Mathematische Annalen

, Volume 337, Issue 3, pp 591–612

Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series

Article

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.

Mathematics Subject Classification (2000)

1F30 11F37 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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