Mathematische Annalen

, Volume 337, Issue 3, pp 591–612 | Cite as

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlgren S., Ono K. (2005): Arithmetic of singular moduli and class polynomials. Compos. Math. 141, 293–312MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Andrews G.E., Askey R., Roy R. (1999): Special Functions. Cambridge University Press, CambridgeMATHGoogle Scholar
  3. 3.
    Boylan M. (2005): (2)-adic properties of Hecke traces of singular moduli. Math. Res. Lett. 12, 593–609MATHMathSciNetGoogle Scholar
  4. 4.
    Bruinier, J.H.: Borcherds products on O(2,ℓ) and Chern classes of Heegner divisors. Springer Lecture Notes, vol. 1780. Springer, Berlin Heidelberg New York (2002)Google Scholar
  5. 5.
    Bruinier J.H., Funke J. (2004): On two geometric theta lifts. Duke Math. J. 125, 45–90MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bruinier J.H., Funke J. (2006): Traces of CM-values of modular functions. J. Reine Angew. Math. 594, 1–33MATHMathSciNetGoogle Scholar
  7. 7.
    Bruinier J.H., Jenkins P., Ono K. (2006): Hilbert class polynomials and traces of singular moduli. Math. Ann. 334, 373–393MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Duke W. (2006): Modular functions and the uniform distribution of CM points. Math. Ann. 334, 241–252MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Edixhoven B. (2005): On the p-adic geometry of traces of singular moduli. Int. J. Number Theory 1(4): 495–498MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hejhal, D.A.: The Selberg trace formula for (PSL 2( \(\mathbb{R}\))). Springer Lecture Notes in Mathematics, vol. 1001. Springer, Berlin Heidelberg New York (1983)Google Scholar
  11. 11.
    Jenkins, P.: Kloosterman sums and traces of singular moduli. J. Number Theory (accepted for publication)Google Scholar
  12. 12.
    Jenkins P. (2005): p-adic properties for traces of singular moduli. Int. J. Number Thoery 1(1): 103–108MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Katok S., Sarnak P. (1993): Heegner points, cycles and Maass forms. Israel J. Math. 84(1–2): 192–227MathSciNetGoogle Scholar
  14. 14.
    Kim C.H. (2004): Borcherds products associated to certain Thompson series. Compos. Math. 140, 541–551MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kim, C.H.: Traces of singular values and Borcherds products (preprint)Google Scholar
  16. 16.
    Kohnen W. (1985): Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271, 237–268MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Maass H. (1959): “Uber die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik. Math. Ann. 138, 287–315MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Miller, A., Pixton, A.: Arithmetic traces of non-holomorphic modular invariants (preprint)Google Scholar
  19. 19.
    Niebur D. (1973): A class of nonanalytic automorphic functions. Nagoya Math. J. 52, 133–145MATHMathSciNetGoogle Scholar
  20. 20.
    Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and q-series. In: CBMS Regional Conference, vol. 102. American Mathematical Society Providence (2004)Google Scholar
  21. 21.
    Osburn, R.: Congruences for traces of singular moduli. Ramanujan J., (accepted for publication)Google Scholar
  22. 22.
    Rouse J. (2006): Zagier duality for the exponents of Borcherds products for Hilbert modular forms, J. Lond. Math. Soc. 73, 339–354MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Zagier, D.: Traces of singular moduli, Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998) (2002), Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, pages 211–244Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

Personalised recommendations