Abstract
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module A has order p or 2p, where p is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components \(F_{\gamma }\) of the vector-valued modular form are antisymmetric in the sense that \(F_{\gamma } = -F_{-\gamma }\) for all \(\gamma \in A\). As an application, we compute restrictions of Doi–Naganuma lifts of odd weight to components of Hirzebruch–Zagier curves.
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Communicated by Jens Funke.
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We thank Jan H. Bruinier and Stephan Ehlen for helpful discussions. M. Schwagenscheidt is supported by the SFB-TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the DFG. B. Williams is supported by the LOEWE research unit Uniformized Structures in Arithmetic and Geometry.
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Schwagenscheidt, M., Williams, B. Twisted component sums of vector-valued modular forms. Abh. Math. Semin. Univ. Hambg. 89, 151–168 (2019). https://doi.org/10.1007/s12188-019-00209-4
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DOI: https://doi.org/10.1007/s12188-019-00209-4