Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T01:02:53.125Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

(1)Estermann, T., On Kloosterman's sum, Mathematika 8 (1961), 83–86.CrossRefGoogle Scholar
(2)Langer, R. E., On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. 33 (1931), 23–64.CrossRefGoogle Scholar
(3)Petersson, H., Über eine Metrisierung der ganzen Modulformen, Jber. Deutsch. Math.-Verein. 49 (1939), 49–75.Google Scholar
(4)Rankin, R. A., The scalar product of modular forms, Proc. London Math. Soc. (3) 2 (1952), 198–217.CrossRefGoogle Scholar
(5)Rankin, R. A., Modular forms and functions (Cambridge, 1977).CrossRefGoogle Scholar
(6)Selberg, A., Über die Fourierkoeffizienten elliptischer Modulformen negativer Dimension, Neuvième Congrès des Mathematiciens Scandinaves, Helsingfors, 1938, pp. 320–2.Google Scholar
(7)Watson, G. N., A treatise on the theory of Bessel functions (Cambridge, 1922).Google Scholar