A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift

TL;DR

This paper establishes a converse theorem for Borcherds products related to lattices of square-free level and anisotropic discriminant groups, extending prior results and analyzing the Kudla-Millson theta lift's injectivity.

Contribution

It generalizes the converse theorem for Borcherds products to broader lattices, including non-unimodular and non-(p,2) types, and proves the injectivity of the Kudla-Millson theta lift in this context.

Findings

Proves a converse theorem for Borcherds products on certain lattices.

Shows the injectivity of the Kudla-Millson theta lift for these lattices.

Refines a theorem on the non-existence of reflective automorphic products.

Abstract

We prove a converse theorem for the multiplicative Borcherds lift for lattices of square-free level whose associated discriminant group is anisotropic. This can be seen as generalization of Bruinier's results in \cite{Br2}, which provides a converse theorem for lattices of prime level. The surjectivity of the Borcherds lift in our case follows from the injectivity of the Kudla-Millson theta lift. We generalize the corresponding results in \cite{BF1} to the aforementioned lattices and thereby in particular to lattices which are not unimodular and not of type $(p,2)$. Along the way, we compute the contribution of both, the non-Archimedean and Archimedean places of the $L^2$-norm of the Kudla-Millson theta lift. As an application we refine a theorem of Scheithauer on the non-existence of reflective automorphic products.