizes, s f is a rational function of j and h = EE 4 E 6 /, where E τ = 4 n σnq n 3 πy σn = m n m is the nonholomorphic weight Eisenstein series. For a CM point like τ Q given in, it was shown by Ramanujan that hτ Q is algebraic [, eq. 3 p. 33]. More precisely, hτ Q Qjτ Q see [8, Thm. A, p. 4]. Using this, we can deduce the remarkable fact that for any f M! s with s and with rational Fourier coefficients, the singular value of the weak Maass form s fτ Q is algebraic. We are thus motivated to study Tr d,d s f for such f. For D a fundamental discriminant let ŝ = s if s D > 0 and ŝ = s otherwise. It is also convenient to set 6 Tr d,df = ŝ d ŝ D ŝ Tr d,d s f. Suppose that f M s! for s has Fourier coefficients an. For D fundamental, define the D th Zagier lift of f to be Z D fτ = a mm s ŝ χ D nnŝ q m D n + L s, χ Da0 + Trd,Dfq d. m>0 d: dd<0 The linear map f Z D f is a negative weight analogue of the Shintani lift on integral weight cusp forms. This follows from our main result, whose proof will be completed in Section 5. Theorem. Suppose that f M! s for an integer s. If D is a fundamental discriminant with s D > 0, we have that Z D f M! 3/ s, while if s D < 0, then Z D f M! s+/. If f has integral Fourier coefficients, then so does Z D f. Here we will not treat the case s =, which requires special considerations and which can be dealt with by the methods of [6]. Furthermore, when s =, 3, 4, 5, 7, Theorem can also be deduced from results of [6]. The first new example occurs when s = 6 and D =, where we have the pair fτ = E 4τ τ =q + 4q 96560 47709536 q + M! 0 Z fτ =q 4 + 56 q + 390 + 5360 q 3 + 464 q 4 + 6540 q 7 + M! 9/. Here ζ 5 96560 = 390 and the first few values of Tr d, f are 3 4 5 f + 3 = 5360 7 5 fi = 464 7 3 5 f + 7 = 6540. Similarly, when D = 3 we have Z 3 fτ = q 8q 3 5360q 5339598080q 4 + M! 3/. The main new difficulty in proving Theorem comes from the existence of cusp forms in M! s. The method of Poincaré series adapts nicely to handle it. A key dividend of the method is the last statement of Theorem, showing that the integrality of coefficients is preserved under the lift.

4 W. DUKE AND PAUL JENKINS Remarks: It follows from Theorem that if s D > 0 then the image Z D f M! 3/ s is determined by its principal part, hence by the principal part of f. Furthermore, a0 is divisible by the denominator of each of the L-values L s, χ D, provided that the Fourier coefficients of f are integral. Using well-known properties of the generalized Bernoulli numbers, one can reproduce the divisibility properties that follow from work of Siegel [3, pp. 54 56]. On the other hand, if s D < 0 then L s, χ D = 0. It can be shown that the Zagier lift is compatible with the Hecke operators. For more details, see the end of Section 5. Using a theta lift, Bruinier and Funke [4] have generalized Zagier s result in various other ways, for instance to higher levels, where the existence of cusp forms in the dual weight is also a complication see also []. As another application of these methods we will give a simple proof of a basic property of the Shimura lift for weakly holomorphic modular forms. For gτ = n bnq n M! s+ with s Z + and fundamental D with s D > 0, define the D th Shimura lift of g by 7 S D gτ = L s, χ Db0 + χ D nn s b m D q m. n m>0 When g is holomorphic this is the usual definition. We will repeatedly use the basic fact that S D g M s if g M s+/ see [6]. Recall that a CM point is a point in H of the form b+ b 4ac for integral a, b, c. The proof of the following result will be completed in Section a 6. In the case D = it is due to Borcherds and follows from a special case of Theorem 4.3 in [] see Example 4.4. Theorem. For g M! with s and D a fundamental discriminant with s D > 0, s+ S D g is a meromorphic modular form of weight s for Γ whose possible poles are of order at most s and occur at CM points.. WEAKLY HOLOMORPHIC FORMS In this section we will define a canonical basis for the space M k! for any k = s + / with s Z in which all basis elements have integral Fourier coefficients. Then we will construct forms in M k! when s using Poincaré series. We begin by recalling the canonical basis for M s! defined in [9] for any s Z. Write s = l + k with uniquely determined l Z and k {0, 4, 6, 8, 0, 4}, so that if l 0, then l is the dimension of the space S s of cusp forms of weight s. For every integer m l, there exists a unique f s,m M s! with a q-expansion of the form f s,m τ = q m + n>l a s m, nq n, and together they form a basis for M! s. The basis element f s,m can be given explicitly in the form f s,m = f s P j,

