TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES

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1 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE Abstract. Suose that 1 mod 4 is a rime, and that O K is the ring of integers of K := Q. A cassica resut of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hibert moduar surface h h/sl O K are weight hoomorhic moduar forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singuar modui over these intersection oints are often weaky hoomorhic weight moduar forms. For the singuar modui of J 1 z = jz 744, we exicity determine these generating functions using cassica Weber functions, and we factorize their norms as roducts of Hibert cass oynomias. We aso exicity comute a such generating functions in the SL Z case for the rimes = 5, 13, and Introduction and Statement of Resuts [ ] 1+ For rimes 1 mod 4, et O K := Z be the ring of integers of the rea quadratic fied K := Q. The grou SL O K, i.e., the grou of matrices with entries in O K and determinant 1, acts on h h, the roduct of two comex uer haf anes h, by α β αz1 + β z γ δ 1, z := γz 1 + δ, α z + β. γ z + δ Here ν denotes the conjugate of ν in Q. The quotient X := h h/sl O K is a non-comact surface with finitey many singuarities. It can be naturay comactified by adding finitey many oints i.e. cuss, and Hirzebruch showed [6] how to resove the singuarities introduced by adding cuss using cycic configurations of rationa curves. The resuting moduar surface Y is a neary smooth comact agebraic surface with quotient singuarities suorted at those oints in h h with a non-trivia isotroy subgrou within PSL O K. In their famous work [7] on these surfaces, Hirzebruch and Zagier introduced a sequence of agebraic curves Z 1, Z, X, and studied the generating functions for Date: August 1, Mathematics Subject Cassification. 11F03, 11F30, 11F37. The authors thank the Nationa Science Foundation for their generous suort. The second author is gratefu for the suort of the David and Lucie Packard, H. I. Romnes, and John S. Guggenheim Feowshis. The third author is gratefu for the suort of a NDSEG Feowshi. 1

2 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE their intersection numbers. They roved the striking fact that these generating functions are weight moduar forms, an observation which aowed them to identify saces of moduar forms with certain homoogy grous for Y. To define these curves, for a ositive integer N, consider the oints z 1, z h h satisfying an equation of the form 1.1 Az 1 z + λz1 λ z + B = 0, where A, B Z, λ O K, and λλ + AB = N. Each such equation defines a curve in h h isomorhic to h, and their union is invariant under SL O K. The Hirzebruch- Zagier divisor Z N is defined to be the image of this union in X. If N = 1, then one easiy sees from 1.1 that Z N is emty. We et denote the cosure of Z in Y. If Z m, Z n denotes the intersection Z N Z m Z n N number of and in Y see [7] for the recise formuation, then Hirzebruch and Zagier roved in [7], for every ositive integer m, that 1. Φ m z := a m 0 + Z m, Z n q n note q = e πiz throughout is a hoomorhic weight moduar form on Γ 0 with Nebentyus. Here a m 0 is a sime constant arising from a voume comutation. More recisey, Φ m z is in the us sace M + Γ 0,, the sace of hoomorhic weight moduar forms Fz = n=0 anqn on Γ 0 with Nebentyus, with the additiona roerty that n=1 n 1.3 an = 0 if = 1. In the resent aer, we require the sace M Γ 0,, the sace of weaky hoomorhic moduar forms of weight on Γ 0 with Nebentyus. Let M + Γ 0, be the subsace of those forms in M Γ 0, that satisfy 1.3. Reca that a function is weaky hoomorhic if its oes if there are any are suorted at cuss. The geometric art of the roof of the moduarity of 1. rovides a concrete descri- Z m Z n. Loosey seaking, the finite oints Z m Z n tion of the intersection oints are identified with CM oints in h which are the roots of Γ 0 m equivaence casses of binary quadratic forms with negative discriminants of the form 4mn x / see Section.1. The vaues of moduar functions at such CM oints are known as singuar modui, and in view of the moduarity of 1., it is natura to consider generating functions for the vaues of singuar modui over the CM oints constituting Z m Z n. Suose that = 1 or that is an odd rime with 1, and et Γ 0 be the rojective image of the extension of Γ 0 by the Fricke invoution W = 0 1 0

