TRACES OF SINGULAR MODULI. Don Zagier

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1 TRACES OF SINGULAR MODULI Don Zagier Introduction. Singular moduli is the classical name for the values assumed by the modular invariant jτ or by other modular functions when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtain instead a result for their traces, and a number of generalizations. The results are closely related to a theorem of Borcherds [] of which we will give a new proof and a generalization. The formula for the traces of the singular values of j can be stated in two equivalent ways. The first one, formulated in, gives these traces as the Fourier coefficients of a certain modular form of weight 3/. The second one, stated in and proved in 3 4, gives them as the solutions of a certain pair of recursion equations. The proof of one of the two recursions is a variant of the classical proof of the Kronecker class number relations and is quite amusing. Borcherds s theorem is recalled in 5 and the relationship to our result is explained. In a few words, it is as follows. There is a sequence of modular forms f d τ of weight /, one for each order in an imaginary quadratic field, and a sequence of modular forms g D τ of weight 3/, one for each order in a real quadratic field. Our formula says that the trace of a singular modulus of discriminant d is the coefficient of q d in the fixed modular form g τ; Borcherds s theorem implies that it is the coefficient of q in the variable modular form f d ; and the relationship of the two results is that, for any d and D, the Dth Fourier coefficient of f d is the negative of the dth Fourier coefficient of g D. In fact, by a relatively straightforward application of Hecke operators one can obtain the full statement of Borcherds s theorem which describes the minimal polynomials of singular moduli, not just their traces, in terms of the n -th Fourier coefficients of f d as n varies from the formula for the traces. This is done in 6. The third part of the paper describes generalizations in three different directions: 7 gives an interpretation of the other Fourier coefficients as traces roughly speaking, the Dth Fourier coefficient of f d, or the dth coefficient of g D, describes the relative traces of singular moduli of discriminant dd from the Hilbert class field of Q dd to its real quadratic subfield Q D; in 8 a generalization to the traces of singular values of Hauptmodules for certain other groups of genus 0 is described they are equal to the Fourier coefficients of Jacobi forms of weight and index >; and in 9 we state a generalization to modular forms of other small half-integral weights there are modular forms of weight k+/ and 3/ k satisfying the same sort of duality relation as the f d and g D, and their Fourier coefficients again can be interpreted as traces, but now of singular values of certain non-holomorphic modular functions. These results can be seen as analogues of the Shimura correspondence between modular forms of integral and half-integral weight. We mention that the first results of this paper were found in 994 and were quoted and applied in the two papers [8] and [9] by Masanobu Kaneko, who used Theorem to obtain a closed formula for the Fourier coefficients of jτ in terms of traces of singular moduli.. The trace of jα. Throughout this paper, d denotes a positive integer congruent to 0 or 3 modulo 4. We denote by Q d the set of positive definite binary quadratic forms Q = [a,b,c] =

2 ax +bxy +cy a, b, c Z of discriminant b 4ac = d, with the usual action of the modular group Γ = PSL,Z. To each Q Q d we associate its unique root α Q in H = upper half-plane. The value of jα Q depends only on the Γ-equivalence class of Q and, as is well-known, is an algebraic integer. More precisely, if h d denotes the class number of d, i.e., the number of Γ- equivalence classes of primitive quadratic forms in Q d, then each of the corresponding h d values of jα Q is an algebraic integer of exact degree h d and they form a full set of conjugates, so that the sum of these numbers is the trace. For instance, we have h 3 = h 4 = h 7 = h 8 = and h 5 =, and the corresponding j-values are j +i 3 j +i 5 +i 7 = 0, ji = 78, j = 3375, ji = 8000, =, j +i =. 4 We want to give a formula for the traces. Actually, it is convenient to make three small changes. First, we replace the function j by the normalized Hauptmodul for Γ = PSL,Z, Jτ = jτ 744 = q q q + τ H, q = e πiτ. Secondly, we sum over all forms in Q d, not just the primitive ones. Finally, we weight the number Jα Q by the factor /w Q, where w Q = Γ Q = or 3 if Q is Γ-equivalent to [a,0,a] or [a,a,a], and otherwise. We therefore define the Hurwitz-Kronecker class numbers Hd and the modular trace function td by Hd := Q Q d /Γ w Q, td := Q Q d /Γ w Q Jα Q d > 0, d 0 or 3 mod 4. For instance, from the examples in we see that H3 = 3, H4 =, H5 = and t3 = = 48, t4 = Further values are given in the following small table. = 49, t5 = = 953. d Hd /3 / 4/3 3/ td d Hd 3 4/3 3 td On the other hand, we define a meromorphic modular form of weight 3/ by the formula gτ := θ τ E 44τ η4τ 6 = q + 48q 3 49q q 7 756q where θ, E 4 and η are the classical modular forms θ τ = n= n q n, E 4 τ = +40 n= n 3 q n, ητ = q/4 qn q n of weights /, 4, and /, respectively. Comparing the first few coefficients of g with the above table suggests the following modular formula for the traces td : n=

3 Theorem. Write the Fourier expansion of gτ as gτ = d Bdqd. Then td = Bd d > 0. 4 The proof of this result will occupy the next three sections.. A recursion for the numbers td. Let θτ be the theta series n Z qn and U 4 the operator c n q n c 4n q n. The fact that gτ is a modular form of weight 3/ having non-zero Fourier coefficients only for q d with d 0 or 3 modulo 4 the so-called Kohnen pluscondition ; see 5 implies that the function gθ U 4 is a holomorphic modular form of weight on the full modular group PSL,Z and hence vanishes identically. Similarly, the image [g,θ] U 4 of the Cohen bracket [g,θ]τ = g τθτ 3gτθ τ under U 4 is a holomorphic modular form of weight 4 on PSL,Z and hence is a multiple of E 4 τ. Writing these two facts out in terms of Fourier coefficients gives the two identities B4n r = 0, r B4n r = 40σ 3 n n 0, r Z r>0 where σ 3 0 = /40 and σ 3 n for n is the sum of the cubes of the positive divisors of n. These formulas can be rewritten B4n = 40σ 3 n r B4n r, B4n = B4n r, r 4n+ r 4n+ and in this form obviously determine all the Bd by recursion: B = 40σ 3 0 =, B0 = B =, B3 = 40σ 3 4B0 = 48, B4 = B3 B0 = 49, etc. Thus to prove 4 it suffices to prove the corresponding identities for td; i.e., Theorem follows from: Theorem. For all integers n we have the identities 4 if n is a square, t4n r = if 4n+ is a square, r < n 0 otherwise 5 and r< n 8n if n is a square, r t4n r = 40σ 3 n+ 4n+ if 4n+ is a square, 0 otherwise. 6 The proofs of equations 5 and 6 are given in the next two sections and are rather pretty. In particular, the proof of 5 is a refinement of the classical proof of Kronecker s class number relation r < nh4n r = { /6 if n is a square, max{d, n/d} otherwise. d n 3. Proof of the first recursion for td. In this section we prove equations 5 and 7. Let Φ n X,Y Z[X,Y] denote the well-known nth modular polynomial, defined by the property Φ n X, jτ = X jm τ τ H, 8 M Γ\M n 3

