A converse theorem for Borcherds products on $X_0(N)$

TL;DR

This paper establishes a converse theorem linking certain meromorphic modular forms on $X_0(N)$ to Borcherds products, and provides criteria for their multiplier system finiteness based on $L$-function derivatives.

Contribution

It proves a converse theorem for Borcherds products on $X_0(N)$ and offers a criterion for the finiteness of their multiplier systems.

Findings

Every suitable Fricke invariant form is a generalized Borcherds product.

Finiteness of multiplier systems relates to vanishing derivatives of $L$-functions.

Results extend to twisted Borcherds products.

Abstract

We show that every Fricke invariant meromorphic modular form for $\Gamma_0(N)$ whose divisor on $X_0(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-function of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.