3 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 3 in PSL R. Suose that fz = n anqn M 0 Γ 0, the sace of weaky hoomorhic moduar functions with resect to Γ 0. Furthermore, suose that a0 = 0. We define the trace of fz over Z Z n by 1.4 Z, Z n tr fτ f :=, #Γ 0 τ τ Z Z n where Γ 0 τ denotes the stabiizer of τ in Γ 0. For these traces, we consider the anaog of the generating functions in 1. defined by 1.5 Φ,f z := A,f z + B,f z + Here we have that A,f z := ɛ m,n 1 ma mn x Z x m mod Z n=1 q x m 4 +, Z n tr f q n. x Z x m mod q x m 4, B,f z := ɛ σ 1 n + σ 1 n/a n q x, n 1 x Z where ɛ = 1/ for = 1, and is 1 otherwise. As usua, σ 1 x denotes the sum of the ositive divisors of x if x is an integer, and is zero if x is not an integer. Using recent works of Zagier [13], and Bruinier and Funke [4], we show that these generating functions are aso moduar forms of weight. In articuar, we obtain a inear ma: Φ, : M 0Γ 0 M Γ 0, where the ma is defined for the subsace of those functions with constant term 0. Theorem 1.1. Suose that 1 mod 4 is rime, and that = 1 or is an odd rime with 1. If fz = n anqn M 0 Γ 0, with a0 = 0, then the generating function Φ,f z is in M Γ 0,. Remark. In [4], Bruinier and Funke estabish that the generating functions for traces of singuar modui are moduar forms in great generaity. In the case of moduar curves, for simicity they work out the detais for X0 for = 1 and for odd rimes. We foow their ead by making this assumtion as we in Theorem 1.1. Remark. If we aow the constant term of fz to be non-zero, non-hoomorhic terms woud be incuded, as in [4]. In articuar, if fz = 1, then we obtain the Hirzebruch- Zagier moduar forms restricted to the finite oints of intersection.

4 4 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE We turn to the robem of exicity comuting natura exames of these moduar forms Φ,f z. Let J 1z = jz 744, where jz is the usua eitic moduar function 1.6 jz = E 4z 3 ηz 4 = q q +, where ηz = q 1/4 n=1 1 qn is Dedekind s eta-function and E 4 z = d 3 q n is the usua Eisenstein series of weight 4. The moduar forms Φ z can be described in terms of ηz, E 4 z, and the cassica Weber functions 1.7 f 1 z = ηz/ ηz and Theorem 1.. If 1 mod 4 is rime, then n=1 d n f z = ηz ηz. Φ z = ηzηze 4zf z f z 4ηz 6 f 1 4z 4 f z f 1 4z 4 f z. Remark. Using the cassica theta functions Θz and Θ odd z see 3.1 and 3.3, the formua in Theorem 1. may be reformuated as Φ z = E 4z ΘzΘ ηz 6 odd z/4 ΘzΘ odd z/4. It turns out that the forms Φ z, the generating functions for the traces of singuar modui on X, are cosey reated to Hibert cass oynomias. The singuar modui jτ, as τ ranges over C D, the equivaence casses of CM oints with discriminant D, are the roots of the Hibert cass oynomia 1.8 H D x = τ C D x jτ Z[x]. Each H D x is an irreducibe oynomia in Z[x] which generates a cass fied extension of Q D. To reate the forms Φ z to Hibert cass oynomias, define N z as the mutiicative norm of Φ 1,J1 z 1.9 N z := Φ M. M Γ 0 \SL Z If Nz is the normaization of N z with eading coefficient 1, then zh 75 jz if = 5, N z = E 4 z z H 3 jzh 507 jz if = 13, z 3 H 4 jzh 867 jz if = 17, z 5 H 7 jz H 53 jz if = 9,

5 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 5 where z = ηz 4 is the usua Deta-function. These exames iustrate a genera henomenon in which N z is essentiay a roduct of certain Hibert cass oynomias. Before we state the genera resut, we fix some notation. Define integers a, b, and c by a := , b := , c := Furthermore, et D be the set of negative discriminants D 3, 4 of the form x 4 16f with x, f 1. Theorem 1.3. Assume the notation above. If 1 mod 4 is rime, then Nz = E 4 zh 3 jz a H 4 jz b z c H 3 jz H D jz. D D For the rimes = 5, 13, and 17, and when = 1, work of Bruinier and Bundschuh [3] make it ossibe to obtain exicit formuas for the traces of every weaky hoomorhic eve 1 function, with constant term 0, on X. There is a natura sequence of moduar functions J m z which forms a basis of such functions. For every ositive integer m et J m z be the unique moduar function on SL Z which is hoomorhic on h with a Fourier exansion of the form 1.13 J m z = q m + c m nq n. In articuar, note that n=1 J 1 z = jz 744 = q q +. Each J m z is a monic degree m oynomia in jz with integer coefficients, and its generating function is given by J m xq m = E 4z E 6 z 1 z jz x, m=0 d n d5 q n. where E 6 z = n=1 To describe Φ for = 5, 13, and 17 we give a basis for M + Γ 0,. In articuar, for m 0 with m 1 there is a unique function K m z with Fourier exansion of the form 1.14 K m z = q m + Oq M + Γ 0,.

