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 [STIX]x1D6E4_0(N) whose divisor on X_0(N) is defined over 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.