A GENERALIZED JACOBI THETA FUNCTION AND QUASIMODULAR FORMS Masanobu Kaneko and Don Zagier In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. f Let M¤(¡1) denote the graded ring of quasimodular forms on the full modular group ¡1 = P SL(2; Z). This is the ring generated by G2, G4, G6, and graded by assigning to each Gk the weight k, where 1 µ ¶ B X X G = ¡ k + dk¡1 qn (k = 2; 4; 6;:::;B = kth Bernoulli number) k 2k k n=1 djn are the classical Eisenstein series, all of which except G2 are modular. (See x1 for a more general and more intrinsic de¯nition of quasi-modular.) We de¯ne a generalization of the classical Jacobi theta function by the triple product Y Y ¡ 2 ¢¡ 2 ¢ £(X; q; ³) = (1 ¡ qn) 1 ¡ en X=8 qn=2 ³ 1 ¡ e¡n X=8 qn=2 ³¡1 ; (1) n>0 n>0 n odd considered as a formal power series in X and q1=2 with coe±cients in Q[³; ³¡1]. (We can also consider q and ³ as complex numbers, in which case the coe±cient of Xn is a holomorphic function of these variables for each n, but we cannot consider the product as a holomorphic function of the third variable X because it diverges rapidly for any X with non-zero real 0 part.) Let £0(X; q) 2 Q[[q; X]] denote the coe±cient of ³ in £(X; q; ³), considered as a Laurent series in ³, and expand £0 as a Taylor series X1 2n £0(X; q) = An(q) X ;An(q) 2 Q[[q]] (2) n=0 in X. (It is easy to see that there are no odd powers of X in this expansion.) The result in question is then f Theorem 1. An 2 M6n(¡1) for all n ¸ 0. 2g¡2 The coe±cient of X in log £0, which as explained in [1] is the generating function counting maps of curves of genus g > 1 to a curve of genus 1, is then also quasi-modular of weight 6g ¡ 6, but we will not discuss this connection further. The proof of Theorem 1 will be given in x2. In x3 and x4 we compute the \highest 3n degree term" (coe±cient of G2 ) in An and comment on the relationship to Jacobi forms. 1 x1. Quasimodular forms. We denote by H = f¿ 2 C j =(¿) > 0g the complex upper half-plane and for ¿ 2 H write q = e2¼i¿ and Y = 4¼=(¿), while D denotes the di®erential 1 d d operator D = = q . (The factors 4¼ and 2¼i are included for convenience and 2¼i d¿ dq to avoid unnecessary irrationalities later.) Recall that a modular form of weight k on a subgroup ¡ of ¯nite index of ¡1 is a holomorphic function f on H satisfying µ ¶ ¡a¿ + b¢ a b f = (c¿ + d)k f(¿) 8¿ 2 H; 2 ¡ c¿ + d c d ¡ ¢ and growing at most polynomially in 1=Y as Y ! 0. If 1 ¸ 2 ¡, then these conditions 0 1 P 1 n=¸ imply that f has a convergent Fourier series expansion f(¿) = n=0 a(n) q at in¯nity. The space ofL holomorphic modular forms of weight k on ¡ is denoted by Mk(¡) and the graded ring k Mk(¡) by M¤(¡). As well as the holomorphic modular forms, there are also functions F (¿) which satisfy the same transformation properties and growth conditions as before but which belong to C[[q1=¸]] [Y ¡1] instead of C[[q1=¸]], i.e. which have the form XM ¡m F (¿) = fm(¿) Y (fm(¿) holomorphic for m = 0; 1;:::;M) (3) m=0 for some integer M ¸ 0 (and necessarily · k=2). We call such a function an almost- c holomorphic modular form of weight k and denote the vector space of them by Mk(¡), while the holomorphic function f0(¿) obtained formally as the \constant term with respect to 1=Y " of f will be called a quasi-modular form of weight k and the space of such functions f c L c f L f denoted by Mk(¡). It is clear that the spaces M¤(¡) = Mk(¡) and M¤(¡) = Mk(¡) c f are graded rings and the map M¤(¡) ! M¤(¡) is a ring homomorphism. As the basic example, if ¡ = ¡1 and we think of the power series Gk 2 Q[[q]] de¯ned in the introduction as functions of ¿ 2 H, then Gk(¿) is a holomorphic modular form of weight k for k > 2 but G2(¿) satis¯es instead µ ¶ ¡a¿ + b¢ c(c¿ + d) a b G = (c¿ + d)2 G (¿) ¡ 8¿ 2 H; 2 ¡ : (4) 2 c¿ + d 2 4¼i c d 1 (A standard proof is to noticeQ that G2 is a multiple of the logarithmic derivative of the Ramanujan function ¢(¿) = q (1 ¡ qn)24, which is a modular form of weight 12.) It is ¤ easy to check that (4) is equivalent to the assertion that the function G2(¿) = G2(¿)+1=2Y is an almost-holomorphic modular form of weight 2, so G2 itself is indeed quasi-modular. 