6 6 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE In Section 4, we wi rovide formuas for K 0 z and K 1 z. Furthermore, we rovide a descrition of each K m z, for m 1 in terms of the action of Hecke oerators on K 1 z. We have the foowing connection between the functions K m z and Φ z. Theorem 1.4. If = 5, 13, or 17, and m 1, then Φ z = σ 1 mk 0 z d d m x d mod x <d K d x /4 z. Remark. Theorem 1.4 shows that the inear ma described in Theorem 1.1 is not necessariy surjective. Secificay, the eading term of Φ z is mq m 4, and this easiy imies that for = 5, K 4 z is not in the image of the inear ma. However, the inear ma is aways injective. If f M 0 Γ 0 is a nonconstant moduar function, then since Γ 0 has ony one cus, f has a oe at. Suose that fz = cq m + Oq m+1, with c 0 and m 1. The formuas for Φ,f z obtained in the roof of Theorem 1.1 easiy imy that Φ,f z = cmq m /4 + Oq 4 m/4 if m is even, cmq 1 m/4 + Oq 5 m/4 if m is odd. and hence Φ,f z 0. In Section.1, we reca the exact reation between the oints in Z m Z n and CM oints see Definition.1, and in Section. we reca works of Zagier, and Bruinier and Funke which describe generating functions for traces of singuar modui on moduar curves as weight 3/ weaky hoomorhic moduar forms. Using these facts, we rove Theorem 1.1 in Section.3. In Section 3 we investigate the traces of J 1 z = jz 744, and we rove Theorems 1. and 1.3. In Section 4, for = 5, 13, and 17 we comute each Φ z using works of Bruinier and Bundschuh, and we rove Theorem 1.4. Acknowedgements The authors thank J. Bruinier, J. Funke, W. Kohnen and the referee for their hefu comments.. The moduarity of Φ,f z.1. Intersection oints on Hibert moduar surfaces as CM oints. The goa of this section is to rovide for = 1 or an odd rime with 1 an interretation Z n of Z Definition.1 beow. as a union of Γ 0 equivaence casses of CM oints. This is given by

7 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 7 For D 0, 1 mod 4, D > 0 we denote by Q D the set of a not necessariy rimitive binary quadratic forms Qx, y = [a, b, c]x, y := ax + bxy + cy with discriminant b 4ac = D. To each such form Q, we et the CM oint α Q be the unique oint in h that satisfies Qα Q, 1 = 0. The grou SL Z acts on Q D in the usua way, i.e., for M = α β γ δ SL Z we define α β [a, b, c] x, y := [a, b, c]αx + βy, γx + δy. γ δ It is easy to see that Q D is invariant under the action of SL Z. For = 1 or an odd rime and D > 0, D 0, 1 mod 4 we define Q [] D to be the subset of Q D with the additiona condition that a. It is easy to show that Q [] D is invariant under Γ 0. Suose that = 1 or is an odd rime with 1. Then, there exists a rime idea O K with norm. Define } α β SL O K, := SL γ δ K : α, δ O K, γ, β 1. In this case there is a matrix A GL + K such that A 1 SL O K, A = SL O K. Define φ : h h/sl O K, h h/sl O K by φz 1, z := Az 1, A z. Let Γ be the stabiizer of z, z : z h} h h in SL O K,. Then Γ = Γ 0 if and Γ = Γ 0 if =. The image of z, z : z h} under φ is Z. Hence, we have a natura ma ψ : h/γ Z. Using work of Hirzebruch and Zagier, we make the foowing definition. Definition.1. If = 1 or an odd rime with 1, and n 1, then define } Z Z n := α Q : Q Q [] 4n x / / Γ 0. x Z x <4n x 4n mod Here the reetition of x and x indicates that Z α Q occurs twice if Q Q [] 4n x / incude x Z x <4n/ x 4n/ mod is a mutiset where a CM oint for x 0. In addition, if > 1 and n, then we Z n α Q : Q Q [] 4n/ x / / Γ 0 },