4 where M n denotes the set of integral matrices of determinant n, with M and M being identified, and denotes the usual action of PGL +,R on H. We exclude for the moment the case when n is a square. Then observing that the function Φ n jτ,jτ vanishes exactly at the points τ H which are fixed by some M M n, and that these are just the points α Q with Q a positive definite quadratic form whose discriminant has the form r 4n for some integer r = trm satisfying r < n, we get the classical identity Φ n X,X = const. r < n H 4n r X, 9 where the constant factor which is always ± and is easily determined does not matter to us and the function H d X is defined by H d X = Q Q d /Γ X jαq /w Q d > 0, d 0 or 3 mod 4. 0 This function is X /3 times a polynomial in X if d/3 is a square, X 78 / times a polynomial in X if d is a square, and a polynomial in X otherwise. Clearly we have the q-expansion H d jτ = Q Q d /Γ q Jα Q +Oq /w Q = q Hd tdq +Oq. On the other hand, taking as representatives for Γ\M n the matrices a b 0 d with ad = n, b d, we get Φ n jτ,jτ = d ad=nb= = ad=nb= j τ j aτ +b d d q e πib/d q a/d +O q >0 = q d q a +O q > ad=n = ad=n ±q max{a,d} ε a q + Oq, where ε a is if a d = which can happen only if 4n+ is a perfect square and 0 otherwise. Comparing this with 9 and, we get both equations 5 and 7 for non-square n. If n is a square, then Φ n X,Y is divisible by Φ X,Y = X Y and we must replace 9 by Φ n X,Y Φ X,Y = const. H 4n r X/ H 4 r X. Y=X r < n When we substitute X = jτ, the left-hand side is ±q A + Oq by essentially the same calculation as before, but omitting the factor corresponding to a = d = n, b = 0. Note that now 4n+ is not a square. Hence r < n t4n r = r < t4 r = Proof of the second recursion for td. The method used to prove 5 was essentially geometric: we identified two different functions up to a constant by observing that they had the same divisors, and then obtained the desired identity by computing the expansions of both 4 r <

5 expressions at infinity. This does not generalize directly to the higher weight identity 6, but does if we think of the logarithmic derivatives of the previous functions as modular forms of weight and replace these by modular forms of higher weight. One could state this geometrically by working on the Kuga variety over H/Γ, or with appropriate local coefficient systems over H/Γ, but we will not do this. More precisely, for each positive integer d 0 or 3 mod 4 we define Λ d τ = d dτ logh djτ = w Q Q Q d /Γ j τ jτ jα Q. This is a meromorphic modular form of weight, holomorphic at infinity and with a simple pole of residue / Γ α at each point α H satisfying a quadratic equation over Z of discriminant d, and is uniquely characterized by these properties since there are no holomorphic modular forms of weight on Γ. The analogue of the logarithmic derivative of equation 9 is then the following: Proposition. Let n be a positive integer which is not a square, τ H. Then where Mτ := E 4 τe 6 τ τ M Γ\M n E 4 Mτ jτ jmτ = 4πi r < n r nλ 4n r τ, aτ +b cτ +d and E 4 Mτ := n 3 cτ +d 4 E 4 Mτ for M = a b c d Mn. Proof. Both sides of the equality are meromorphic modular forms of weight, holomorphic at infinity, and with only simple poles, so it suffices to compare the residues. Let α H be a fixed point of the action of M = a b c d Mn, where we can suppose that c > 0. The number λ = cα+d is a quadratic algebraic integer of trace r = trm and norm n, and E4 τe 6 τ E 4 Mτ Res τ=α = E 4αE 6 α n 3 λ 4 E 4 α τ jτ jmτ α nλ j α = λ 3 πiλ λ. We leave to the reader the modifications necessary if α is a root of E 4 or E 6, i.e. a point with a non-trivial stabilizer in Γ. By combining the matrices M and nm = d b c a, which have the same fixed points but conjugate values of λ, we can replace the expression λ 3 /λ λ by λ3 λ 3 /λ λ = r n. The proposition follows. The rest of the proof is similar to the calculation in the last section. Denote by S n τ the expression on the left-hand side of and write its expansion at infinity as C 0 +C q+. From and the expansion πi Λ dτ = w Q Q Q d /Γ q +Oq q Jα Q +Oq = Hd+tdq +Oq, we obtain C 0 = n r H4n r, C = r < n n r t4n r. r < n On the other hand, again taking as a set of representatives for Γ\M n the matrices a b 0 d with 0 b < d and ad = n, we find that S n τ = ad=n a3 S a,d τ where S a,d τ = E 4τE 6 τ E 4 aτ +b/d τ d j τ j aτ +b/d. 0 b<d 5

