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A generalized Jacobi theta function and quasimodular forms. (English) Zbl 0892.11015
Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands, during the last week of April 1994. Basel: Birkhäuser. Prog. Math. 129, 165-172 (1995).
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)

##### MSC:
 11F11 Holomorphic modular forms of integral weight 14K25 Theta functions and abelian varieties 14H42 Theta functions and curves; Schottky problem 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11F55 Other groups and their modular and automorphic forms (several variables)