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A geometrical approach to the theory of Jacobi forms. (English) Zbl 0724.11028
Contrary to the analytic treatment by M. Eichler and D. Zagier [The theory of Jacobi forms. Prog. Math. 55 (1985; Zbl 0554.10018)] the author gives a geometrical approach to the theory of Jacobi forms. Therefore let \(\Gamma\) be a subgroup of finite index in \(SL_ 2({\mathbb{Z}})\) acting without fixed points. At first basic facts about the elliptic modular surface \(X_{\Gamma}\) are recalled. Then the space of Jacobi cusp forms with respect to \(\Gamma\) can be described as the space of global sections of a particular subsheaf of a certain line bundle on \(X_{\Gamma}\). Since the line bundle is ample, the Kodaira- Vanishing-Theorem and the Riemann-Roch-Theorem lead to an explicit formula for the dimension of the space of Jacobi cusp forms.
Reviewer: A.Krieg (Münster)

MSC:
11F50 Jacobi forms
14C40 Riemann-Roch theorems
14F17 Vanishing theorems in algebraic geometry
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