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Modular forms of orthogonal type and Jacobi theta-series

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Abstract

In this paper, we study Jacobi forms of half-integral index for any even integral positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A 1=〈2〉). We construct Jacobi forms of singular (respectively, critical) weight in all dimensions n≥8 (respectively, n≥9). We give the Jacobi lifting for Jacobi forms of half-integral indices and we obtain an additive lifting construction of new reflective modular forms which are natural generalizations to O(2,n) (n=4, 5 and 6) of the Igusa modular form Δ 5.

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Acknowledgements

This work was supported by the grant ANR-09-BLAN-0104-01 and Labex CEMPI (Lille). The authors are grateful to the Max-Planck-Institut für Mathematik in Bonn for support and for providing excellent working conditions.

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Authors and Affiliations

  1. Korteweg-de Vries Instituut, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE, Amsterdam, The Netherlands

    F. Cléry

  2. Laboratoire Paul Painlevé, University Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    V. Gritsenko

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  1. F. Cléry
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  2. V. Gritsenko
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Correspondence to V. Gritsenko.

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Communicated by U. Kühn.

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Cléry, F., Gritsenko, V. Modular forms of orthogonal type and Jacobi theta-series. Abh. Math. Semin. Univ. Hambg. 83, 187–217 (2013). https://doi.org/10.1007/s12188-013-0080-4

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