Abstract
We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan’s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series.
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Research of the author was supported by an NSF Mathematical Sciences Postdoctoral Fellowship. Part of this work was done while supported by the chair in Analytic Number Theory at Ecole Polytechnique Fédérale de Lausanne.
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Rhoades, R.C. Linear relations among Poincaré series via harmonic weak Maass forms. Ramanujan J 29, 311–320 (2012). https://doi.org/10.1007/s11139-012-9377-7
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DOI: https://doi.org/10.1007/s11139-012-9377-7