6 If a < d, then we can calculate the q-expansion of S a,d τ as S a,d τ = q 40+Oq +40ζq a/d +40σ 3 ζ q a/d + d q ζ q a/d +Oq a/d ζ d = = 40σ 3 lq m+al m/d 40q +Oq l,m 0 l m mod d = + 40δ a, σ 3 n+δ a,d q + Oq, since the only pairs l,m which contribute to the powers q 0 and q are 0,0,,, d,0 if a = and 0,d if a = d. Similarly, for a > d we find S a,d τ = l 0, m l+m 0 mod d = δ a,d+ q + Oq, 40σ 3 lq m+al m/d 40q +Oq since this time only the pair l,m = 0,d contributes. Summing over all divisors a of d, we find that the first two Fourier coefficients of S n τ are given by C 0 = 0<a< n a n a 3, { 3n+ if 4n+ =, C = 40σ 3 n 0 otherwise, where the last term comes from the two factorizations of n as ad with a d = ±. Comparing the formula for C with the one given above, we obtain equation 6 or more precisely, the difference between times equation 6 and n times equation 5, but we have already proved 5 for nonsquare n. The proof when n is a square is similar, except that we must also compute the Fourier expansion of S a,d τ in the case when a = d; we leave this calculation to the reader. By comparing the two expressions obtained for C 0 we also get the well-known identity r < nn r H4n r = d n { n/ if n is a square, mind, n/d 3 0 otherwise, which together with 7 recursively determines the Hurwitz-Kronecker class numbers Hn. 5. Relation to Borcherds s theorem. By equation, the Hurwitz-Kronecker class number Hd and the modified-trace-of-j function td can be interpreted as the first two Fourier coefficients of the logarithmic derivative of H d jτ, where H d X is the nearly polynomial function 0. Borcherds s theorem describes the full Fourier expansion of the logarithmic derivative of H d jτ in terms of certain Fourier coefficients of a meromorphic modular form f d τ of weight /. In this section we state his result and show how to deduce from it the formula for td given in Theorem. In 6 we show how, conversely, Borcherds s formula follows from Theorem and some simple considerations about Hecke operators in integral and half-integral weight, and in the following section we show how similar ideas can be used to identify the remaining Fourier coefficients of f d τ in terms of traces of singular moduli. To define the functions f d τ, we need some basic facts from the theory of modular forms of half-integral weight. For any integer k, denote by M k+/ Γ 0 4 the space of functions which are holomorphic in H and at the cusps and which transform under the action of Γ 0 4 like θτ k+, 6

7 where θτ is the Jacobi theta-function n qn already used in. The Kohnen plus-space is the subspace consisting of functions whose Fourier expansion at infinity has the form cnq n where cn is non-zero only for integers n satisfying k n 0 or modulo 4. We will denote this space simply by M k+/, and consider its elements to be modular forms of level one, because of Kohnen s basic result that M k+/ is isomorphic as a Hecke module to the space M k = M k Γ of holomorphic modular forms of weight k on the full modular group. The definition of the action of the Hecke algebra on M k+/ will be recalled in the next section. From the isomorphism M k+/ M k, or directly, we see that the space M k+/ is onedimensional spanned by θ itself for k = 0 and is zero for k = or k < 0. To get non-trivial spaces in these low half-integral weights, we have to allow modular forms with poles. Let M k+/! Γ 04 be defined like M k+/ Γ 0 4 but with the condition holomorphic at the cusps relaxed to meromorphic at the cusps Borcherds [] calls such forms nearly holomorphic and define a plussubspace M k+/! of it by the same condition on the Fourier coefficients as before where now cn may be non-zero for a finite number of negative values of n. Equivalently, a function fτ belongs to M k+/! if and only if fτ 4τn belongs to M k+n+/ for some integer n > 0. The space M! k+/ is infinite-dimensional for every k. In particular, for every integer d 0 with d 0, 3 mod 4 there is a unique modular form f d M /! having a Fourier development of the form f d τ = q d + D>0AD,dq D, 3 and the functions f 0, f 3, f 4, f 7,... form a basis of M /!. The existence and uniqueness of the forms f d =Lemma 4. of [] are easy to see. Indeed, the uniqueness is clear since the difference of any two functions satisfying the definition of f d would be an element of M / with constant term 0 and hence would vanish. For the existence, we first construct f 0 and f 3 by hand f 0 τ is just θτ itself, and a non-trivial linear combination of f 3 and f 0 can be obtained e.g. as [θτ, E 0 4τ]/ 4τ, where E 0 τ = E 4 τe 6 τ is the Eisenstein series of weight 0 on Γ and [θτ, E 0 4τ] = θτe 04τ 5θ τe 0 4τ the Cohen bracket of θτ and E 0 4τ. Then for each d 4 we obtain f d τ by multiplying f d 4 τ by j4τ to get a plus -form of weight / with leading coefficient q d and then subtracting off multiples of f d to successively kill the coefficients of q d for 0 d < d. The process is effective and we find, for instance, that the Fourier expansions of the first few f d begin as follows: f 0 = + q + q 4 + q 9 + q 6 + Oq 5, f 3 = q 3 48q +675q q q q 9 +Oq, f 4 = q 4 +49q q q q q 9 +Oq, f 7 = q 7 49q q q q 8 +Oq 9, f 8 = q q +6456q q q 8 +Oq 9, f = q 335q q q q 8 +Oq 9. Theorem 3 Borcherds [Bo, p. 04]. Let d > 0, d 0 or 3 mod 4. Then H d jτ = q Hd q n An,d. 4 Example. From equation 4 for d = 3 and the above expansion of f 3 we find n= jτ /3 = q /3 q 48 q 675 q Comparing equation 4 with, we obtain : 7

8 Corollary. td = A,d for all d > 0. This corollary and Theorem give two apparently different ways to calculate td: Theorem gives all of the td as the Fourier coefficients of a single modular form gτ of weight 3/, while the Corollary gives each td as a coefficient of a different modular form f d of weight /. To see how the two formulas are related, we define a second sequence of modular forms, analogous to the f d but now in weight 3/ rather than /, namely for each positive integer D congruent to 0 or modulo 4 we define g D τ as the unique form in M 3/! having a Fourier expansion of the form g D τ = q D + d 0BD,dq d. 5 For D = this is just the function gτ defined in 3, so B,d is the same as the coefficient which was denoted Bd in. We can construct g 4 either by applying the weight 3/ Hecke operator T to g cf. 6 or in the same way as f 3 was constructed from f 0 [g τ,e 0 τ]/ 4τ is a linear combination of g τ, g 4 τ, and g τj4τ, and then get the forms g D τ for D > 4 inductively by multiplying g D 4 τ by j4τ and subtracting a suitable linear combination of g D τ with D < D; and the uniqueness of the g D follows from the fact that M 3/ = {0}. Here are the beginnings of the Fourier developments of the first few g D : g = q +48q 3 49q 4 +49q 7 756q q 53008q +Oq 5, g 4 = q 4 675q q q q 8 +Oq, g 5 = q q q q q 8 +Oq, g 8 = q q q q q 8 +Oq, g 9 = q q q q q 8 +Oq. Comparing these expansions with those of the f d given above, we notice that the coefficients occurring are the same up to sign. The next theorem, which by virtue of the above corollary reduces to 4 for D =, says that this is true in general. Theorem 4. AD,d = BD,d for all D and d. Proof. Consider the function Fτ, τ = f 0τ g 4 τ +f 3 τ g τ j4τ j4τ τ, τ H. It is meromorphic, with a simple pole whenever 4τ and 4τ are Γ-equivalent. If Iτ and Iτ are big enough to be precise, bigger than /4, then this is the case if and only if 4τ and 4τ differ by an integer. On the other hand, we can easily check that n n f 0 τ + g4 τ+f 3 τ + g τ = +in j 4τ n = 0,,, 3, 4 4 πi sotheresidueoff withrespecttoτ, keepingτ fixedatτ = τ +n/4n Zequals+i n /8πi. This is the same as the residue of the elementary function F 0 τ,τ = q 4 +q q 3 /q 4 q 4, where q j = e πiτ j, so F F 0 is a holomorphic function of q,q in a neighbourhood of 0,0 C, i.e. Fτ, τ = q4 +q q 3 q 4 q4 + CD,dq D q d 8 D>0, d 0

