The Quantum Variance  of the Modular Surface

Sarnak, P.; Zhao, P.

doi:10.24033/asens.2406

The Quantum Variance of the Modular Surface
[La variance quantique de la surface modulaire]

Sarnak, P. ; Zhao, P.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1155-1200.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé
Abstract

Nous calculons la variance des observables des états quantiques du Laplacien sur la surface modulaire dans la limite semiclassique. Nous montrons que cette forme hermitienne est diagonalisée par les représentations irréductibles du quotient modulaire et sur chacune de ces représentations, elle est égale à la variance classique du flot géodésique après insertion d'une subtile valeur spécifique de la fonction $L$ correspondante.

The variance of observables of quantum states of the Laplacian on the modular surface is calculated in the semiclassical limit. It is shown that this hermitian form is diagonalized by the irreducible representations of the modular quotient and on each of these it is equal to the classical variance of the geodesic flow after the insertion of a subtle arithmetical special value of the corresponding $L$ -function.

Publié le : 2019-11-12

MR Zbl

DOI : 10.24033/asens.2406

Classification : 11F72, 11M36, 37D40.
Keywords: Quantum variance, modular surface, $L$-function.
Mot clés : Variance quantique, surface modulaire, fonction $L$.

@article{ASENS_2019__52_5_1155_0,
     author = {Sarnak, P. and Zhao, P.},
     title = {The {Quantum} {Variance}  of the {Modular} {Surface}},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1155--1200},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
     number = {5},
     year = {2019},
     doi = {10.24033/asens.2406},
     mrnumber = {4057780},
     zbl = {1455.11076},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2406/}
}

TY  - JOUR
AU  - Sarnak, P.
AU  - Zhao, P.
TI  - The Quantum Variance  of the Modular Surface
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2019
SP  - 1155
EP  - 1200
VL  - 52
IS  - 5
PB  - Société Mathématique de France. Tous droits réservés
UR  - http://www.numdam.org/articles/10.24033/asens.2406/
DO  - 10.24033/asens.2406
LA  - en
ID  - ASENS_2019__52_5_1155_0
ER  -

%0 Journal Article
%A Sarnak, P.
%A Zhao, P.
%T The Quantum Variance  of the Modular Surface
%J Annales scientifiques de l'École Normale Supérieure
%D 2019
%P 1155-1200
%V 52
%N 5
%I Société Mathématique de France. Tous droits réservés
%U http://www.numdam.org/articles/10.24033/asens.2406/
%R 10.24033/asens.2406
%G en
%F ASENS_2019__52_5_1155_0

Sarnak, P.; Zhao, P. The Quantum Variance  of the Modular Surface. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 5, pp. 1155-1200. doi : 10.24033/asens.2406. http://www.numdam.org/articles/10.24033/asens.2406/

Bibliographie
Cité par

Anantharaman, N.; Zelditch, S. Patterson-Sullivan distributions and quantum ergodicity, Ann. Henri Poincaré, Volume 8 (2007), pp. 361-426 (ISSN: 1424-0637) | DOI | MR | Zbl

Anantharaman, N.; Zelditch, S. Intertwining the geodesic flow and the Schrödinger group on hyperbolic surfaces, Math. Ann., Volume 353 (2012), pp. 1103-1156 (ISSN: 0025-5831) | DOI | MR | Zbl

Bateman, H., McGraw-Hill, 1954 | MR

Bernstein, J.; Reznikov, A. Periods, subconvexity of $L$ -functions and representation theory, J. Differential Geom., Volume 70 (2005), pp. 129-141 http://projecteuclid.org/euclid.jdg/1143572016 (ISSN: 0022-040X) | MR | Zbl

Bump, D., Lecture Notes in Math., 1083, Springer, 1984, 184 pages (ISBN: 3-540-13864-1) | DOI | MR | Zbl

Eckhardt, B.; Fishman, S. et al. Approach to ergodicity in quantum wave functions, Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics, Volume 52 (1995), pp. 5893-5903

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., McGraw-Hill Book Company-Toronto-London, 1954, 451 pages | MR | Zbl

Garrett, P. B. Standard archimedean integrals for $G L (2)$ (preprint http://www.math.umn.edu/~garrett/ )

Garrett, P. B. Decomposition of Eisenstein series: Rankin triple products, Ann. of Math., Volume 125 (1987), pp. 209-235 (ISSN: 0003-486X) | DOI | MR | Zbl

Gross, B. H.; Kudla, S. S. Heights and the central critical values of triple product $L$ -functions, Compos. math., Volume 81 (1992), pp. 143-209 (ISSN: 0010-437X) | Numdam | MR | Zbl

Gradshteyn, I. S.; Ryzhik, I. M., Academic Press, 1965

Ghosh, A.; Reznikov, A.; Sarnak, P. Nodal domains of Maass forms I, Geom. Funct. Anal., Volume 23 (2013), pp. 1515-1568 (ISSN: 1016-443X) | DOI | MR | Zbl

Hejhal, D. A., Lecture Notes in Math., 548, Springer, 1976, 516 pages | MR | Zbl

Harris, M.; Kudla, S. S. The central critical value of a triple product $L$ -function, Ann. of Math., Volume 133 (1991), pp. 605-672 (ISSN: 0003-486X) | DOI | MR | Zbl

Hoffstein, J.; Lockhart, P. Coefficients of Maass forms and the Siegel zero, Ann. of Math., Volume 140 (1994), pp. 161-181 (ISSN: 0003-486X) | DOI | MR | Zbl

