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Developments in the theory of Jacobi forms. (English) Zbl 0745.11029

Automorphic functions and their applications, Int. Conf., Khabarovsk/USSR 1988, 167-185 (1990).
[For the entire collection see Zbl 0727.00003.]
The paper under review contains a survey about the theory of Jacobi forms. The author starts with the basic properties of Jacobi forms as for instance described in the monograph of M. Eichler and D. Zagier [The theory of Jacobi forms (Prog. Math. 55) (1985; Zbl 0554.10018)]. Then connections with Siegel modular forms of degree 2 are described, i.e. the Maaß Spezialschar as well as a type of Dirichlet series arising from the Petersson inner product of the attached Jacobi forms. Moreover the space of Jacobi forms of weight \(k\) and index \(m\) turns out to be isomorphic to a certain subspace of elliptic modular forms of weight \(2k-2\) on \(\Gamma_ 0(m)\). Finally the author introduces skew-holomorphic Jacobi forms. The obey a similar transformation property, are holomorphic in the variable \(z\in\mathbb{C}\) and satisfy the heat equation. The author announces analogous results for skew- holomorphic Jacobi forms. In particular the isomorphy with elliptic modular forms of weight \(2k-2\) becomes more natural, if one uses the direct sum of the spaces of holomorphic and skew-holomorphic Jacobi forms.
Reviewer: A.Krieg (Münster)

MSC:

11F50 Jacobi forms
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F11 Holomorphic modular forms of integral weight
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