9 in this neighbourhood for some coefficients CD,d. Here we can write D > 0 rather than D 0 in the summation because both F and F 0 obviously vanish on the q -axis; moreover, it is clear that CD,d is 0 unless D and d satisfy the usual congruences D 0, mod 4, d 0, 3 mod 4. Expanding F 0 in a geometric series in the region q < q, we find that the Taylor expansion of F with respect to τ is given by Fτ, τ = d 0 d 0,3 mod 4 D>0 D 0, mod 4 q d + D>0 D 0, mod 4 CD,dq D q d τ H, Iτ. Since F is a plus-modular form of weight / with respect to τ, the expression in parentheses also is one, so it equals f d τ. Hence CD,d = AD,d. Similarly, from the expansion Fτ, τ = q D + CD,dq d q D τ H, Iτ d 0 d 0,3 mod 4 and the fact that F is a plus-modular form of weight 3/ with respect to τ we deduce that CD,d = BD,d. This completes the proof. Notice that the Fourier expansions of F just given also yield formulas for all f d τ d 0 as linear combinations of f 0 τ and f 3 τ with coefficients in Q[j4τ], namely f d τ = [d/4] n=0 Coeff q d [d 3/4] g4 τ j4τ n+ j4τ n f 0 τ + n=0 Coeff q d g τ j4τ n+ j4τ n f 3 τ, and similarly for all g D τ D > 0 as linear combinations over Q[j4τ] of g τ and g 4 τ. Remark. As Masanobu Kaneko has pointed out to me, there is an easier proof of Theorem 4. It is easily checked that the constant term of the q-expansion of fg vanishes for any f M! / and g M! 3/ because fg U 4 is a modular form of weight on SL Z with a pole only at infinity, and every such form is the derivative of a polynomial in j. Applying this to f = f d, g = g D immediately gives the assertion. We have retained the original proof because of the supplementary information it gives about the functions f d and g D. 6. Hecke operators. Theorems and 4 together clearly imply the corollary to Borcherds s theorem, since one has td = B,d = A,d. In fact, they imply the full theorem in a more or less straightforward way, as we now explain. To obtain the full minimal polynomial H d X of the algebraic number jα Q, we need not just its trace but also the traces of all of its powers or at least of all of its powers up to the class number. Instead of working with the traces of j or of J, we work with the functions J m, defined for every integer m 0 as the unique holomorphic function on H/Γ with a Fourier expansion beginning q m +Oq. For m = 0 this is the constant function and for m = it is the function Jτ defined in, while the next three values have Fourier developments starting J τ = q q q q 3 +, J 3 τ = q q q q 3 +, J 4 τ = q q q q 3 +. The uniqueness of these functions is obvious the difference of any two functions satisfying the definition of J m would be a holomorphic cusp form of weight 0 and hence 0 and the existence 9

10 also, since J m can clearly be obtained as a monic polynomial of degree m in jτ, e.g. J τ = jτ 488jτ+59768, J 3 τ = jτ 3 3jτ jτ , J 4 τ = jτ 4 976jτ jτ jτ The minimal polynomials of all jα Q are then determined if we know the numbers t m d := Q Q d /Γ w Q J m α Q 6 for all d and all m or even just for 0 m Hd. To get them, we must generalize Theorems and 4 to include Hecke operators. For any integer m let A m D,d and B m D,d denote the coefficient of q D in f d Tm and the coefficient of qd in g D 3 Tm, respectively, where Tm and 3 Tm denote the action of the mth Hecke operator Tm sometimes denoted Tm or T + m on the space M k!, k = / or 3/. The formulas for this action at least for holomorphic modular forms, but nothing changes if we allow the pole at infinity are given in many places, e.g. [0], Prop. or, in the equivalent language of Jacobi forms, [4], Th In particular, for m = p prime we have A p D,d = pap D, d + D AD, d + Ap D, d 7 p and B p D,d = BD,p d + d BD, d + pbd, p d 8 p with the convention that Ap D,d = 0 unless p D is an integer congruent to 0 or modulo 4, and similarly for BD,p d, while for D = and m arbitrary we have A m,d = n mnan, d. 9 Theorem 5. With the above notations, we have i H d jτ = q Hd exp m=t m d qm m ii t m d = B m,d for all m and d; iii A m D,d = B m D,d for all m, D and d. for all d; Note that parts ii and iii of this theorem are the generalizations of Theorems and 4 to m, while Theorem 3 is obtained by combining i, ii, the case D = of iii, and equation 9. Proof. i This identity follows directly from the definitions 0 and 6 of H d τ and t m d together with the identity jτ jz = q exp J m z qm m m= q = e πiτ, Iτ 0, 0 an identity which has been discovered independently by a number of authors and which can be proved easily by noting that the logarithmic derivative with respect to τ of either side is, for fixed 0

