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The vanishing of Poincaré series

Published online by Cambridge University Press:  20 January 2009

R. A. Rankin
R. A. Rankin
Affiliation:
Department of MathematicsUniversity of Glasgow
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Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series

.

Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} and

The summation is over all matrices

with different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 2 , June 1980 , pp. 151 - 161
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

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