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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. Z.Rudnick and P.Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161, 1994, 195-213. MR 95m:11052
  • 3. D.Zagier, The Eichler-Selberg trace formula on $SL_{2}({\mathbf Z})$, Appendix to Introduction to Modular Forms by S.Lang, Springer, Berlin, 1976, 44-54; errata in Lecture Notes in Math., Vol. 627, Springer-Verlag, 171-173, 1977. MR 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

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

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