11 τ with sufficiently large imaginary part, a Γ-invariant meromorphic function of z whose only pole in the standard fundamental domain is a simple one of residue at z = τ and which vanishes at infinity. ii We first observe that the function J m is the composite of J with the mth Hecke operator Tm acting in degree 0, i.e., J m τ is the sum of JMτ with M ranging over Γ\M n. This is obvious, since J 0 Tm is a modular function with a pole q m +Oq at infinity and no other poles. It follows that t m d is the sum of the values of J α summed over all α S d Tm, where S d H/Γ is the set of α Q, Q Q d /Γ counted with multiplicity /q Q. But S d Tm is a combination of sets S dm /n with n ranging over the divisors of m. For example, S d Tp for p prime is equal to S dp + d Sd +ps d/p. The proof of this is an exercise, carried out, e.g., on p pages 90 9 of [5]. Assertion ii for m = p then follows from Theorem and equation 8. The general case can be proved similarly, or one can use the multiplicative properties of Hecke operators which are the same in integral and half-integral weight and induction on the number of prime factors of m. iii Again we need only consider the case when m = p is prime. Formula 7 for the Fourier coefficients of f d Tp is true for all D, negative as well as positive, so from the defining property AD,d = δ D, d D < 0 of f d we deduce that A p D,d = pδ D, d/p + d δd, d + δ D, dp p for D < 0, i.e., the principal part of f d Tp at infinity is + d pq d/p q d + q dp with p the first term to be omitted unless d/p is a discriminant. It follows that f d Tp equals pf d/p + d fd +f dp andhencethatitsdthfouriercoefficienta p D,disequalto pad,d/p + p d AD,d+AD,dp. Comparing with 8, we find that iii follows from Theorem 4. One p could also give a dual proof by using 8 to show that g D 3 Tp = pg Dp + D gd +g D/p, and p there is yet a third proof along the lines of the proof given for the special case Theorem 4 by looking at the singular part at infinity of the images of the two-variable modular form Fτ,τ under the actions of Tp on τ in weight / and τ in weight 3/. This completes the proof of Theorem 5 and of Borcherds s product formula 4 from our more elementary formula 4. In the remaining sections of the paper we describe without detailed proofs generalizations of these results in various different directions. 7. First generalization: other discriminants. The theorems we have discussed so far give an interpretation of the numbers A m D,d = B m D,d as traces when D is equal to or, more generally, whenever D is a perfect square. Is there a similar interpretation for the other values? The answer is yes, but now we have to take twisted traces in which the conjugates of jα Q are multiplied by a genus character before summing. We recall the definition of these characters. Let D and d be a positive and a negative discriminant, respectively. For ease of exposition we suppose them to be fundamental and coprime. Comments on other situations are given in Remark 3 below. Then the genus character χ = χ D, d assigns to any quadratic form Q of discriminant = d p dd a value ± defined by χq = D p dividing Dd. This is independent of the choice of p. where p is any prime represented by Q and not Theorem 6. Let D > and d < 0 be coprime fundamental discriminants, and χ = χ D, d the

12 associated genus character. Then AD,d = D Q Q dd /Γ χqjα Q. Example. Take D = 5, d = 3, the first non-trivial case. There are two Γ-equivalence classes of quadratic forms of discriminant 5, represented by the forms Q = [,,4] and Q = [,,] with χq =, χq =. The corresponding values of jα Q are given in. We see that the right-hand side of is jα Q jα Q / 5 = 85995, in agreement with the value A5,3 = B5,3 = given in 5. Remarks.. In formula we could omit the usual factor w Q because it is always for D and d as in the theorem, and similarly we could write jα Q rather than Jα Q because summing with the non-trivial character χ kills the additive constant by which j and J differ.. The right-hand side of is a priori a rational integer. Indeed, the sum of jα Q as Q ranges overtheformswithχq = +is, byclassfieldtheory, thetraceofthealgebraicintegerjα Q from the Hilbert class field of Q dd to its real quadratic subfield Q D. If we write this number as λ = a+b D with a and b in Z, then the sum over all jα Q is equal to Tr Q D/Q λ = a and therefore the weighted sum χqjα Q is equal to a+λ = b D. 3. If D and d are not coprime, then the statement of the theorem is unaltered but χq must be defined as 0 for imprimitive forms Q which are divisible by some prime dividing both D and d. Thus for instance the values A8,4 = and A5,5 = are obtained as jα [,0,8] jα [3,,3] / 8 and jα [,,9] jα [3,3,7] / 5, respectively. The cases when D and d are not assumed to be fundamental can be given by a suitable modification of the reader can find the right formulas easily by a little numerical experimentation or can be obtained from the case of fundamental discriminants by suitable applications of Hecke operators. We can also give a generalization of the Borcherds product expansion 4, but with a difference. For D and d as in the theorem we define a rational function H D,d X by H D,d X = X jαq χq. Q Q Dd /Γ For the same reasons as in Remark, this belongs to Q DX and is mapped by the nontrivial element of GalQ D/Q to its inverse. It follows that the power series expansion of logh D,d jτ, which begins with, has a logarithm belonging to DQ[[q]]. Theorem 7. Let D, d and H D,d X be as above. Then where H D,d jτ = P D q n An D,d P D t := exp D n= r= D t r r r The function P d t is a rational function of t with coefficients in Q D. 3 Q D[[t]]. 4 Examples. For D = we clearly have P t = t, so that 4 reduces to 4 in this case. The next values of P D are P 5 t = φt+t φ t+t, P 8t = t+t + t+t, P t = 3t+t + 3t+t,

13 where φ = + 5/ is the golden ratio and φ = 5/ its Galois conjugate. A typical example of the theorem D = 8, d = 3 is the expansion jτ jτ = q n +q n n= + q n +q n A8n,3 of the ratio of jτ j 6 and jτ j 3/, whereas Borcherds s product expansion 4 would give the Fourier expansion of the product of the same two expressions in the form jτ jτ = q q n An,4, Theorem 7 follows from Theorem 6 in the same way as explained in 6 for the special case D =. If we denote by td,d the sum occurring in the right-hand side of twisted trace and by t m D,d the corresponding sum with j replaced by J m, then we can use Hecke operators in integral and half-integral weight to show that generalizes to t m D,d = A m D,d D for all m, D and d. 5 n= From equations 0 and 5 we find H D,d jτ = exp t m D,d qm m m= = exp D A m D,d qm m m=, and combining this with the formula A m D,d = n m n D An D, d, m/n which is the generalization of 9 to fundamental discriminants D >, gives 3. For the final statement of Theorem 7 we use the Gauss sum identity D D D = e πinr/d to get r n P D t = 0<n<D 0<n<D ζ n D t D n = N Qζ D /Q D ζd t ζd t ζd = exp πi/d. 6 N QζD /Q Digression on Hilbert modular surfaces, the Doi-Naganuma lifting, and twisted Borcherds products. To prove Theorem 6, two approaches are possible although to be honest I have not actually carried out either one in detail. One way would be to generalize the proof which Borcherds gave in [] for the case D =. This proof is based on the circle method and uses explicit calculations of the Fourier developments of Poincaré series, so that it is a kind of multiplicative analogue of the method which was introduced in [3] to construct the Doi-Naganuma lifting from elliptic to Hilbert modular forms and applied in [] for the usual Shimura lifting from modular forms of half-integral to modular forms of integral weight. But in keeping with the spirit of the rest of this paper, one would like to give a more elementary proof, and also to see the twisted traces td,d for a fixed D and variable d appear naturally as the Fourier coefficients of a single modular form g D rather than a collection of modular forms {f d }. The method used to prove Theorem in this paper was to reduce it to the set of recursions given in Theorem and then prove these by 3

