# Twisted component sums of vector-valued modular forms

## TL;DR

This paper establishes isomorphisms between spaces of vector-valued modular forms and scalar-valued forms using twisted sums, extending previous work to antisymmetric components and applying results to special modular lift restrictions.

## Contribution

It generalizes existing isomorphisms to antisymmetric vector-valued modular forms for specific finite quadratic modules, broadening the understanding of their structure.

## Key findings

- Constructed isomorphisms for modules of order p or 2p.
- Extended Bruinier and Bundschuh's work to antisymmetric cases.
- Computed restrictions of Doi-Naganuma lifts to Hirzebruch-Zagier curves.

## 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.