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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Equidistribution of Hecke eigenforms on the modular surface


Author: Wenzhi Luo
Journal: Proc. Amer. Math. Soc. 131 (2003), 21-27
MSC (2000): Primary 11F11, 11F25
Published electronically: May 8, 2002
MathSciNet review: 1929018
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Abstract: For the orthonormal basis of Hecke eigenforms in $S_{2k}(\Gamma (1))$, one can associate with it a probability measure $d\mu _{k}$ on the modular surface $X = \Gamma (1) \backslash {\mathbf H}$. We establish that this new measure tends weakly to the invariant measure on $X$ as $k$ tends to infinity, and obtain a sharp estimate for the rate of convergence.


References [Enhancements On Off] (What's this?)

  • 1. W.Luo and P.Sarnak, Mass equidistribution for Hecke eigenforms, preprint, 2001.
  • 2. Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), no. 1, 195–213. MR 1266075 (95m:11052)
  • 3. D. Zagier, Correction to: “The Eichler-Selberg trace formula on 𝑆𝐿₂(𝑍)” (Introduction to modular forms, Appendix, pp. 44–54, Springer, Berlin, 1976) by S. Lang, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 171–173. Lecture Notes in Math., Vol. 627. MR 0480354 (58 #522)

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Additional Information

Wenzhi Luo
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: wluo@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06619-4
PII: S 0002-9939(02)06619-4
Keywords: Hecke eigenform, automorphic kernel
Received by editor(s): August 6, 2001
Published electronically: May 8, 2002
Additional Notes: This research was partially supported by NSF grant DMS-9988503, the Alfred P. Sloan Foundation Research Fellowship and the Seed Grant from the Ohio State University
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2002 American Mathematical Society