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 5 where f s = f s, l = l E k and P is a polynomial of degree m + l. As shown in [9], the basis elements have the following generating function: 8 f s,m zq m = f szf s τ = f s,m τr m, jτ jz m l m l+ where r = ez. It follows from this that the coefficients a s m, n are integral and satisfy the duality relation 9 a s m, n = a s n, m. In order to formulate a similar result for M k! when k = s+/ with s Z, let l be defined by s = l + k as above. By the Shimura correspondence given in [4] one finds that the maximal order of a nonzero f M k! at i is { l A = s, if l is odd; l, otherwise. If B < A is the next admissible exponent we can construct functions in M k! of the form f k τ = q A + Oq B+4 and f k τ = q B + Oq B+4. Writing s = a + b, where b {6, 8, 9, 0,,, 3, 4, 5, 6, 7, 9}, f k and fk can be given explicitly in the form f k τ = 4τ a f b+/ τ and f k τ = 4τ a f b+/τ, where the forms f b+/, fb+/ M b+/ are given in the appendix and have integral Fourier coefficients. Using them it is easy to construct a unique basis for M k! consisting of functions of the form 0 f k,m τ = q m + n>a a k m, nq n, where m A satisfies s m 0, mod 4. Here f k, A = f k and f k, B = fk. This can be done recursively; f k,m τ is obtained by multiplying f k,m 4 τ by j4τ and then subtracting a suitable linear combination of the forms f k,m τ with m < m. We also have the following generating function, whose proof is similar to Zagier s proof of the k = / case in [6]: f k,m zq m = f kzf k τ + f k zf kτ = f k,m τr m. j4τ j4z m m By and the fact that f k and fk have integral Fourier coefficients, we have the following result. Proposition. The Fourier coefficients a k m, n defined in 0 are integral and satisfy the duality relation a k m, n = a k n, m for all m, n Z. Another way to construct weakly holomorphic forms is by Poincaré series. In this paper we only need them for k = s + / where s. Set jγ, τ = θγτ/θτ for γ Γ 0 4. For m Z define the Poincaré series P k,m τ = emγτjγ, τ k, γ Γ \Γ 0 4

6 W. DUKE AND PAUL JENKINS where Γ is the subgroup of translations in Γ 0 4. For k 5/ this is absolutely convergent and represents a weakly holomorphic form of weight k for Γ 0 4, but it is not in M k! since its Fourier coefficients are not supported on n with s n 0, mod 4. When m = 0 the Poincaré series is an Eisenstein series that Cohen [7] projected to a form in M k and whose Fourier coefficients are expressed in terms of the values of Dirichlet L-functions at s. When m > 0, Kohnen see [5] showed how to obtain in this way cusp forms in S k. It was observed in [5] that a similar procedure works for m < 0. Petersson [] had explicitly computed the Fourier expansions of P k,m in terms of Bessel functions and Kloosterman sums, and the projections g k,m of P k,m to M k! have Fourier expansions that are simple modifications of these. To give them, for m, n Z and c Z + with c 0 mod 4 let 3 K k m, n; c = amod c c a ε k a e ma+na c be the Kloosterman sum, where c a is the extended Legendre symbol and { if a mod 4 ε a = i if a 3 mod 4. Also, let δ odd n = if n is odd and δ odd n = 0 otherwise. Proposition. Suppose that k = s + / where s. Then, for any nonzero integer m with s m 0, mod 4, there exists a form g k,m M k! with Fourier expansion g k,m τ = q m + b k m, nq n n s n 0,4 where for s 0, mod 4 the coefficient b k m, n is given explicitly by the absolutely convergent sum b k m, n = πi k n k m + δodd c c I k 4π mn, if m < 0; K 4 k m, n; c c J k 4π mn, if m > 0. c c 04 c>0 A similar formula holds when m = 0 that can be further evaluated to give Cohen s formulas. Also, a modified version holds when s = see [5]. Of course, g k, m can be expressed in terms of the basis elements f k,m. If there are no nonzero cusp forms in M s, then g k, m = f k,m for all m. In general, however, 4 g k, m f k,m S k is a nonzero cusp form. It seems likely that the Fourier coefficients b k m, n of g k,m are irrational, even transcendental, in general. 3. WEAK MAASS FORMS Next we will show that for f M! s with s Z +, the function s f is a weak Maass form and compute its Fourier expansion. Recall that s was defined in 5. Then we express s f s,m in terms of certain Poincaré series. We need the following result which, in essence, is due to Maass see also [7, p. 50].