14 a geometric method. Specifically, to prove Theorem we used the polynomial Φ n X,Y, which is the function on P P = H/Γ H/Γ =: X whose zero-divisor is the graph of the Hecke correspondence T n X. The recursions 5 and 6 are then connected with the geometric fact that the intersection of T n with the diagonal T = H/Γ X is the union over r < n of the CM-cycles S 4n r H/Γ. We would like to see what replaces this when D >. The obvious replacement for the surface X when one is dealing with a non-square positive discriminant D, say the discriminant of a real quadratic field K, is the Hilbert modular surface X D = H H/SL,O K, and the usual replacement for the graph of the Hecke correspondence in X is the curve also denoted T n in X D which was studied in [7]. The result T n T = S 4n r on X then generalizes nicely to a similar formula on X D essentially T n T = S 4n r /D, at least for D prime, which we now assume. The problem is that T n is not in general the zero-set of any function on X D, because its homology class on the surface X D is not in general trivial. Indeed, the main result of [7] says that this homology class as n varies is the nth coefficient of a modular form with coefficients in H X D of level D and Nebentypus and that this modular D form describes i.e., is the kernel function for the Naganuma map [] from M Γ0 D, to D M SL,OK. This map is injective and the dimension of M Γ0 D, grows linearly with D D, so it becomes increasingly difficult to find linear combinations of the curves T n which are are homologically trivial. However, as well as the Naganuma lifting from M k Γ0 D, to Mk SL,O K, there is also D the lifting constructed earlier by Doi and Naganuma [3] from M k SL,Z to Mk SL,O K, and in[4, pp ] it was shown that it, too, could be described geometrically in terms of intersections numbers with certaindivisors T n onx D. But sincetherearenocusp forms of weight on SL,Z, this means that all of the divisors T n are homologically trivial and hence since X D is known to be simply connected principal, so that they can be obtained as the divisors of certain Hilbert modular functions Ψ D n. It is these functions which are the analogues of the functions Φ n jτ,jτ in the case D =. The definition of the curves T n, which we will not repeat here, is well suited to our twisted traces td,d and twisted Heegner cycles S D,d = χq[α Q ], because T n is defined as the difference of two components of T nd which are distinguished by the value of the Legendre symbol /D. Using this one finds easily that T n T = r S D,4n r and hence that the restriction of to T is r H D,4n r j, where we have identified T H/Γ with the j-line, and it is this Ψ D n which replaces the classical identity 9. We now have an amusing calculation. Write J m τ = q m + n>0 c mnq n. The coefficients c m n are given by c m n = d m,n md cmn/d, where cn = c n is the coefficient of q n in Jτ. It follows that c m n = mn c n m and that the function S m τ := q m n>0 c nmq n is a multiple of J mτ and hence is the obviously unique modular form in M! Γ with a Fourier expansionoftheformq m +Oq. Nowconsiderthefunctiong D θ U 4, whered asusualispositive and congruent to 0 or modulo 4. From g D = q D εd+oq εd = or 0 according as D is or is not a square we find g D θ U 4 = f qf D/4 +Oq, where the summation runs over all integers f with f < D, f D mod. Also, g D θ U 4 is a modular form of weight on Γ and is nearly holomorphic. Hence by our previous remark g D θ U 4 = f S D f /4, and comparing the coefficients of q n on both sides we obtain the identity r <4n AD, 4n r = f <D, f D mod D f c n + ε4n+d. 4 Combining this with the above identity Ψ D n τ,τ = r <4n H D,4n r jτ and equation 3, 4

15 we deduce after a short calculation that Ψ D n τ,τ = ν d, Trν>0 P D q Trν c n Dνν, where the product is over all all elements ν in the inverse different d = D / O K with Trν positive and Dνν either positive or else equal to n. This strongly suggests the identity Ψ D n τ,τ = P D e πiντ +ν τ c n Dνν 7 ν d /{±} in a suitable neighborhood of infinity, which is a kind of twisted Borcherds product expansion for the Hilbert modular function Ψ D n. This identity is similar apart from the twisting to an identity proved by Bruinier [] for the usual curves T n, and could doubtless be proved the same way, although I have not gone through the details. Indeed, the calculation should be much easier here because in Bruinier s case only certain linear combinations of the T n were homologically trivial and therefore representable as the divisor of a Hilbert modular function, and correspondingly the Poincaré series of weight, which here are trivial because Γ has genus 0, were more complicated because of the the presence of holomorphic cusp forms on Γ 0 D and Nebentypus of weight. If we take the logarithm of 7 and substitute the definition 4 of the power series P D t, then we find that the coefficient of e πiντ +ν τ in log Ψ D n τ,τ is equal to r D Dνν cn. r r r This formula is identical with a n = c n and k = 0 to the formula for the νth Fourier coefficient of the classical Doi-Naganuma lifting of a holomorphic modular form f = a n q n M k Γ cf. [3]. We can thus interpret the logarithm of Ψ D n as the Doi-Naganuma lifting of the nearly holomorphic modular form J n M 0Γ,! and 8 as saying that such a multiplicative version of the Doi-Naganuma lifting exists. Again, this is the analogue of the corresponding but harder result proved by Bruinier for the case of the Naganuma lifting. An interesting problem would be to show that the Hilbert modular function Ψ D n is defined over Q or K and to compute its values at CM points, as was done in [5] for the case D =. In analogy with the results of [5], we would expect these values to factor completely into small primes in a way which can be described explicitly. 8. Second generalization: other groups. Another evident way to generalize Theorem is to replace the function jτ by a modular function of higher level, and in particular by the Hauptmodul associated to other groups of genus 0. We discuss examples of this in this section. We will consider only the case when the genus 0 group is the group Γ 0N the extension of Γ 0 N by the group of all Atkin-Lehner involutions W p, p N. We will look in detail at the first five values N =, 3, 4, 5 and 6 although there are many further values of N for which Γ 0N has genus 0. The corresponding Hauptmodules with apologies for the mixed German/English spelling! are j z = ηz ηz ηz ηz 4 = q +437q +9656q +4000q q q q 6 +, ηz η3z j3z = η3z ηz = q +783q +867q q q q q 6 +, 8 8 ηz η4z j4z = η4z ηz = q +76q +048q +0q q q q 6 +, 5

