Quantum unique ergodicity for ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$

Abstract

We eliminate the possibility of “escape of mass” for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for Hecke-Maass forms on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash \mathbb{H}$.

  • [1] Go to document P. Dusart, Autour de la fonction qui compte le nombre de nombre premiers, 1998.
    @misc{1,
      author={Dusart, P.},
      TITLE={Autour de la fonction qui compte le nombre de nombre premiers},
      NOTE={ e-print},
      URL={http://www.unilim.fr/laco/theses/1998/T1998\_01.pdf},
      YEAR={1998},
     }
  • [2] R. Holowinsky, "Sieving for mass equidistribution," Ann. of Math., vol. 172, pp. 1499-1516, 2010.
    @article{2,
      author={Holowinsky, R.},
      TITLE={Sieving for mass equidistribution},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1499--1516},
      }
  • [3] R. Holowinsky and K. Soundararajan, "Mass equidistribution of Hecke eigenforms," Ann. of Math., vol. 172, pp. 1517-1528, 2010.
    @article{3,
      author={Holowinsky, R. and Soundararajan, K.},
      TITLE={Mass equidistribution of Hecke eigenforms},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1517--1528},
      }
  • [4] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002, vol. 53.
    @book {4, MRKEY = {1942691},
      AUTHOR = {Iwaniec, Henryk},
      TITLE = {Spectral Methods of Automorphic Forms},
      SERIES = {Grad. Stud. Math.},
      VOLUME = {53},
      EDITION = {Second},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2002},
      PAGES = {xii+220},
      ISBN = {0-8218-3160-7},
      MRCLASS = {11F72 (11F12 11F37)},
      MRNUMBER = {2003k:11085},
      ZBLNUMBER = {1006.11024},
      }
  • [5] Go to document D. Jakobson, "Quantum unique ergodicity for Eisenstein series on ${ PSL}_2({\bf Z})\backslash { PSL}_2({\bf R})$," Ann. Inst. Fourier $($Grenoble$)$, vol. 44, iss. 5, pp. 1477-1504, 1994.
    @article {5, MRKEY = {1313792},
      AUTHOR = {Jakobson, Dmitry},
      TITLE = {Quantum unique ergodicity for {E}isenstein series on {${\rm PSL}\sb 2({\bf Z})\backslash {\rm PSL}\sb 2({\bf R})$}},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {44},
      YEAR = {1994},
      NUMBER = {5},
      PAGES = {1477--1504},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {11F72 (33C80 81Q50)},
      MRNUMBER = {96b:11068},
      MRREVIEWER = {Jonathan Huntley},
      URL = {http://www.numdam.org/item?id=AIF_1994__44_5_1477_0},
      ZBLNUMBER = {0820.11040},
      }
  • [6] Go to document E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006.
    @article {6, MRKEY = {2195133},
      AUTHOR = {Lindenstrauss, Elon},
      TITLE = {Invariant measures and arithmetic quantum unique ergodicity},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {163},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {165--219},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F72 (37A45 37D40)},
      MRNUMBER = {2007b:11072},
      MRREVIEWER = {Ze{é}v Rudnick},
      DOI = {10.4007/annals.2006.163.165},
      ZBLNUMBER={1104.22015},
      }
  • [7] E. Lindenstrauss, "Adelic dynamics and arithmetic quantum unique ergodicity," in Current Developments in Mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 111-139.
    @incollection {7, MRKEY = {2459293},
      AUTHOR = {Lindenstrauss, Elon},
      TITLE = {Adelic dynamics and arithmetic quantum unique ergodicity},
      BOOKTITLE = {Current Developments in Mathematics, 2004},
      PAGES = {111--139},
      PUBLISHER = {Int. Press, Somerville, MA},
      YEAR = {2006},
      MRCLASS = {58J51 (37A45 37D40)},
      MRNUMBER = {2459293},
      MRREVIEWER = {Daniel Bump},
      ZBLNUMBER = {1186.58022},
      }
  • [8] Go to document W. Z. Luo and P. Sarnak, "Quantum ergodicity of eigenfunctions on ${ PSL}_2(\Bbb Z)\backslash \Bbb H^2$," Inst. Hautes Études Sci. Publ. Math., iss. 81, pp. 207-237, 1995.
    @article {8, MRKEY = {1361757},
      AUTHOR = {Luo, Wen Zhi and Sarnak, Peter},
