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A NOTE ON FOURIER COEFFICIENTS OF POINCARÉ SERIES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 December 2010

Emmanuel Kowalski ,
Abhishek Saha  and
Jacob Tsimerman
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland (email: kowalski@math.ethz.ch)
Abhishek Saha
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland (email: abhishek.saha@math.ethz.ch)
Jacob Tsimerman
Affiliation:
Princeton University, Fine Hall, Princeton NJ 08540, U.S.A. (email: jtsimerm@math.princeton.edu)
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Abstract

We give a short and “soft” proof of the asymptotic orthogonality of Fourier coefficients of Poincaré series for classical modular forms as well as for Siegel cusp forms, in a qualitative form.

MSC classification

Secondary: 11F30: Fourier coefficients of automorphic forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Type
Research Article
Information
Mathematika , Volume 57 , Issue 1 , January 2011 , pp. 31 - 40
Copyright
Copyright © University College London 2011

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References

[1]Gottschling, E., Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades. Math. Ann. 138 (1959), 103–124.Google Scholar
[2]Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53), American Mathematical Society (Providence, RI, 2004).Google Scholar
[3]Klingen, H., Introductory Lectures on Siegel Modular Forms (Cambridge Studies in Advanced Mathematics 20), Cambridge University Press (Cambridge, 1990).Google Scholar
[4]Kowalski, E., Saha, A. and Tsimerman, J., Local spectral equidistribution for Siegel modular forms and applications. Preprint, 2010, arXiv:1010.3648.Google Scholar
[5]Maass, H., Über die Darstellung der Modulformen n-ten Grades durch Poincarésche Reihen. Math. Ann. 123 (1951), 125–151.Google Scholar
[6]Sarnak, P., Statistical properties of eigenvalues of the Hecke operator. In Analytic Number Theory and Diophantine Problems (Progess in Mathematics 60), Birkhäuser (Boston, MA, 1987), 75–102.Google Scholar
[7]Serre, J.-P., Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p. J. Amer. Math. Soc. 10 (1997), 75–102.Google Scholar
[8]Siegel, C. L., Symplectic geometry. Amer. J. Math. 65 (1943), 1–86; www.jstor.org/stable/2371774.Google Scholar