On the converse theorem for Borcherds products

TL;DR

This paper establishes a new converse theorem for Borcherds' multiplicative theta lift, introduces a newform theory for vector valued modular forms, and derives bounds for Picard groups and differential forms of modular varieties.

Contribution

It presents a novel converse theorem, develops a newform theory for vector valued modular forms, and provides bounds for geometric invariants of modular varieties.

Findings

Improved converse theorem for Borcherds' lift

Development of a newform theory for vector valued modular forms

Lower bounds for Picard groups and differential forms

Abstract

We prove a new converse theorem for Borcherds' multiplicative theta lift which improves the previously known results. To this end we develop a newform theory for vector valued modular forms for the Weil representation, which might be of independent interest. We also derive lower bounds for the ranks of the Picard groups and the spaces of holomorphic top degree differential forms of modular varieties associated to orthogonal groups.