      TITLE = {Quantum ergodicity of eigenfunctions on {${\rm PSL}\sb 2(\bold Z)\backslash \bold H\sp 2$}},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {81},
      YEAR = {1995},
      PAGES = {207--237},
      ISSN = {0073-8301},
      CODEN = {PMIHA6},
      MRCLASS = {11F72 (11M41 58F19)},
      MRNUMBER = {97f:11037},
      MRREVIEWER = {Yiannis Petridis},
      URL = {http://www.numdam.org/item?id=PMIHES_1995__81__207_0},
      ZBLNUMBER = {0852.11024},
      }
  • [9] J. Marklof, "Arithmetic quantum chaos," in Encyclopedia of Math. Phys., Francoise, J. -P., Naber, G. L., and Tsou, S. T., Eds., Oxford: Elsevier Science, 2006, vol. 1, pp. 212-220.
    @incollection{9,
      author={Marklof, J.},
      TITLE={Arithmetic quantum chaos},
      BOOKTITLE={Encyclopedia of Math. Phys.},
      VOLUME={1},
      EDITOR={Francoise, J.-P. and Naber, G. L. and Tsou, S. T.},
      PUBLISHER={Elsevier Science},
      ADDRESS={Oxford},
      YEAR={2006},
      PAGES={212--220},
      }
  • [10] Go to document Z. Rudnick and P. Sarnak, "The behaviour of eigenstates of arithmetic hyperbolic manifolds," Comm. Math. Phys., vol. 161, iss. 1, pp. 195-213, 1994.
    @article {10, MRKEY = {1266075},
      AUTHOR = {Rudnick, Ze{é}v and Sarnak, Peter},
      TITLE = {The behaviour of eigenstates of arithmetic hyperbolic manifolds},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {161},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {195--213},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {11F72 (11F32 11F37 81Q50)},
      MRNUMBER = {95m:11052},
      MRREVIEWER = {Jens Bolte},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104269797},
      ZBLNUMBER = {0836.58043},
      }
  • [11] P. Sarnak, "Arithmetic quantum chaos," in The Schur Lectures (1992), Ramat Gan: Bar-Ilan Univ., 1995, vol. 8, pp. 183-236.
    @incollection {11, MRKEY = {1321639},
      AUTHOR = {Sarnak, Peter},
      TITLE = {Arithmetic quantum chaos},
      BOOKTITLE = {The {S}chur Lectures (1992)},
      VENUE={{T}el {A}viv},
      SERIES = {Israel Math. Conf. Proc.},
      VOLUME = {8},
      PAGES = {183--236},
      PUBLISHER = {Bar-Ilan Univ.},
      ADDRESS = {Ramat Gan},
      YEAR = {1995},
      MRCLASS = {11F72 (11M41 81Q50)},
      MRNUMBER = {96d:11059},
      MRREVIEWER = {Jens Bolte},
      ZBLNUMBER = {0831.58045},
      }
  • [12] K. Soundararajan, "Weak subconvexity for central values of $L$-functions," Ann. of Math., vol. 172, pp. 1469-1498, 2010.
    @article{12,
      author={Soundararajan, K.},
      TITLE={Weak subconvexity for central values of $L$-functions},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1469--1498},
      }
  • [13] Go to document T. C. Watson, Rankin triple products and quantum chaos, ProQuest LLC, Ann Arbor, MI, 2002.
    @book {13, MRKEY = {2618394},
      AUTHOR = {Watson, Thomas Crawford},
      TITLE = {Rankin triple products and quantum chaos},
      NOTE = {Thesis (Ph.D.), Princeton University},
      PUBLISHER = {ProQuest LLC, Ann Arbor, MI},
      YEAR = {2002},
      PAGES = {81},
      ISBN = {978-0493-52934-9},
      MRCLASS = {Thesis},
      MRNUMBER = {2618394},
      URL = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3039710},
      }
  • [14] Go to document S. Zelditch, "Mean Lindelöf hypothesis and equidistribution of cusp forms and Eisenstein series," J. Funct. Anal., vol. 97, iss. 1, pp. 1-49, 1991.
    @article {14, MRKEY = {1105653},
      AUTHOR = {Zelditch, Steven},
      TITLE = {Mean {L}indelöf hypothesis and equidistribution of cusp forms and {E}isenstein series},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {97},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {1--49},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {11F72 (58G15 58G25)},
      MRNUMBER = {92h:11046},
      MRREVIEWER = {Jonathan Huntley},
      DOI = {10.1016/0022-1236(91)90014-V},
      ZBLNUMBER={0743.58034},
      }

Authors

Kannan Soundararajan

Stanford University
Department of Mathematics
450 Serra Mall, Building 380
Stanford CA 94305-2125
United States