Half-Integral Weight Modular Forms and Modular forms for Weil representations

TL;DR

This paper establishes an isomorphism between specific half-integral weight modular form spaces and vector-valued modular forms, generalizing known results and enabling explicit constructions like Borcherds lifts.

Contribution

It generalizes the isomorphism between modular forms for a0a0a0(4) and Weil representations, proving Zagier duality and explicitly constructing Borcherds lifts.

Findings

Established a new isomorphism between complex-valued and vector-valued modular forms.

Proved the Zagier duality within this framework.

Explicitly constructed Borcherds lifts for the forms.

Abstract

We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $\Gamma_0(4)$ with Kohnen's plus condition and modular forms for the Weil representation associated to the discriminant form for the lattice with Gram matrix $(2)$. With such an isomorphism, we prove the Zagier duality and write down the Borcherds lifts explicitly.