8 8 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE where each oint with non-zero x is taken with mutiicity, and a oint where x = 0 is taken with mutiicity. To justify our definition we argue as foows. Hirzebruch and Zagier [7],. 66 show that if t h, n 1 and ψt Z Z n, then at + λ λ t + b = 0 for a, b, λ Z Z 1 with λλ + ab = n. This foows as a resut of considering the inverse image φ 1 Z h h/sl O K,. Write λ = c + d 1+, for c, d Z. We have that the discriminant of the equation above is d 4ab. However, this imies that c + d 4n = d 4ab. Thus, the discriminant is of the form x 4n/. From Hirzebruch and Zagier s Theorem 3 [7],. 77, comuting the number of transverse intersections of Z and Z n, we see that each z h with discriminant of the form x 4n/ occurs with the aroriate mutiicity... Generating functions for traces of singuar modui on moduar curves. Throughout we et be 1 or an odd rime. Motivated by Borcherds work [1] on the infinite roduct exansions of certain automorhic forms on orthogona grous, Zagier [13] comuted the generating functions for the traces of the J m z singuar modui, as we as severa other casses of moduar functions. If m, D are ositive integers and D is a discriminant, then Zagier defined the trace of the singuar modui of discriminant D for J m z by.1 t m D := Q Q D /PSL Z 1 ω Q J m α Q, where ω Q is the order of the stabiizer of Q in PSL Z. He roved the striking fact that these generating functions are essentiay weight 3/ weaky hoomorhic moduar forms. We now reca some of Zagier s generating functions. Foowing Kohnen [8], for integers k et M + k+ Γ be the sace of weaky hoomorhic weight k + 1 moduar forms on Γ 0 4 with a Fourier exansion of the form. anq n. n 1 k n 0,1 mod 4 Zagier s trace generating functions are described in terms of a secia sequence of weight 3/ forms g m z. For ositive m 0, 1 mod 4, g m z is the unique form in

9 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 9 M + 3 Γ 0 4 with a Fourier exansion of the form.3 g m z = q m + Bm, nq n. Zagier roved that the forms g m z determine the generating functions for traces and twisted traces of singuar modui on SL Z. For exame, his work shows that.4 g 1 z = ηz E 4 4z ηzη4z = 6 q 1 t 1 Dq D = q q 3. D>0 n=0 To state his more genera resut, for ositive integers m, et.5 B m 1, D := the coefficient of q D in g 1 z Tm, where Tm is the usua Hecke oerator on M + 3 Γ 0 4. Zagier s formuae for the traces t m D are given by the foowing theorem Theorem 5 of [13]. Theorem.. If m 1 and 0 < D 0, 3 mod 4, then t m D = B m 1, D. Recenty, Bruinier and Funke [4] have generaized Zagier s resuts to incude traces of singuar modui of moduar functions on grous which do not necessariy ossess a Hautmodu. A articuary eegant exame of their work aies to moduar functions on Γ 0. Suose that fz = n anqn M 0 Γ 0 has constant term a0 = 0. The discriminant D trace is given by.6 t f D := #Γ 0 fα Q. Q Q Q D, /Γ 0 1 Here Γ 0 Q is the stabiizer of Q in Γ 0. Foowing Kohnen [8], we et for ɛ ±1} M +,ɛ Γ k be the sace of those weight k + 1 weaky hoomorhic moduar forms fz = n anqn on Γ 0 4 whose Fourier coefficients satisfy 1.7 an = 0 whenever 1 k k n n, 3 mod 4 or = ɛ. Bruinier and Funke s generaization of Zagier s work Theorem 1.1 of [4] gives the foowing theorem. Theorem.3. If = 1 or is an odd rime and fz = n anqn M 0 Γ 0, with a0 = 0, then G f, z := m,n 1ma mnq m + σ 1 n + σ 1 n/ a n + t fdq D n 1 D>0 is an eement of M +,+ 3 Γ 0 4. Remark. Theorem.3 recovers Zagier s Theorem. when = 1 and f = J m.

