Twisted component sums of vector-valued modular forms

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.