A CONVERSE THEOREM FOR BORCHERDS PRODUCTS ON $X_{0}(N)$. (1st December 2020)

• Record Type: Journal Article
• Authors: BRUINIER, JAN HENDRIK; SCHWAGENSCHEIDT, MARKUS
• Is Part Of: Nagoya mathematical journal. Volume 240(2020)
• Page Start: 237
• Page End: 256
• Publication Date: 2020-12-01
• Subjects: 11F27, -- 11F37; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathematics; Wiskunde; Electronic journals; Periodicals; 510.5
• DOI: 10.1017/nmj.2019.3
• Languages: English
• ISSNs: 0027-7630

Abstract

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.