10 10 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE.3. Proof of Theorem 1.1. To rove Theorem 1.1 we require the cassica Jacobi theta function.8 Θz = x Z q x = 1 + q + q 4 + q 9 +. It is we known that Θz M1 Γ 0 4. Suose that fz = n anqn M 0 Γ 0 satisfies the hyotheses of Theorem 1.1. By Definition.1 and Theorem.3, a straightforward cacuation reveas that.9 Φ,f z = ɛ G f, zθz U4 U + V, where for d 1 the oerators Ud and V d are defined on forma ower series by.10 anq n Ud := adnq n, and.11 anq n V d := anq dn. It is we-known for exame, see [8, 1] that V mas G f, z M3 Γ 0 4 to the sace M3 Γ 0 4,. Since Θz M1 Γ 0 4, it foows that G f, zθz is in M Γ 0 4,. Now, we ay the oerator U twice to G f, zθz. Since 4 and has conductor, Lemma 1 of [9] imies that G f, zθz U is in M Γ 0,. Since the non-zero coefficients of G f, z are suorted on exonents n 0, 3 mod 4, it foows that the non-zero coefficients of G f, zθz are suorted on exonents n 0, 1, 3 mod 4. In articuar, the non-zero coefficients of G f, zθz U are suorted on exonents n 0 mod. Lemma 4 i of [9] then imies that G f, zθz U U = G f, zθz U4 is in M Γ 0,. The theorem foows since it is we-known that U + V mas G f, zθz U4 to M Γ 0,. Remark. Stricty seaking, Lemmas 1 and 4 of [9] are ony stated for integer weight cus forms. However, it is sime to check that their roofs aso hod for weaky hoomorhic integer weight forms.

11 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES The secia case of J 1 z = jz 744 Here we examine the moduar forms Φ z, the trace generating functions for J 1 z = jz 744 on the Hibert moduar surface X. In articuar, we rove Theorem 1. which describes these forms in terms of the cassica Weber functions, and Theorem 1.3 which reates these forms to roducts of Hibert cass oynomias Proof of Theorem 1.. To rove Theorem 1., we work directy with Zagier s identity.4. We reca the foowing cassica theta function identities: 3.1 Θz = ηz5 ηz η4z = q x = 1 + q + q 4 +, x Z 3. Θ 0 z = ηz ηz = 1 x q x = 1 q + q 4 q 9 +, x Z and 3.3 Θ odd z = η16z η8z = x 0 q x+1 = q + q 9 + q 5 + q Proof of Theorem 1.. By.4,.9, and 3., we have that Φ z = g 1 zθz U4 Θ0 ze 4 4z = Θz U4. η4z 6 By the definition of U4, it is straightforward to rewrite this exression as Φ z = 1 3 Θ0 ze 4 4z 1 ν Θz 4 η4z ν=0 = 1 3 Θ0 z + ν/4e 4 z + ν Θz + ν/4. 4 ηz + ν 6 ν=0 Using the fact that E 4 z + ν = E 4 z, and that we obtain Φ z = E 4z 4ηz 6 By 3. and 3.3, one finds that ηz + ν 6 = i ν ηz 6, 3 i ν Θ 0 z + ν/4θz + ν/4. ν=0 Φ z = E 4z 4ηz 6 x,y Z q x+y/4 1 x 3 ν=0 i νx +y ν ν.

12 1 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE Since we have that it foows that 3 i νx +y ν ν = ν=0 Φ z = E 4z ηz 6 x,y Z 0 if x y mod, 4 if x y mod, q y+1 +4x /4 x,y Z q 4y +x+1 /4 = E 4z ηz 6 ΘzΘ odd z/4 ΘzΘ odd z/4. The caimed formua now foows easiy from 1.7, 3.1, and Proof of Theorem 1.3. Here we rove Theorem 1.3, the descrition of N z in terms of roducts of Hibert cass oynomias. Proosition 3.1. If 1 mod 4 is rime, then 3.4 N z = E 4z a z c F jz, where F x Z[x] is a monic oynomia with 5 5/1 if 1 mod 1, degf x = 5 1/1 if 5 mod 1. Proof. From Theorem 1.1 one easiy sees that N z M +SL Z. Lemma.34 of [10] then imies that N z has the desired factorization 3.4. Since Φ z M + Γ 0,, Lemma 3 of [3] imies that 3.5 Φ z W = 1 Φ z U. This imies in articuar that N z has integer coefficients with eading coefficient one. Since jz has integer coefficients with eading coefficient 1, it foows that F x is a monic oynomia with integer coefficients. To comete the roof, it suffices to comute the degree of F x which is equivaent to comuting the order of N z at z =. For this notice that.4 and.9 give 3.6 Φ z = q + Θz U4 = q q +1 + U4 = q 1/4 +. We now use as a set of reresentatives for the coset sace Γ 0 \SL Z the matrices } } : 0 s s