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 7 Proposition 3. Suppose that fτ = n anqn M s! for integral s. Then s f is a weak Maass form for Γ with eigenvalue s s. Explicitly, we have 5 s fτ = πy a nn s Is πnye nx n>0 + π s s Γs y s a0 + y an n s Ks π n yenx, where I and K are the usual Bessel functions. Proof. By induction it is readily shown that for n > 0 n 0 s s e nτ = n s s + m! m!s m! 4πnym e nτ. m=0 Standard formulas for Bessel functions with half-integral parameter see e.g. [] yield s e nτ = n s y πi s πny + s K s πny enx s enτ = s n s y Ks s = s π s Γs y s. πnyenx, These formulas easily give 5, thus finishing the proof of Proposition 3. We next express the weak Maass form s f s,m associated to the basis element f s,m in terms of certain Poincaré series, when s and s = l + k as before. For m Z with m 0 consider the Poincaré series see [0] 6 F m τ, s = π m s γ Γ \Γ em Re γτim γτ Is π m Im γτ, which converges absolutely for Re s >. Here Γ is the subgroup of translations in Γ. Clearly F m γτ, s = F m τ, s for γ Γ and F m τ, s = s sf m τ, s. Proposition 4. For integral s we have for m l + 7 s f s,m τ = F m τ, s + a s m, nf n τ, s. 0<n<l+ Proof. To prove Proposition 4, we need the Fourier expansion of F m. This can be found, for instance, in [0]. Let ξs = π s Γs/ζs. Then we have 8 where F m τ, s = π m s y 9 cm, n; s = c>0 Is π m yemx + 4πσ s m s ξs y s + 4π m s c K 0 m, n; c n 0 cm, n; sy Ks π n yenx, { I s 4π mn c if mn < 0 J s 4π mn c if mn > 0

8 W. DUKE AND PAUL JENKINS and K 0 m, n; c = e ma+nā c amod c is the usual Kloosterman sum, the * restricting the sum to a, c =. Consider the Maass form φτ = s f s,m τ F m τ, s + a s m, nf n τ, s. By Proposition 3 and 8 we have that φτ = c0y s + n 0 0<n<l+ cny Ks π n yenx. where the cn are explicitly computable in terms of the c s m, n and the a s m, n. Since φ L Γ\H with eigenvalue s s, it must be equal to 0. In the case s = the Poincaré series F m τ, is defined through analytic continuation see e.g. [0] and Proposition 4 continues to hold in the modified form f 0,m τ = j m τ = F m τ, 4σm for m. 4. PRELIMINARY FORMULAS FOR THE TRACE For the proof of Theorem, we will need to compute the trace of s f s,m in terms of the coefficients of the basis elements f s+/,m. In view of Proposition 4, we are reduced to computing Tr d,d F m, s, where F m τ, s is the Poincaré series defined in 6. When D = s = it was shown in [8] that this trace may be expressed in a simple way in terms of a certain exponential sum. In general we need the exponential sum introduced in [5]: S m d, D; c = b mod c b Dd mod c where χ is defined in 3 and c 0 mod 4. Clearly We have the following identity. χ c, b, b Dd 4 c e mb c S m d, D; c = S m d, D; c = S m d, D; c. Proposition 5. Let s and m 0. Suppose that D is fundamental and that dd < 0. Then Tr d,d F m, s = π m s 4π d 4 D 4. c 0 mod 4 Proof. We have the absolutely convergent expression Tr d,d F m, s = π m s 0 Q = π m s d 4 D 4 χq ω Q a= γ Γ \Γ a Is where the sum over γ is over all γ Γ \Γ with, c Sm d, D; ci s m dd c emre γτ Q Im γτ Q Is π m Im γτ Q π m dd a [ Q χq ω Q e m Re γτ Q ], γ Im γτ Q = Dd a.

