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A CONVERSE THEOREM FOR BORCHERDS PRODUCTS ON $X_{0}(N)$

  • JAN HENDRIK BRUINIER (a1) and MARKUS SCHWAGENSCHEIDT (a2)

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.

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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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