13 Moreover we have TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES = W 1 s 1/ s/. 0 1 Therefore, by.4,.9, and 3.5 it foows, since UU4 = U4U, that Φ z W = 1 Φ z U = +. Together with 3.6, this imies that Using 3.4, we get F jz = N z = q 1/4 +. q 5 5/1 + if 1 mod 1, q 5 1/1 + if 5 mod 1. The concusion about degf x foows from the fact that jz = q To rove Theorem 1.3, it suffices to comute the factorization of F x over Z[x]. Loosey seaking, F x catures the divisor of the moduar form N z in h. To comute the oints in the divisor, we sha make use of Theorem 1.. Since ηz is non-vanishing on h, the factors of F x ony arise from the zeros of the norm of E 4 z and of f 1 4z 4 f z f 1 4z 4 f z. To determine these zeros and their corresonding mutiicities, we first reca some cassica facts about cass numbers. Let h D denote the cass number of rimitive ositive definite binary quadratic forms with discriminant D. These cass numbers have the roerty that h D = degh D x. The foowing we known fact sha ay an imortant roe for exame, see age 69 of [7]. Proosition 3.. If D 0 is a fundamenta discriminant and f 1, then h D 0 f ω D 0 f = h D 0 ω D 0 f D0 1, f rime where ω D is haf the number of units in the imaginary quadratic order of discriminant D. We sha require the foowing cass number reation to rove Theorem 1.3. Lemma 3.3. If 1 mod 4 is rime, then <s< f ts, h ω s 4 16f s 4 16f = 5 1, where ts, := t is the argest integer for which t s 4 and s 4/t 0, 1 mod 4.

14 14 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE Proof. We use the Eicher-Seberg trace formua for exame, see [5], [11], giving the trace of T, where T is the usua Hecke oerator, on S Γ This trace is zero, since S Γ 0 16 = 0. The Eicher-Seberg trace formua for this case gives the identity f 1 φ 1/f c + 1, f, 16, 1 s <4 f ts, bs, f, cs, f, 16, = 0, where φ is Euer s totient. The number bs, f, = hs 4/f /ωs 4/f, and the cs, f, 16, count soutions to quadratic congruences moduo owers of these are exicity given in [11]. It is straightforward to rove that cs, f, 16, is equa to, 6, 8, 4, 6 if s 4 1 mod 8, 4 mod 3, 16 mod 18, 80 mod 18, 0 mod 64, f resectivey. Otherwise it is equa to 0. From this it is straightforward to see that 1 1 f 1 φ 1/fc + 1, f, 16, = 6. Hence, it suffices to consider the third term of 3.7. We have that bs, f, cs, f, 16, = 5. s <4 f ts, From the consideration above it foows that c0, f, 16, = 0. Aso, bs, f, cs, f, 16, = b s, f, c s, f, 16, and hence bs, f, cs, f, 16, = 5. 0<s< f ts, Now, if we write ts, = αs ms where m is odd and α 1, we have bs, i g, cs, i g, 16, = 5. 0<s< 0 i αs g ms It suffices to rove for a g that α α bs, i g, cs, i g, 16, = 4 i=0 i=0 s 4 s 4 h /ω. 16 i g 16 i g If i α 3, we have that that cs, i g, 16, = 6. In this case, Proosition 3. imies that bs, i s g, = 4h 4 s /ω 4. This gives i g i g s bs, i g, cs, i 4 g, 16, = 4h i g /ω s 4 i g.