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 9 Consider the sum in brackets in 0. For fixed a > 0, the values of are γτ Q run over the mod a-incongruent solutions to the quadratic congruence b dd mod 4a with multiplicity w Q as γ and Q run over their respective representatives. Thus we have e m Re γτ Q = S md, D; 4a. Q χq ω Q γ Substitute this in 0 and write c for 4a to finish the proof. We need to express the traces in terms of the Fourier coefficients of modular forms. This is done by applying an identity originally due to Salié in a special case to transform the sum of exponential sums in Proposition 5 into a sum of Kloosterman sums. This sum may then be interpreted in terms of the Fourier coefficients of half-integral weight Poincaré series. This technique goes back to Zagier [5], who applied it in the context of base-change. Kohnen [5] applied it to the Shimura lift of cusp forms. More recently, this method has proven to be fruitful in the context of weakly holomorphic forms. In [8] it was applied to give a new proof of Zagier s original identity for traces of singular moduli. The technique has since been extended in various ways in [3] and [3]. In particular, the following formula for the trace of F m τ, s in terms of the coefficients b k m, n of half integral weight Poincaré series was given in [3] when m = and s D < 0. Proposition 6. Suppose that m 0, s and dd < 0 with D fundamental. Then ε d s s D Tr d,d F m, s = ε d s D s m s χ Dnn s b s+ χ Dnn s b s+ d, m D n, if s D > 0, m D n, d, if s D < 0, where the sum is over the positive divisors of m, ε = s+, and b s+/ was defined in Proposition. Proof. Recall the Kloosterman sum associated to modular forms of half integral weight defined in 3. It is clear that replacing k with k + does not change this sum; each K s+ m, n, c is equal to K m, n; c or K 3 m, n; c, depending on whether s is even or odd, respectively. In fact, we have the relations K m, n; c = i K 3 m, n; c = K n, m; c. We have the following identity for the Kloosterman sums, which can be proved by a slight modification of the proof of Kohnen in [5, Prop. 5, p. 58] see also [8], [3] and [4]. Lemma. For integers m 0 and c > 0 with 4 c, an integer d with d 0, mod 4 and D a fundamental discriminant, we have the identity S m d, D; c = i, c 4 + δodd c n χ 4n D n K c d, m D n ; c n.

0 W. DUKE AND PAUL JENKINS By Proposition 5 and Lemma we quickly derive that Tr d,d F m, s = π i m s d 4 D 4 χ D nn c 04 c + δ odd c 4 K d, m D n ; c I s 4π m Dd /n c. Comparison with Proposition and use of finishes the proof of Proposition 6. 5. THE ZAGIER LIFT In this section we give the proof of Theorem. The following proposition gives an explicit formula for the Zagier lift of f M! s when s D > 0. In its proof we make repeated use of the classical Shimura lift, integral and half-integral weight duality from 9 and, and the fact that the constant term of a form in M! must vanish. Write s = l + k with k {0, 4, 6, 8, 0, 4} as above. Proposition 7. Suppose that s is an integer and that fτ = n anqn M! s. Suppose that D is a fundamental discriminant with s D > 0. Then the D th Zagier lift of f is given by Z D f = m>0 a m χ D nn s f 3 s, m D n. Proof. Recall that when s D > 0 the Zagier lift was defined by Z D fτ = a m χ D nn s q m D n + L s, χ Da0 + m>0 d: dd<0 Tr d,dfq d, where Trd,Df = s d s s D Tr d,d s f. We prove Proposition 7 by comparing the Fourier coefficients of Z D f with those of the function on the right hand side of, which we will denote simply by F. We do this for the positive coefficients, the principal parts and the constant terms separately. Consider first the positive coefficients. By Propositions 4 and 6, we have for m > l that Trd,Df s,m = χ D nn s b s+ d ; m D 3 n + l a s m, j h j j= χ D hh s b s+ d ; j D h. From 4 we have the cusp form Cτ = g s+, d τ f s+, d τ = n cnqn. Thus b s+ d, j D h = as+ d, j D h + c j D h. However, S D C, the D th Shimura lift of C, is a cusp form of weight s with jth coefficient h j χ D hh s c j D h. The contribution to Tr d,d f s,m in 3 from coefficients of C, which is χ D nn s c m D + n l a s m, j h j j= χ D hh s c j D, h