16 j 5z = 6 6 ηz η5z η5z ηz = q +34q +760q +3345q 3 +56q q q 6 +, ηzηz η3zη6z ηzη3z ηzη6z j6z = = η3zη6z ηzηz ηzη6z ηzη3z = q +79q +35q +43q q q q 6 +. Each of these functions takes on algebraic integer values at CM points in the upper half-plane, and we want to give a formula for their traces. For N > one must consider Heegner points rather than arbitrary CM points. The easiest way to describe these is to say that they are the images of the roots α Q H of quadratic forms QX,Y = ax + bxy + cy with a 0 mod N and a,b,c,n =. The last condition is automatic if we assume that the d is not divisible as a discriminant by the square of any prime dividing N, an assumption we make from now on. This requires in particular that the discriminant dofqiscongruenttoasquaremodulo4n. Theclassofα Q moduloγ 0 Nhasasecondinvariant, apart from its class in Q d /Γ, namely the value of b mod N. If we fix this class, i.e., choose an integer β mod N with β d mod 4N and consider only forms Q = [a,b,c] with a 0 mod N and b β mod N, then the number of Γ 0 N-equivalence classes of such Q, counted as usual with multiplicity /w Q, is Hd, and the natural map from the quotient Q d,n,β /Γ 0 N with the obvious notation to Q d /Γ is a bijection. We could then define a trace t N,β d as the sum of the values of jn α Q with Q running over a set of representatives for Q d,n,β /Γ 0 N. In our case, however, we are looking at the smaller quotient of H by Γ 0 N, and here the value of t N,β d is independent of β. This is why we chose to restrict to this case. We can therefore define t N d = j N αq, w Q Q where the sum runs over Γ 0N-representatives of forms Q = [a,b,c] satisfying N a. Here are some numerical examples with N = and N = 3: t 4 = j +i = 5, t 7 = j = 3, t 8 = j = 5, and t 3 3 = 3 j = 4, t 3 = j3+ =. 6 In analogy with Theorem, we want to express these traces as the coefficients of some modular forms of weight 3/. In fact, the results will be more easily formulated in the language of Jacobi forms rather than modular forms of half-integral weight. If we were looking at Hauptmodules for groups Γ 0 N rather than Γ 0N, then we would actually have to work with Jacobi forms, since the trace would then depend on both the discriminant d and on a choice of square-root β of d mod 4N, as explained above. In our case it would be possible to work just with modular forms, but using Jacobi forms makes things simpler. We begin by recalling some basic definitions and facts from the theory of Jacobi forms, referring to [4] for a more detailed exposition. For any integers k and N > 0, the space J k,n of Jacobi forms of weight k and index N on the full modular group is defined as the set of holomorphic functions φτ,z on H C which are in a suitable sense both modular with respect to the action τ,z aτ +b/cτ +d,z/cτ +d of Γ on H C and elliptic with respect to the translations of z by elements of the lattice Zτ +Z, 6

17 and which satisfy a suitable condition of holomorphy at infinity. The transformation equations imply that φ has a Fourier development of the form φτ,z = n,r Z cn,rq n ζ r q = e πiτ, ζ = e πiz in which the coefficient cn,r depends only on the quantity 4Nn r and on r mod N. The holomorphy condition at infinity is that cn,r vanishes unless 4Nn r 0. If we relax the condition to merely requiring that cn,r = 0 if n < 0, we obtain the space of weak Jacobi forms, denoted J k,n in [4], and if we drop it entirely we obtain the space of nearly holomorphic Jacobi forms, which we will denote J! k,n. The structure of the ring J, of all weak Jacobi forms was determined in [4], Theorem 9.3 and 9.4: its even weight subring is the free polynomial algebra over M = M Γ = C[E 4 τ,e 6 τ] on two generators a = φ, τ,z J, and b = φ 0, τ,z J 0,, while the full ring is an extension of this by a further element c J, with 43c = ab 3 3E 4 a 3 b+ E 6 a 4. It follows from this description that the rings J! ev, and J!, are given by a similar formula, but with M Γ replaced by M! Γ = C[E 4,E 6, ]/E 3 4 E 6 = 78. The Fourier expansions of a and b begin a =ζ +ζ + ζ +8ζ +8ζ ζ q + ζ 3 ζ +39ζ q + 8ζ 3 56ζ +5ζ q 3 + b =ζ +0+ζ + 0ζ 64ζ ζ +0ζ q + ζ 3 +08ζ 53ζ q + 64ζ ζ 75ζ q 3 +, where the... means that the coefficients of negative powers of ζ have been omitted the transformation equation of Jacobi forms imply that cn, r = k cn,r for a Jacobi form of weight k. The above-mentioned periodicity property of the Fourier coefficients is clearly visible in these expansions. If N =, then the periodicity property of the Fourier coefficients mentioned above implies that cn,r = C4n r for some function d Cd, and also that k is even. There is then an isomorphism between J k, and M k /, and between J! k, and M! k /, sending cn,rq n ζ r to Cdq d. For example, the two forms g and g 5 in M 3/ correspond to the Jacobi forms φτ,z = E 4 a = ζ +ζ + ζ +48ζ q +, φ 5 τ,z = 5 E E E 4 a + 4E 6 6 b = ζ +ζ q + 0q 0 +ζ ζ q +. in J, and J!,, respectively. Generalizing the function φ = φ there are Jacobi forms φ N J,N N 6 uniquely characterized by the requirement that they have Fourier coefficients cn,r = B N 4nN r which depend only on the discriminant r 4Nn except in the case when N = 4 and r 0 mod 4, in which case cn,r = r/4 B 4 6n r, with B N =, B N 0 =. In particular, the Fourier development of φ N begins ζ +ζ +Oq. The representations of the forms φ N in terms of the generators a = φ,, b = φ 0, of J, are as 7