15 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 15 Hence, it suffices to consider α i α. These may be argued on a case by case basis. We give the argument when D = s 4/ i g 1 mod 8. The other cases are simiar and sighty simer. Suose that D 1 mod 8. If α = 0, then s 4 1 mod 8 and hence s is odd, so s mod 8, a contradiction. If α = 1, then s 4 4D 4 mod 16. Thus, s mod 16, a contradiction. Hence, α. Now, we have that cs, α g, 16, =, cs, α 1, 16, = 6, and cs, α g, 16, = 8. We aso have by Proosition 3. that bs, α g, 16, = bs, α 1 g, 16, = h D/ω D and bs, α g, 16, = h D/ω D. Thus, α i=α as desired. bs, i g, cs, i g, 16, = 4h D ω D s 4 = 4h 16 α 4 Now we determine the factor of F x arising from E 4 z. Proosition 3.4. If 1 mod 4 is rime, then where I x Z[x] has degi x = F x = H 3 x I x, /ω 1/1 if 1 mod 1, 5/1 if 5 mod 1. s 4 16 α 4 Proof. Since E 4 ω = 0 for ω = e π/3 = 1+ 3 it foows that E 4 z is zero for z := ω/. Since z has discriminant 3, by the integraity of F x and the irreducibiity of H 3 x, it foows that H 3 x F x in Z[x]. By Proosition 3., we have that 1/3 if 1 mod 1, degh 3 x = + 1/3 if 5 mod 1, and so the caimed formua for degi x foows from Proosition 3.1. In view of Proosition 3.1 and Proosition 3.4, to rove Theorem 1.3 it suffices to determine the oynomia I x. To this end, we begin by observing that I x is the oynomia which encodes the divisor of the norm of f 1 4z 4 f z f 1 4z 4 f z. To study this divisor, we first reca the foowing moduar transformation roerties. Proosition 3.5. If a c d b SL Z with b c 0 mod 4 and gz := f 1 4z 4 f z, then g az+b cz+d = gz. The roof of Theorem 1.3 is comete once we estabish the foowing emma. Lemma 3.6. If 1 mod 4 is a rime, then we have I x = H 3 x a H 4 x b H D x. D D,

16 16 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE Proof. By Proosition 3.5, z H is a root of gz gz if az+b cz+d with b c 0 mod 4. This eads to the quadratic equation = z for a b c d SL Z c 4 z + d a z b 4 4 = 0. In view of Lemma 3.3 and since the Hibert cass oynomias are irreducibe, we simy need to show that for a negative discriminant of the form D := x 4 with x, f Z 16f there exist two integra binary quadratic forms Q 1 := c 1 4f x + d 1 a 1 xy b 1 4f 4f y Q := c 4f x + d a xy b 4f 4f y, which are inequivaent under Γ 0 with discriminants D such that a 1 b 1 c 1 d 1, a b c d SL Z with b 1 b c 1 c 0 mod 4. We can easiy show that there exist a 1, a, d 1, and d such that a 1 d 1 a d 1 mod 16f and d 1 + a 1 = d + a = x. Moreover we can choose a 1, a, d 1, and d such that d 1 a 1 d a 0 mod 4f. We et b 1 = b = 4f, c 1 = a 1 d 1 1/4f, and c = a d 1/4f. Then a 1, a, b 1, b, c 1, c, d 1, and d are integra with b 1 c 1 b c 0 mod 4 and a 1 d 1 b 1 c 1 = a d b c = 1. It is we-known that if α 1 x + α xy + α 3 y and β 1 x + β xy + β 3 y are two integra rimitive binary quadratic forms with α 3, β 3 > 0, which are equivaent under Γ 0, then α β mod. This fact imies that Q 1 and Q are not equivaent under Γ 0 since d 1 a 1 4fx mod and d a 4fx 4f 4f mod. Here 4f denotes the inverse of 4f mod. 4. Traces of J m z for = 5, 13, and 17 In this section, we give formuas for the K m z and rove Theorem 1.4. For = 5, 13, and 17 and m 0 with m 1 there is a unique Let K m z = q m + Oq M + E 1 z = Γ 0, n/d d q n, d n 1, + d n=1 d n n=1 L E z =. d q n, be the two Eisenstein series in M Γ 0, corresonding to the cuss 0 and. Here Ls, χ denotes the usua Dirichet series with Dirichet character χ. Then, we can

