Formulas for the coefficients of half-integral weight harmonic Maaß forms

Recently, Bruinier and Ono proved that the coefficients of certain weight - 1 / 2 harmonic weak Maaß forms are given as “traces” of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight 3 / 2 + k , k even, an...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 277; no. 3-4; pp. 769 - 795
Main Author Alfes, Claudia
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2014
Subjects
Mathematics
Mathematics and Statistics
11F30
11F37
11F20
11F67
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Summary:Recently, Bruinier and Ono proved that the coefficients of certain weight - 1 / 2 harmonic weak Maaß forms are given as “traces” of singular moduli for harmonic weak Maaß forms. Here, we prove that similar results hold for the coefficients of harmonic weak Maaß forms of weight 3 / 2 + k , k even, and weight 1 / 2 - k , k odd, by extending the theta lift of Bruinier–Funke and Bruinier–Ono. Moreover, we generalize these results to include twisted traces of singular moduli using earlier work of the author and Ehlen on the twisted Bruinier–Funke-lift. Employing a general duality result between weight k and 2 - k , we obtain formulas for all half-integral weights. We also show that the non-holomorphic part of the theta lift in weight 1 / 2 - k , k odd, is connected to the vanishing of the special value of the L -function of a certain derivative of the lifted function.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-014-1278-6