M24-twisted Product Expansions are Siegel Modular Forms

TL;DR

This paper investigates the modularity of certain product expansions derived from elliptic genera of K3 surfaces, establishing conditions under which they are Siegel modular forms and expressing their powers as Borcherds products.

Contribution

It identifies when these twisted product expansions are Siegel modular forms and provides a representation of their powers as rescaled Borcherds products.

Findings

Non-composite predicted levels imply modularity.

Such product expansions are Siegel modular forms.

Their powers can be expressed as rescaled Borcherds products.

Abstract

Cheng constructed product expansions from twists of elliptic genera of symmetric powers of K3 surfaces that are related to M_24 moonshine. We study which of them are Siegel modular forms. If the predicted level is non-composite, they are modular, and their powers can be represented as products of rescaled Borcherds products.