17 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 17 exress K 0 as a inear combination of these two Eisenstein series. 4.1 K 0 z = L 1, E 1 z + E z. In addition, for = 5 and = 13, we can exress K 1 4. K 5 1 z = 5E5 z 55E 5 1 ze5 z, E 5 1 z and 4.3 K 13 1 z = E13 z 3E 13 1 ze 13 z. E 13 1 z For = 17, we need M 17 z = σ 1 n 17σ 1 n/17q n the Eisenstein series for M Γ 0 17 and n=1 S 17 z = q q q 4 q 5 +, the cus form associated to the eitic curve X Then, 4.4 K 17 1 z = E17 zm 17 z S 17 z + E17 z 4 z in terms of E 1 z and E z. 17E17 1 z. 4 A the K m z can be exressed in terms of K 1 z using Hecke oerators. For this descrition, suose that m 1 and m 1. Let K m z = q m + a m nq n M Γ 0, n=1 be the unique form such that a m n = 0 if n = 1. It is straightforward to verify the foowing facts about the K m z and K m z. Lemma 4.1. If gcdm, = 1, then 1 m K 1 z T m = K m z K m z m m = 1, = 1, where T m is the usua Hecke oerator on M Γ 0,. Lemma 4.. If gcdm, = 1 and b 1, then 1 K K m z U z K b z = m b m b K m z U b m m = 1, = 1.

18 18 KATHRIN BRINGMANN, KEN ONO, AND JEREMY ROUSE Lemma 4.3. If gcdm, = 1 and b 1, then 1 K K z + K z U b m z = m b m b K m z m m = 1, = 1. Using these emmas, one can determine the q-exansions of a the K m z and in terms of the q-exansion of K 1 z. Now we rove Theorem 1.4. K m z Proof of Theorem 1.4. From 3.5 it foows that a weaky hoomorhic moduar form f M + Γ 0, is hoomorhic at infinity if and ony if it is hoomorhic at zero. Since S Γ 0, = 0 for = 5, 13 and 17, this imies that f M + Γ 0, is determined by its rincia art and constant term. The rincia art of Φ z comes from that of A z by 1.5. The definitions of J m z and A,f z imy that A z = 1 = d m d d m d x d mod 4 x <d x d mod x <d q x d 4. q x d 4 + x d mod x <d q x d 4 It coincides with the rincia term of the right hand side of the identity in Theorem 1.4. Simiary, the constant term of Φ z comes from that of B z and is ceary σ 1 m. Again, this coincides with the constant term on the right hand side of the identity in Theorem 1.4. This roves the theorem. References [1] R. E. Borcherds, Automorhic forms on O s+, R and infinite roducts, Invent. Math , ages [] J. H. Bruinier, Borcherds roducts on O, and Chern casses of Heegner divisors, Sringer Lect. Notes, 1780, Sringer-Verag, Berin, 00. [3] J. H. Bruinier and M. Bundschuh, On Borcherds roducts associated with attices of rime discriminant, Ramanujan J. 7, 003, ages [4] J. H. Bruinier and J. Funke, Traces of CM-vaues of moduar functions, rerint. [5] H. Hijikata, A. Pizer, and T. Shemanske, The basis robem for moduar forms on Γ 0 N, Proc. Jaan Acad. Ser. A Math. Sci , ages [6] F. Hirzebruch, Hibert moduar surfaces, L Enseign. Math , ages [7] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hibert moduar surfaces and moduar forms with Nebentyus, Invent. Math , ages [8] W. Kohnen, Newforms of haf-integra weight, J. Reine Angew. Math , ages 3 7. [9] W. C. Li, Newforms and functiona equations, Math. Ann , ages [10] K. Ono, The web of moduarity: arithmetic of the coefficients of moduar forms and q-series, CBMS Regiona Conference Series in Mathematics, No. 10, Amer. Math. Soc., Providence, R.I., 004.

19 TRACES OF SINGULAR MODULI ON HILBERT MODULAR SURFACES 19 [11] J. Rouse, Vanishing and Non-Vanishing of Traces of Hecke Oerators, to aear in Trans. Amer. Math. Soc. [1] G. Shimura, On moduar forms of haf integra weight, Ann. of Math , ages [13] D. Zagier, Traces of singuar modui, Motives, oyogarithms and Hodge theory, Part I Irvine, CA, , Int. Press Lect. Ser., 3, I, Int. Press, Somervie, MA, ages Deartment of Mathematics, University of Wisconsin, Madison, Wisconsin E-mai address: bringman@math.wisc.edu E-mai address: ono@math.wisc.edu E-mai address: rouse@math.wisc.edu Deartment of Mathematics, University of Wisconsin, Madison, Wisconsin 53706