2 Another easy consequence of (4) is that the expressions D(G2) + 2G2 and D(f) + 2kG2f (f 2 Mk) are holomorphic modular forms (of weights 4 and k + 2, respectively), from which it follows that the ring of quasi-modular forms is closed under di®erentiation and, in particular, that all derivatives of holomorphic modular forms or of G2 are quasi-modular. A converse to this and some other simple properties of quasi-modular forms are contained in the following proposition, whose proof (by induction on the degree of almost-holomorphic forms with respect to 1=Y ) we omit. 2 Proposition 1. Let ¡ ½ ¡1 be a subgroup of ¯nite index of the full modular group. Then: c f (a) The \constant term map" M¤(¡) ! M¤(¡) is an isomorphism of rings; f (b) M¤(¡) = M¤(¡) ­ C[G2], i.e. every quasi-modular form on ¡ can be written uniquely as a polynomial in G2 with coe±cients whichL are modular forms on ¡; f i © h k=2¡1 i (c) For (even) k > 0 we have Mk(¡) = 0·i·k=2 D Mk¡2i(¡) D G2 , i.e., any quasi-modular form has a unique representation as a sum of derivatives of modular forms and of G2. x2. Expansions of £(X; q; ³). As well as the variables X, q, and ³, we introduce further variables w and Z de¯ned by w = eX , ³ = eZ . We will also follow the physicists' practice of using the same letter to denote a function expressed in terms of di®erent independent variables, denoting for instanceQ the Eisenstein series Gk by either Gk(¿) or Gk(q) and the Dedekind eta-function q1=24 (1 ¡ qn) by either ´(¿) or ´(q), and writing £(X; ¿; Z) for 0 the function de¯ned by (1). Finally, we denote by ¡2 the group (usually denoted ¡ (2)) ¡ ¢ P 2 of matrices a b 2 ¡ with b even and by θ(¿) the theta-series (¡1)rqr =2, a modular c d 1 r form of weight 1/2 on ¡2. The ¯rst result is: Proposition 2. The function £(X; ¿; Z) has an expansion of the form X Xj Zl £(X; ¿; Z) = θ(¿) Hj;l(¿) (5) j; l¸0 j! l! where H0;0(¿) = 1 and each Hj;l(¿) is quasimodular of weight 3j + l on ¡2. Proof. From (1) and the identity θ(¿) = ´(¿=2)2=´(¿) we ¯nd µ ¶ X ¡ ¢ £(X; ¿; Z) 1 2 2 log = ¡ en rX=8³r ¡ 2 + e¡n rX=8³¡r qnr=2 θ(¿) r n; r>0 n odd X (X=8)j Zl = ¡2 Á (¿) (6) j;l j! l! j; l¸0 j´l (mod 2) j+l>0 with ( X 22jD2jF (1) (¿) if l > j, l+j¡1 2j nr=2 l¡j Áj;l(¿) = r n q = (7) j+l¡1 j+l¡1 (2) ¸ n; r>0 2 D Fj¡l+2(¿) if j l, n odd (1) (2) where Fk and Fk (k = 2; 4;::: ) are the two Eisenstein series 1 µ ¶ ¡¿ ¢ X X F (1)(¿) = G ¡ G (¿) = (n=d)k¡1 qn=2 ; k k 2 k n=1 djn; 2-d 1 µ ¶ ¡¿ ¢ B X X F (2)(¿) = G ¡ 2k¡1G (¿) = (2k¡1 ¡ 1) k + dk¡1 qn=2 k k 2 k 2k n=1 djn; 2-d of weight k on ¡2. Since each of these is quasi-modular (indeed, actually modular except (1) for F2 ) of weight k on ¡2, and since the mth derivative of a quasi-modular form of weight f k is quasi-modular of weight k + 2m, it follows that in both cases Áj;l 2 M3j+l(¡2). The result now follows by exponentiating, since quasi-modular forms form a graded ring. 3 The next identity is an analogue of the Jacobi triple product identity. Set Y ¡ 2 ¢¡ 2 ¢ H(w; q; ³) = q¡1=24 1 ¡ wn =8qn=2³ 1 ¡ w¡n =8qn=2³¡1 n>0; n odd 0 and denote by H0(w; q) the coe±cient of ³ in H(w; q; ³) as a Laurent series in ³. Proposition 3. The function H(w; q; ³) has the expansion X r r r3=6 r2=2 r H(w; q; ³) = (¡1) H0(w; w q) w q ³ : r2Z Proof. From the product expansion of H we ¯nd Y ¡ 2 ¢¡ ¡ 2 ¢ 1=2 ¡ 1 (n+2) n+2 ¡ (n 2) n¡2 ¡1 H(w; wq; w q³) = (wq) 24 1 ¡ w 8 q 2 ³ 1 ¡ w 8 q 2 ³ n odd 1 ¡ w¡1=8q¡1=2³¡1 = w¡1=24 H(w; q; ³) 1 ¡ w1=8 q1=2 ³ = ¡w¡1=6 q¡1=2 ³¡1 H(w; q; ³) ; and this means that if we write the Laurent expansion of H with respect to ³ in the form X r r3=6 r2=2 r H(w; q; ³) = (¡1) Hr(w; q) w q ³ ; r2Z r then Hr+1(w; q) = Hr(w; wq) and hence by induction Hr(w; q) = H0(w; w q) for all r.