A CONVERSE THEOREM FOR BORCHERDS PRODUCTS ON

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,...

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Published in:Nagoya mathematical journal 2020-12, Vol.240, p.237-256
Main Authors: BRUINIER, JAN HENDRIK, SCHWAGENSCHEIDT, MARKUS
Format: Article
Language:English
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Summary: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.
ISSN:0027-7630
2152-6842
DOI:10.1017/nmj.2019.3