18 follows: φ = a E 4 b E 6 a, φ 3 = a E 4 b E 6 ab+e 4a, φ 4 = 3 a E 4 b 3 3E 6 ab +3E 4a b E 4 E 6 a 3, φ 5 = 4 a E 4 b 4 4E 6 ab 3 +6E 4a b 4E 4 E 6 a 3 b+4e 6 3E 3 4a 4, φ 6 = 5 a E 4 b 5 5E 6 ab 4 +0E 4a b 3 0E 4 E 6 a 3 b +8E 6 3E 3 4a 4 b E 4E 6 a 5, so that each φ N τ,z has the form N ν= f N,ντ φ, τ,z ν φ0, τ,z N ν where f N,ν τ is a holomorphic modular form of weight ν+ on SL,Z with constant term ν N N ν. Using these expansions one can compute the Fourier coefficients B N d. Here is a table for 0 < d < 30. d B d B 3 d B 4 d B 5 d B 6 d The empty entries correspond to d which are not congruent to squares modulo 4N. Comparing these numbers with the examples of traces t N d given at the beginning of the section, we are led to the following result, which can be proved in a way analogous to the proof of Theorem. Theorem 8. t N d = B N d for all d and N. 9. Third generalization: other weights. The results discussed so far concerned the modular forms jτ and J m τ of weight 0, the modular forms f d of weight /, and the modular forms g D of weight 3/. It is natural to wonder whether there are similar results for other weights. In fact our story so far involved not only the weights 0, /, and 3/, but also the weight, although in an invisible form: the existence of the Hauptmodul, a special modular form of weight 0, depended on the fact that the modular curve being studied had genus 0 and hence that the space of modular forms of the complementary weight contained no cusp forms. We will find analogous results for all other pairs of half-integral weights k +, 3 k for which there are no non-zero cusp forms of weight k on SL,Z, i.e., for k =, 3, 4, 5, and 7. We look first at the case k =. For each nonpositive discriminant d there is a unique form u d M! 5/ such that u d = q d +Oq at infinity, and for each positive discriminant D there is a unique form v D M! / such that v D = q D +O at infinity. The Fourier expansions of the 8

19 first few of these functions begin u 0 = 0q 70q 4 48q 5 0q 8 50q 9 40q 40q 3 +Oq 6, u 3 = q 3 +64q 3384q q q q 9 +Oq, u 4 = q 4 08q 3976q q q q 9 +Oq, u 7 = q 7 +53q 44538q q q 8 +Oq 9, u 8 = q 8 808q 375q q q 8 +Oq 9, v = q +0 64q 3 +08q 4 53q q 8 75q +406q +Oq 5, v 4 = q q q q q 8 +Oq, v 5 = q q q q q 8 +Oq, v 8 = q q q q 7 +Oq 8, v 9 = q q q q 7 +Oq 8. These values suggest the following result, whose proof is entirely analogous to that of Theorem 4. Theorem 9. For all D and d the coefficient of q D in u d is the negative of the coefficient of q d in v D. The corresponding result holds also for the other cases k = 3, 4, 5, and 7. We will come back to this at the end of the section, but first we want to know whether there is any analogue of Theorem for the new coefficients, and specifically whether the coefficients of the form v and then later of the other forms v D and u d, in analogy with the generalization of Theorem given in 7 can be interpreted as traces of some modular function. The answer is yes, but we have to work with non-holomorphic modular functions. This is in a sense quite natural. The values k =, 3, 4, 5, 7 for which S k+/ = {0} are, by virtue of the Shimura-Kohnen correspondence, precisely those for which S k Γ = {0}, as already mentioned, so their special behaviour has to do with the pair of integral weights k and k, rather than and 0 as in the case we studied originally. In particular, for these values of k there is a unique nearly holomorphic modular form F k M! k Γ having a Fourier development starting q + O. We would like to take the sum of the values of this function at all points of Q d /Γ, but of course this makes no sense since F k is not Γ-invariant. However, there is a well-known non-holomorphic differentiation operator which increases the weight of a modular form by at the cost of making it mildly non-holomorphic, and by applying this the right number of times to the function F k we get a Γ-invariant function which we then can evaluate on Q d /Γ. For k = the function F k is given by F τ = E 4τE 6 τ = q q 85980q q 3 τ and its non-holomorphic derivative is the function Kτ = πi τ + F τ = E τe 4 τe 6 τ+3e 4 τ 3 +E 6 τ, πiτ 6 τ where E 4 and E 6 are the usual Eisenstein series of weights 4 and 6 on SL Z and E τ = 4 n= nq n q n 3 πiτ = 3 πiτ 4q 7q 96q 3 68q 4 9

20 the non-holomorphic Eisenstein series of weight on Γ. It is well-known that, although E z is not holomorphic, it has the same arithmetic property as a holomorphic modular form with rational Fourier coefficients, namely that its quotient by a meromorphic modular form of weight has algebraic values belonging to the usual class field at CM points in the upper half-plane. It follows that Kτ is algebraic at all CM points. Here are a few sample values. K + 3 K = 576, K i = 864, K = 359, K = 6464, =, K =. 4 Note that in all of these examples, the values of Kα Q are algebraic integers, but this is not always the case. For instance, the values of Kα Q for the three inequivalent quadratic forms of discriminant 3 are the roots of the polynomial X X X in which the coefficients of X and X 0 trace and norm are integral and even divisible by 3, but the coefficient of X has a denominator 3. This is because the values of E 4 z/ z /3 = 3 jz and E 6 z/ z / = jz 78 for z Q d are algebraic integers, but E z/ z /6 in general has a denominator d. For instance, the values of E z/ z /6 for z Q 3 with an appropriate choice of z /6 are equal to 3α/ 3 with α a root of α 3 6α +85α 4 = 0. Comparing the values of Kτ just given with the coefficients of the function v given above, we are led to formulate the following theorem, which is analogous to Theorem and can be proved in a similar way: Theorem 0. For all positive integers d congruent to 0 or 3 mod 4 we have Q Q d /Γ where βd denotes the coefficient of q d in the function v M! /. w Q Kα Q = dβd, 8 All of the extensions of Theorem which we discussed in the previous sections carry over to the new situation. For instance, we can consider the image K m = K Tm of K under the mth Hecke operators m, which has the form p m jτkτ + jτjτ 78p mjτ for a certain monic polynomial p m j of degree m p =, p = j 504, p 3 = j 48j +805,..., determined by the requirement that p m jτf τ = q m +O. Then 8 generalizes to Q Q d /Γ w Q K m α Q = dβ m d, 9 where β m d is the coefficient of q d in the image of v under Tm, given for m = p prime by β p d = p 3 βp d + p d βd+βp d. p 0