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On the non-vanishing of Poincaré series

Published online by Cambridge University Press:  20 January 2009

J. Lehner
J. Lehner
Affiliation:
Department of MathematicsUniversity of PittsburghPittsburgh, Pennsylvania 15260, U.S.A.
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Let M = SL(2, Z) be the classical modular matrix group. One form of the Poincaré series on M is

here z ∈ H={z = x + iy: y >0}, q ≧ 2 and m ≧ 1 are integers, and the summation is over a complete system of matrices (ab: cd) in M with different lower row. The problem of the identical vanishing of the Poincaré series for different values of m and q goes back to Poincaré.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 2 , June 1980 , pp. 225 - 228
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Lehner, J., Discontinuous groups and automorphic functions (Amer. Math. Soc. Mathematical Surveys no. 8, 1964).CrossRefGoogle Scholar
(2)Lehner, J., On automorphic forms of negative dimension, Illinois J. of Math. 8 (1964), 395–407.CrossRefGoogle Scholar
(3)Rankin, R. A., The vanishing of Poincaré series, Proc. Edinburgh Math. Soc. 23 (1980), 151–161.CrossRefGoogle Scholar