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS can be interpreted as the constant term of S D Cf s,m M,! which must be zero. Thus we have Trd,Df s,m = χ D nn s a s+ d, m D 4 n 5 + l a s m, j h j j= χ D hh s a s+ d, j D h. By duality, Tr d,d f s,m is the coefficient of q d in the Fourier expansion of χ D nn s f 3 s, m D n l a s j, m h j j= For an arbitrary form f = amq m M! s we have χ D hh s f 3 s, j D h. f = m>l a mf s,m, and so Tr d,df = m>l a mtr d,df s,m is the coefficient of q d in 6 a m m>l χ D nn s f 3 s, m D n l a s j, m h j j= χ D hh s f 3 s, j D. h For j l we have, once again using that the constant of a form in M! vanishes, that a j = m>l a ma s j, m. Thus the form in 6 simplifies to F. Next consider the principal parts. The properties of the basis elements given in Section show that f 3 s, m D n = 0 if for some C that depends only on the weight 3 s. We use this and the Fourier expansion m D n < C f 3 s, m D n τ = q m D n + h a m D 3 s n, h q h to write the negative powers of q appearing in the Fourier expansion of F as χ D nn s q m D 7 n a m m>0 a m χ D hn s q m D n + m>0, m D n <C m,h>0 a m χ D nn s a 3 s m D n, h q h. The first sum is the principal part of Z D f, so we must prove that the expression in the second line of 7, call it S, vanishes. By duality, S = a m χ D nn s q m D n + χ D nn s a s+ h, m D n q h. m>0 h>0, m D n <C

W. DUKE AND PAUL JENKINS Now for any h > 0, the coefficient of q m in the Shimura lift S D f s+, h of the cusp form f s+, h is given by χ D nn s a s+ h, m D n + {, if m D n = h, 0, otherwise. The last term here arises from the initial q h in the Fourier expansion of f s+, h, since h, h is zero by definition. From this, it is clear that the coefficient of q h in S for a s+ each h > 0 can be interpreted as the constant term of S D f s+, hf M,! so S = 0. Finally we evaluate the constant term of F, again using duality, as m>0 a m Since s we have by [6] χ D nn s a 3 s m D n, 0 = m>0 S D f s+,0τ = L s, χ D + m>0 a m χ D nn s a s+ χ D nh s a s+ 0, n D q m h and the constant term of S D f s+,0f M! is L s, χ Da0 + a m χ D nn s a s+ 0, m D = 0, n m>0 as desired. This concludes the proof of Proposition 7. We also need the corresponding statement if s D < 0. 0, m D n. Proposition 8. Suppose that s is an integer and that f M! s has Fourier coefficients an. Suppose that D is a fundamental discriminant with s D < 0. Then the D th Zagier lift of f is given by 8 Z D f = m>0 a mm s χ D nn s f s+, m D n + g, where g S s+/ is the unique cusp form whose Fourier coefficients bn match those of Z D f for the first l positive values of n with s n 0, mod 4. Proof. Using Propositions 4 and 6 as before, we find that Trd,Df s,m =m s χ D nn s b s+ m D 9 ; d n + l a s m, jj s h j j= Thus, the trace Trd,D f of an arbitrary form χ D hh s b s+ j D ; d. h is given by m>l a m f = amq m = m>l a mf s,m m s χ D nn s b s+ m D n ; d

INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 3 + l a s m, jj s h j j= χ D hh s b j D s+ ; d h = m>0 a mm s χ D nn s b m D s+ ; d, n where we have simplified as before. This is just the coefficient of q d in the modular form F M! given by s+ F = m>0 Now since g s+, m D n f s+, m D n S s+ F = m>0 a mm s χ D nn s g s+, m D n. from 4, we find that a mm s χ D nn s f s+, m D + g n for a certain cusp form g, and, arguing as in Proposition 7, the principal part of F matches the principal part of Z D f. Since the constant term and positive coefficients of F match those of Z D f, Proposition 8 now follows. The first statement of Theorem follows from Propositions 7 and 8. The statement on integrality follows from Proposition in the case s D > 0. Otherwise we can reduce to this case using the following identity, which holds if s D < 0 and D is fundamental with s D > 0: Tr m D,D f = ms µaχ D a χ D bab s Tr m ab D,D f. a m b ma This identity is a consequence of the following lemma. Lemma. For D and D fundamental discriminants with DD < 0 and m Z + we have Tr m D,D = µaχ D a χ D btr m ab D,D. a m b ma Lemma is obtained by writing the trace as a sum of sums over primitive quadratic forms, noting that χ D = χ D for such primitive forms, and applying Möbius inversion. We now briefly indicate how one shows that the Zagier lift is compatible with the Hecke operators. If k Z > 0 and p is a prime, the weight k Hecke operator k T p acts on a modular form fτ = n anqn M k! by f k T p = apn + p k a n q n. p n If k Z 0, we multiply this by p k so that k T p preserves the integrality of Fourier coefficients. When 0 < s Z, the half integral weight Hecke operator s+/ T p acts on a form gτ = n bnqn M s+/! by g s+/ T p = n bp n + s n p s bn + p s b n q n. p p Again, for s 0, we normalize this by multiplying by p s.

4 W. DUKE AND PAUL JENKINS It is straightforward to see that for any prime p, Z D f 3/ ŝ T p = Z D f s T p. In the case that s D > 0, we need only use the explicit Fourier expansion of the Zagier lift to compare principal parts. If s D < 0, though, we must also show that 30 Tr s n,df s T p = Tr s np,d f + s n p s n p p s Tr s n,df + p s Tr s n/p,d f for the first l positive values of n with s n 0, mod 4. To see that this holds, we argue as in the proof of [6, Theorem 5ii] to show that Tr s n,d s f 0 T p equals Tr s np,d s f + Tr s n,d s f + ptr s n/p,d s f, and use the fact that if k < 0, then k f k T p = p k f k+ T p to obtain equation 30. 6. THE SHIMURA LIFT In this final section we prove Theorem. For this we need the following two propositions. Proposition 9. Suppose that s Z + and τ H. As a function of z H, s fs zf s τ jτ jz is a meromorphic modular form of weight s with poles of order at most s that only occur at points equivalent to τ under Γ. Proof. Observe first that if f has weight k and g has weight 0 then by 4 k fg = g k f + f Dg. Apply this repeatedly with gτ = jτ jz n for n < s. We derive that s f s τ s g n τ = jτ jz jz jτ n for g n M! 0, from which the result follows easily. Theorem is a consequence of Proposition 9 together with the following explicit formula for the D th Shimura lift of f s+/, d. Write s = l + k as above. Proposition 0. Suppose that s, s D > 0 and that dd < 0. Then S D f s+/, d z = Trd,D fs zf s τ + fz, jτ jz where f M s is the unique holomorphic form whose Fourier coefficients an match those of S D f s+/, d for n = 0,..., l. Proof. By 7 we have, writing r = ez, 3 n= S D f s+/, d z = L s, χ Da s+/ d, 0 + m>0 χ D nn s a s+/ d, m D n r m.

By 4 and 3 we have 3 33 INTEGRAL TRACES OF SINGULAR VALUES OF WEAK MAASS FORMS 5 m>l Tr d,df s,m r m =S D f s+/, d z L s, χ Da s+/ d, 0 0<m l + χ D nn s a s+/ d, m D n l j= m>l a s m, jr m h j By integral weight duality 9 the term in 33 is l j= χ D hh s a s+ d, j D h a s j, mr m m>l h j l = s, j z r j=f j h j so by 3 we get after some cancellation that r m χ D hh s a s+ d, j D h. χ D hh s a s+ d, j D h, 34 m>l Tr d,df s,m r m = S D f s+/, d z fz. The identity of Proposition 0 follows from 8 and 34, at least when Im z > max Q Im τ Q. Proposition 0 now follows by analytic continuation. Acknowledgements: After this paper was written we learned that results similar to some of those presented here were obtained independently in a recent preprint by Miller and Pixton [9]. Additionally, we thank the referee for some helpful comments. 7. APPENDIX Table gives explicit formulas for the first two basis elements f b+, f of weight b+/ b+ for all b {6, 8, 9, 0,,, 3, 4, 5, 6, 7, 9} as polynomials in the weight / theta function θ = n Z qn and the weight Eisenstein series on Γ 0 4 given by F z = σn + q n+. n=0 Both θ and F have integral Fourier coefficients. The space of holomorphic modular forms on Γ 0 4 of weight s + / is generated by the forms F n θ s+ 4n, where 0 n s+ see [7]. Thus, to construct these basis elements 4 we examine the Fourier expansion of the form f = s+ 4 n=0 AnF n θ s+ 4n and choose the coefficients An so that f is in the plus space M! and has the appropriate s+ leading terms in its Fourier expansion. Table shows that all of the An are integral for

