Let \(\widetilde M_* =M_*\otimes_\mathbb{C} \mathbb{C} [G_2]\) be the graded ring of quasi-modular forms for the group \(SL(2, \mathbb{Z})\) (here \(M_*\) denotes the (graded) ring of modular forms, and \(G_2= -{1\over 24} +\sum_{n\geq 1} \sigma(n) q^n\) with \(\sigma(n): = \sum_{d | n}d)\). Let \[\Theta (X,q,\zeta) =\prod_{n>0} (1-q^n) \prod_{n>0} (1-e^{n^2 X/8} q^{n/2} \zeta)(1- e ^{-n^2X/8} q^{n/2} \zeta^{-1}),\] and let \(\Theta_0 (X,q)= \sum^\infty_{n=0} A_n(q) X^{2n}\), \(A_n(q) \in\mathbb{Q} [[q]]\), be the constant term in the Laurent series \(\theta (X,q, \zeta)= \sum^\infty_{n= -\infty} \Theta_n(X,q) \zeta^n\).
By a direct computation, the authors prove that \(A_n(q) \in\widetilde M_{6n}\) for \(n\geq 0\). The coefficient of \(X^{2g-2}\) in the power series expansion of \(\log\Theta_0\) is a quasimodular form of weight \(6g-6\). The authors mention that this coefficient is equal to the generating function counting maps of curves of genus \(g>1\) to a curve of genus 1, and comment on the relation of their construction to the theory of Jacobi forms.
For the entire collection see [Zbl 0827.00037].

Reviewer: B.Z.Moroz (Bonn)