Ichino, A. Trilinear forms and the central values of triple product $L$ -functions, Duke Math. J., Volume 145 (2008), pp. 281-307 (ISSN: 0012-7094) | DOI | MR | Zbl

Iwaniec, H. Small eigenvalues of Laplacian for $Γ_{0} (N)$ , Acta Arith., Volume 56 (1990), pp. 65-82 (ISSN: 0065-1036) | DOI | MR | Zbl

Iwaniec, H., Graduate Studies in Math., 17, Amer. Math. Soc., 1997, 259 pages (ISBN: 0-8218-0777-3) | DOI | MR | Zbl

Jakobson, D. QUE for Eisenstein Series on ${PSL}_{2} (Z) ∖ {PSL}_{2} (R)$ , Annales de l'Institut Fourier, Volume 44 (1994), pp. 1477-1504 | Numdam | MR | Zbl

Jakobson, D. Equidistribution of cusp forms on ${PSL}_{2} (𝐙) ∖ {PSL}_{2} (𝐑)$ , Ann. Inst. Fourier, Volume 47 (1997), pp. 967-984 (ISSN: 0373-0956) | DOI | Numdam | MR | Zbl

Jutila, M. The spectral mean square of Hecke $L$ -functions on the critical line, Publ. Inst. Math. (Beograd), Volume 76(90) (2004), pp. 41-55 (ISSN: 0350-1302) | DOI | MR | Zbl

Kowalski, E.; Michel, P. Zeros of families of automorphic $L$ -functions close to 1, Pacific J. Math., Volume 207 (2002), pp. 411-431 (ISSN: 0030-8730) | DOI | MR | Zbl

Knapp, A. W., Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, Amer. Math. Soc., 1994, pp. 393-410 | MR | Zbl

Kuznetsov, N. V. Petersson's Conjecture for Cusp Forms of Weight Zero and Linnik's Conjecture, Math. USSR Sbornik, Volume 29 (1981), pp. 299-342 | MR | Zbl

Lindenstrauss, E. Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math., Volume 163 (2006), pp. 165-219 (ISSN: 0003-486X) | DOI | MR | Zbl

Luo, W.; Rudnick, Z.; Sarnak, P. The variance of arithmetic measures associated to closed geodesics on the modular surface, J. Mod. Dyn., Volume 3 (2009), pp. 271-309 (ISSN: 1930-5311) | DOI | MR | Zbl

Luo, W.; Sarnak, P. Quantum variance for Hecke eigenforms, Ann. Sci. École Norm. Sup., Volume 37 (2004), pp. 769-799 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Luo, W. Z.; Sarnak, P. Quantum ergodicity of eigenfunctions on ${PSL}_{2} (𝐙) ∖ 𝐇^{2}$ , Inst. Hautes Études Sci. Publ. Math., Volume 81 (1995), pp. 207-237 (ISSN: 0073-8301) | Numdam | MR | Zbl

Luo, W. Values of symmetric square $L$ -functions at 1, J. reine angew. Math., Volume 506 (1999), pp. 215-235 (ISSN: 0075-4102) | DOI | MR | Zbl

Liu, J.; Ye, Y. Subconvexity for Rankin-Selberg $L$ -functions of Maass forms, Geom. Funct. Anal., Volume 12 (2002), pp. 1296-1323 (ISSN: 1016-443X) | DOI | MR | Zbl

Michel, P.; Venkatesh, A. The subconvexity problem for $GL (2)$ , Publications Math. IHÉS, Volume 111 (2010), pp. 171-271 | DOI | Numdam | MR | Zbl

Ratner, M. The central limit theorem for geodesic flows on $n$ -dimensional manifolds of negative curvature, Israel J. Math., Volume 16 (1973), pp. 181-197 (ISSN: 0021-2172) | DOI | MR | Zbl

Ratner, M. The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, Volume 7 (1987), pp. 267-288 (ISSN: 0143-3857) | DOI | MR | Zbl

Sarnak, P., The Schur Lectures, 2003

Selberg, A., Springer, 1989, 711 pages (ISBN: 3-540-18389-2) | MR | Zbl

Soundararajan, K. Quantum unique ergodicity for ${SL}_{2} (ℤ) ∖ ℍ$ , Ann. of Math., Volume 172 (2010), pp. 1529-1538 (ISSN: 0003-486X) | MR | Zbl

Venkatesh, A. Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., Volume 172 (2010), pp. 989-1094 (ISSN: 0003-486X) | DOI | MR | Zbl

Watson, T. Central Value of Rankin Triple L-function for Unramified Maass Cusp Forms (2004)

Woodbury, M. Trilinear forms and subconvexity of the triple product $L$ -function (preprint http://www.math.columbia.edu/~woodbury/research/ ) | MR

Woodbury, M., L-functions and automorphic forms (Contrib. Math. Comput. Sci.), Volume 10, Springer, 2017, pp. 275-298 | DOI | MR | Zbl

Zelditch, S. Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987), pp. 919-941 (ISSN: 0012-7094) | DOI | MR | Zbl

Zelditch, S. Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series, J. Funct. Anal., Volume 97 (1991), pp. 1-49 (ISSN: 0022-1236) | DOI | MR | Zbl

Zelditch, S. On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Volume 160 (1994), pp. 81-92 http://projecteuclid.org/euclid.cmp/1104269516 (ISSN: 0010-3616) | DOI | MR | Zbl

Zhao, P. Quantum variance of Maass-Hecke cusp forms, Comm. Math. Phys., Volume 297 (2010), pp. 475-514 (ISSN: 0010-3616) | DOI | MR | Zbl

Cité par Sources :