Quantum variance for Hecke eigenforms
Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 37 (2004) no. 5, p. 769-799
@article{ASENS_2004_4_37_5_769_0,
     author = {Luo, Wenzhi and Sarnak, Peter},
     title = {Quantum variance for Hecke eigenforms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 37},
     number = {5},
     year = {2004},
     pages = {769-799},
     doi = {10.1016/j.ansens.2004.08.001},
     zbl = {02136290},
     mrnumber = {2103474},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2004_4_37_5_769_0}
}
Luo, Wenzhi; Sarnak, Peter. Quantum variance for Hecke eigenforms. Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 37 (2004) no. 5, pp. 769-799. doi : 10.1016/j.ansens.2004.08.001. http://www.numdam.org/item/ASENS_2004_4_37_5_769_0/

[1] Bump D., Automorphic Forms on GL 3,R, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. | MR 765698 | Zbl 0543.22005

[2] Bump D., The Rankin-Selberg method: a survey, in: Number Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 49-109. | MR 993311 | Zbl 0668.10034

[3] Deshouillers J.-M., Iwaniec H., Kloosterman sum and Fourier coefficients of cusp forms, Invent. Math. 70 (1982) 219-288. | MR 684172 | Zbl 0502.10021

[4] Miller, S., Dueñez, E., The low-lying zeros of a GL 6 family of L-functions, preprint, 2004.

[5] Eckhardt B., Fishman S., Keating J., Agam O., Main J., Müller K., Approach to ergodicity in quantum wave functions, Phys. Rev. E 52 (1995) 5893-5903.

[6] Feingold M., Peres A., Distribution of matrix elements of chaotic systems, Phys. Rev. A (3) 34 (1) (1986) 591-595. | MR 848319

[7] Gelbart S., Jacquet H., A relation between automorphic representations of GL 2 and GL 3, Ann. Sci. École Norm. Sup. 11 (1978) 471-552. | Numdam | MR 533066 | Zbl 0406.10022

[8] Gradshtein I.S, Ryzhik I.M., Tables of Integrals, Series and Products, Academic Press, New York, 1965. | MR 1773820 | Zbl 0918.65002

[9] Helgason S., Differential Geometry and Symmetric Spaces, Pure and Applied Mathematics, vol. XII, Academic Press, New York, 1962. | MR 145455 | Zbl 0111.18101

[10] Hoffstein J., Lockhart P., Coefficients of Maass forms and the Siegel zero, appendix by Goldfeld, D., Hoffstein, J., Lieman, D. An effective zero free region, Annals of Math. 140 (1994) 161-181. | MR 1289494 | Zbl 0814.11032

[11] Iwaniec H., Introduction to the Spectral Theory of Automorphic Forms, Revista Matematica Iberoamericana, Madrid, 1995. | MR 1325466 | Zbl 0847.11028

[12] Iwaniec H., Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, AMS, 1997. | MR 1474964 | Zbl 0905.11023

[13] Iwaniec H., Luo W., Sarnak P., Low lying zeros of families of L-functions, Inst. Hautes Études Sci. Publ. Math. 91 (2001) 55-131. | Numdam | MR 1828743 | Zbl 1012.11041

[14] Iwaniec H., Sarnak P., Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. (2000) 705-741, Special Volume, Part II. | MR 1826269 | Zbl 0996.11036

[15] Iwaniec H., Sarnak P., The nonvanishing of central values of automorphic L-functions and Siegel's zero, Israel J. Math. A 120 (2000) 155-177. | MR 1815374 | Zbl 0992.11037

[16] Jakobson D., Equidistribution of cusp forms on PSL 2 Z PSL 2 R, Ann. Inst. Fourier (Grenoble) 47 (3) (1997) 967-984. | Numdam | MR 1465794 | Zbl 0868.43011

[17] Katok S., Sarnak P., Heegner points, cycles and Maass forms, Israel J. Math. 84 (1993) 193-227. | MR 1244668 | Zbl 0787.11016

[18] Katz N., Sarnak P., Random Matrices, Frobenius Eigenvalues and Monodromy, Colloquium Publications, vol. 45, AMS, 1999. | MR 1659828 | Zbl 0958.11004

[19] Keating J.P., Snaith N.C., Random matrix theory and L-functions at s=1/2, Comm. Math. Phys. 214 (1) (2000) 91-110. | MR 1794267 | Zbl 1051.11047

[20] Kurlberg, P., Rudnick, Z., On the distribution of matrix elements for the quantum cat map, preprint, 2003. | MR 2150390

[21] Kuznetsov N.V., Petersson's conjecture for cusp forms of weight zero and Linnik's conjecture, sums of Kloosterman sums, Mat. Sb. 111 (1980) 334-383. | MR 568983 | Zbl 0427.10016

[22] Luo W., Values of symmetric square L-functions at 1, J. Reine Angew. Math. 506 (1999) 215-235. | MR 1665705 | Zbl 0969.11018

[23] Luo W., Sarnak P., Quantum ergodicity of eigenfunctions on PSL 2 ZH, Inst. Hautes Études Sci. Publ. Math. 81 (1995) 207-237. | Numdam | MR 1361757 | Zbl 0852.11024

[24] Luo W., Sarnak P., Mass equidistribution for Hecke eigenforms, Comm. Pure Appl. Math. 56 (7) (2003) 874-891. | MR 1990480 | Zbl 1044.11022

[25] Ratner M., The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature, Israel J. Math. 16 (1973) 181-197. | MR 333121 | Zbl 0283.58010

[26] Ratner M., The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems 7 (1987) 267-288. | MR 896798 | Zbl 0623.22008

[27] Salié H., Uber die Kloostermanschen Summen Su,v;q, Math. Z. 34 (1931) 91-109. | MR 1545243 | Zbl 0002.12801

[28] Sarnak, P., Arithmetic quantum chaos, The R.A. Blyth Lecture, University of Toronto, 1993. | MR 1321639

[29] Watson, T., Rankin triple products and quantum chaos, Thesis, Princeton University, 2001.

[30] Serre J.-P., A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, New York, 1973. | MR 344216 | Zbl 0256.12001

[31] Terras A., Harmonic Analysis on Symmetric Spaces and Applications I, Springer-Verlag, New York, 1985. | MR 791406 | Zbl 0574.10029

[32] Zelditch S., Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55 (1987) 919-941. | MR 916129 | Zbl 0643.58029

[33] Zelditch S., On the rate of quantum ergodicity, I. Upper bounds, Comm. Math. Phys. 160 (1) (1994) 81-92. | MR 1262192 | Zbl 0788.58043