A CONVERSE THEOREM FOR BORCHERDS PRODUCTS ON $X_{0}(N)$. (1st December 2020) - BRUINIER, JAN HENDRIK; SCHWAGENSCHEIDT, MARKUS
Abstract : We show that every Fricke-invariant meromorphic modular form for $\unicode[STIX]{x1D6E4}_{0}(N)$ 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 $L$ -functions of certain weight  $2$ newforms. We also prove similar results for twisted Borcherds products.