Abstract
Zagier [23] proved that the generating functions for the traces of level 1 singular moduli are weight 3/2 modular forms. He also obtained generalizations for “twisted traces”, and for traces of special non-holomorphic modular functions. Using properties of Kloosterman-Salié sums, and a well known reformulation of Salié sums in terms of orbits of CM points, we systematically show that such results hold for arbitrary weakly holomorphic and cuspidal half-integral weight Poincaré series in Kohnen’s Γ0(4) plus-space. These results imply the aforementioned results of Zagier, and they provide exact formulas for such traces.
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Bringmann, K., Ono, K. Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series. Math. Ann. 337, 591–612 (2007). https://doi.org/10.1007/s00208-006-0048-0
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DOI: https://doi.org/10.1007/s00208-006-0048-0