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Schlagwörter:
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Zusammenfassung:
Let \widetilde M* =M*\otimes\bbfC \bbfC [G2] be the graded ring of quasi-modular forms for the group SL(2, \bbfZ) (here M* denotes the (graded) ring of modular forms, and G2= -1\over 24 +\sumn≥ 1 σ(n) q^n with σ(n): = \sumd | nd). Let \Theta (X,q,ζ) =\prodngt;0 (1-q^n) \prodngt;0 (1-e^n^2 X/8 q^n/2 ζ)(1- e ^-n^2X/8 q^n/2 ζ^-1), and let \Theta0 (X,q)= \sum^∞n=0 An(q) X^2n, An(q) \in\bbfQ [[q]], be the constant term in the Laurent series θ (X,q, ζ)= \sum^∞n= -∞ \Thetan(X,q) ζ^n.\par By a direct computation, the authors prove that An(q) \in\widetilde M6n for n≥ 0. The coefficient of X^2g-2 in the power series expansion of \log\Theta0 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 ggt;1 to a curve of genus 1, and comment on the relation of their construction to the theory of Jacobi forms.