6 W. DUKE AND PAUL JENKINS TABLE f 3 f 3 f 7 f 7 f 9 f 9 f f f 3 f 3 f 5 f 5 f 7 f 7 f 9 f 9 f 3 f 3 f 33 f 33 f 35 f 35 f 39 f 39 = F θ 9 8F θ 5 + 3F 3 θ = q + Oq 4 = θ 3 6F θ 9 + 56F θ 5 = + Oq 4 = F θ 3 6F θ 9 + 5F 3 θ 5 + 8F 4 θ = q + Oq 4 = θ 7 34F θ 3 + 340F θ 9 86F 3 θ 5 = + Oq 4 = F 3 θ 7 6F 4 θ 3 = q 3 + Oq 4 = θ 9 38F θ 5 + 456F θ 67F 3 θ 7 = + Oq 4 = F θ 7 34F θ 3 + 336F 3 θ 9 800F 4 θ 5 + 5F 5 θ = q + Oq 4 = θ 4F θ 7 + 588F θ 3 9F 3 θ 9 + 496F 4 θ 5 = + Oq 4 = F 3 θ F 4 θ 7 64F 5 θ 3 = q 3 + Oq 4 = θ 3 46F θ 9 + 736F θ 5 4600F 3 θ + 8096F 4 θ 7 = + Oq 4 = F 4 θ 9 6F 5 θ 5 = q 4 + Oq 5 = F θ 4F θ 7 + 584F 3 θ 3 808F 4 θ 9 + 79F 5 θ 5 + 048F 6 θ = q + Oq 5 = F 3 θ 5 3F 4 θ + 7F 5 θ 7 56F 6 θ 3 = q 3 + Oq 4 = θ 7 54F θ 3 + 080F θ 9 9576F 3 θ 5 +34048F 4 θ 675F 5 θ 7 = + Oq 4 = F 4 θ 3 36F 5 θ 9 + 30F 6 θ 5 = q 4 + Oq 5 = F θ 5 50F θ + 896F 3 θ 7 6664F 4 θ 3 +667F 5 θ 9 307F 6 θ 5 + 89F 7 θ = q + Oq 5 = F 4 θ 5 30F 5 θ + 4F 6 θ 7 = q 4 + Oq 7 = F 3 θ 9 38F 4 θ 5 + 444F 5 θ 408F 6 θ 7 04F 7 θ 3 = q 3 + Oq 7 = F 4 θ 7 3F 5 θ 3 + 7F 6 θ 9 56F 7 θ 5 = q 4 + Oq 5 = F θ 9 58F θ 5 + 7F 3 θ 84F 4 θ 7 +56064F 5 θ 3 755F 6 θ 9 4096F 7 θ 5 + 3768F 8 θ = q + Oq 5 = F 4 θ 9 38F 5 θ 5 + 440F 6 θ 408F 7 θ 7 = q 4 + Oq 7 = F 3 θ 3 46F 4 θ 9 + 74F 5 θ 5 440F 6 θ +563F 7 θ 7 4096F 8 θ 3 = q 3 + Oq 7 = F 4 θ 3 46F 5 θ 9 + 70F 6 θ 5 4064F 7 θ + 3584F 8 θ 7 = q 4 + Oq 7 = F 3 θ 7 54F 4 θ 3 + 068F 5 θ 9 90F 6 θ 5 +8608F 7 θ 644F 8 θ 7 6384F 9 θ 3 = q 3 + Oq 7 the first two basis elements of each half integral weight, so it follows that all of the f b+ and f have integral Fourier coefficients. b+

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