A Generalized Jacobi Theta Function and Quasimodular Forms

Abstract

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]. Let M * (Γ 1) denote the graded ring of quasimodular forms on the full modular group Γ 1 = P SL(2, Z). This is the ring generated by G 2 , G 4 , G 6 , and graded by assigning to each G k the weight k, where G k = − B k 2k + ∞ n=1 d|n d k−1 q n (k = 2, 4, 6,. .. , B k = kth Bernoulli number) are the classical Eisenstein series, all of which except G 2 are modular. (See §1 for a more general and more intrinsic definition of quasi-modular.) We define a generalization of the classical Jacobi theta function by the triple product Θ(X, q, ζ) = n>0 (1 − q n) n>0 n odd 1 − e n 2 X/8 q n/2 ζ 1 − e −n 2 X/8 q n/2 ζ −1 , (1) considered as a formal power series in X and q 1/2 with coefficients in Q[ζ, ζ −1 ]. (We can also consider q and ζ as complex numbers, in which case the coefficient of X n 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 part.) Let Θ 0 (X, q) ∈ Q[[q, X]] denote the coefficient of ζ 0 in Θ(X, q, ζ), considered as a Laurent series in ζ, and expand Θ 0 as a Taylor series Θ 0 (X, q) = ∞ n=0 A n (q) X 2n , A n (q) ∈ Q[[q]] (2) in X. (It is easy to see that there are no odd powers of X in this expansion.) The result in question is then Theorem 1. A n ∈ M 6n (Γ 1) for all n ≥ 0. The coefficient of X 2g−2 in log Θ 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 §2. In §3 and §4 we compute the "highest degree term" (coefficient of G 3n 2) in A n and comment on the relationship to Jacobi forms.