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THE QUANTITATIVE DISTRIBUTION OF HECKE EIGENVALUES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  10 April 2014

YINGNAN WANG
YINGNAN WANG*
Affiliation:
College of Mathematics and Computational Science, Shenzhen University, Shenzhen, Guangdong 518060, PR China email [email protected]
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Abstract

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In this paper, we prove that the Sato–Tate conjecture for primitive Maass forms holds on average. We also investigate the rate of convergence in the Sato–Tate conjecture and establish some estimates of the discrepancy with respect to the Sato–Tate measure on the average of primitive Maass forms.

Keywords

Sato–Tate conjectureMaass formsHecke eigenvaluesquantitative distribution

MSC classification

Secondary: 11F25: Hecke-Petersson operators, differential operators (one variable) 11F30: Fourier coefficients of automorphic forms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 1 , August 2014 , pp. 28 - 36
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Akiyama, S. and Tanigawa, Y., ‘Calculation of values of L-functions associated to elliptic curves’, Math. Comp. 68 (1999), 1201–1231.Google Scholar
Barnet-Lamb, T., Geraghty, D., Harris, M. and Taylor, R., ‘A family of Calabi-Yau varieties and potential automorphy II’, Publ. Res. Inst. Math. Sci. 47 (2011), 29–98.Google Scholar
Barton, J. T., Montgomery, H. L. and Vaaler, J. D., ‘Note on a Diophantine inequality in several variables’, Proc. Amer. Math. Soc. 129 (2000), 337–345.Google Scholar
Conrey, J. B., Duke, W. and Farmer, D. W., ‘The distribution of the eigenvalues of Hecke operators’, Acta Arith. 78 (1997), 405–409.Google Scholar
Deligne, P., ‘La conjecture de Weil I’, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.Google Scholar
Kim, H. and Sarnak, P., ‘Appendix 2 in H. Kim, Functoriality for the exterior square of G L 4and the symmetric fourth of G L 2’, J. Amer. Math. Soc. 16 (2003), 139–183.CrossRefGoogle Scholar
Lau, Y.-K. and Wang, Y., ‘Quantitative version of the joint distribution of eigenvalues of the Hecke operators’, J. Number Theory 131 (2011), 2262–2281.Google Scholar
Nagoshi, H., ‘Distribution of Hecke eigenvalues’, Proc. Amer. Math. Soc. 134 (2006), 3097–3106.Google Scholar
Vaaler, J. D., ‘Some extremal functions in Fourier analysis’, Bull. Amer. Math. Soc. 12 (1985), 183–216.CrossRefGoogle Scholar