Equidistribution of cusp forms on PSL 2 (𝐙) PSL 2 (𝐑)
Annales de l'Institut Fourier, Volume 47 (1997) no. 3, p. 967-984

We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on PSL 2 (Z) PSL 2 (R). This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on PSL 2 (Z)H.

Nous donnons la preuve d’une version microlocale d’un résultat de W. Luo et P. Sarnak concernant la répartition asymptotique des fonctions de Wigner associées aux formes paraboliques sur PSL 2 (Z) PSL 2 (R).

@article{AIF_1997__47_3_967_0,
     author = {Jakobson, Dmitri},
     title = {Equidistribution of cusp forms on ${\rm PSL}\_2({\bf Z})\backslash {\rm PSL}\_2({\bf R})$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {47},
     number = {3},
     year = {1997},
     pages = {967-984},
     doi = {10.5802/aif.1588},
     zbl = {0868.43011},
     mrnumber = {99c:11063},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1997__47_3_967_0}
}
Jakobson, Dmitri. Equidistribution of cusp forms on ${\rm PSL}_2({\bf Z})\backslash {\rm PSL}_2({\bf R})$. Annales de l'Institut Fourier, Volume 47 (1997) no. 3, pp. 967-984. doi : 10.5802/aif.1588. http://www.numdam.org/item/AIF_1997__47_3_967_0/

[CdV] Y. Colin De Verdière, Ergodicité et fonctions propres du laplacien, Comm. Math. Phys, 102 (1985), 497-502. | MR 87d:58145 | Zbl 0592.58050

[GR] I.S. Gradshteyn and I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, 4th edition, 1980.

[Ja94] D. Jakobson, Quantum Unique Ergodicity for Eisenstein Series on PSL2(ℤ) PSL2(ℝ), Annales de l'Institut Fourier, 44-5 (1994), 1477-1504. | Numdam | MR 96b:11068 | Zbl 0820.11040

[Ja95] D. Jakobson, Thesis, Princeton University, 1995.

[Kuz] N. Kuznetsov, Peterson's conjecture for cusp forms of weight zero and Linnik's conjecture; sums of Kloosterman sums, Mat. Sb., 111 (1980), 334-383. | MR 81m:10053 | Zbl 0427.10016

[LS] M. Luo and P. Sarnak. Quantum Ergodicity of Eigenfunctions on PSL2(ℤ)\ℍ, IHES Publ., 81 (1995), 207-237. | Numdam | Zbl 0852.11024

[MOS] W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Springer, 1966. | MR 38 #1291 | Zbl 0143.08502

[Sn74] A.I. Shnirelman, Ergodic Properties of Eigenfunctions, Uspekhi Mat. Nauk, 29-6 (1974), 181-182.

[Sn93] A.I. Shnirelman, On the Asymptotic Properties of Eigenfunctions in the Regions of Chaotic Motions, Addendum to V. F. Lazutkin's book KAM Theory and Semiclassical Approximations, Springer, 1993.

[S1] J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, 1966. | Zbl 0135.28101

[Ze87] S. Zelditch, Uniform distribution of Eigenfunctions on compact hyperbolic surfaces, Duke Math. Journal, 55 (1987), 919-941. | MR 89d:58129 | Zbl 0643.58029

[Ze91] S. Zelditch, Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, Jour. of Funct. Analysis, 97 (1991), 1-49. | MR 92h:11046 | Zbl 0743.58034

[Ze92] S. Zelditch, Selberg Trace Formulas and Equidistribution Theorems for Closed Geodesics and Laplace Eigenfunctions: Finite Area Surfaces, Mem. AMS, 90 (No. 465), 1992. | Zbl 0753.11023