Mass equidistribution for Hecke eigenforms

Abstract

We prove a conjecture of Rudnick and Sarnak on the mass equidistribution of Hecke eigenforms. This builds upon independent work of the authors.

  • [1] Go to document S. Gelbart and H. Jacquet, "A relation between automorphic representations of ${ GL}(2)$ and ${ GL}(3)$," Ann. Sci. École Norm. Sup., vol. 11, iss. 4, pp. 471-542, 1978.
    @article {1, MRKEY = {533066},
      AUTHOR = {Gelbart, Stephen and Jacquet, Herv{é}},
      TITLE = {A relation between automorphic representations of {${\rm GL}(2)$} and {${\rm GL}(3)$}},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {11},
      YEAR = {1978},
      NUMBER = {4},
      PAGES = {471--542},
      ISSN = {0012-9593},
      CODEN = {ENAQAF},
      MRCLASS = {10D40 (12A67 22E55)},
      MRNUMBER = {81e:10025},
      MRREVIEWER = {G. I. Ol{\cprime}shanski{\u\i}},
      URL = {http://www.numdam.org/item?id=ASENS_1978_4_11_4_471_0},
      ZBLNUMBER = {0406.10022},
      }
  • [2] Go to document J. Hoffstein and P. Lockhart, "Coefficients of Maass forms and the Siegel zero," Ann. of Math., vol. 140, iss. 1, pp. 161-181, 1994.
    @article {2, MRKEY = {1289494},
      AUTHOR = {Hoffstein, Jeffrey and Lockhart, Paul},
      TITLE = {Coefficients of {M}aass forms and the {S}iegel zero},
      NOTE = {With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {140},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {161--181},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11F37 (11F12 11F30 11F66)},
      MRNUMBER = {95m:11048},
      MRREVIEWER = {Lynne H. Walling},
      DOI = {10.2307/2118543},
      ZBLNUMBER = {0814.11032},
      }
  • [4] R. Holowinsky, "Sieving for mass equidistribution," Ann. of Math., vol. 172, pp. 1499-1516, 2010.
    @article{4,
      author={Holowinsky, R.},
      TITLE={Sieving for mass equidistribution},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1499--1516},
      }
  • [5] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002, vol. 53.
    @book {5, 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},
      }
  • [6] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
    @book {6, MRKEY = {2061214},
      AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
      TITLE = {Analytic Number Theory},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      VOLUME= {53},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {xii+615},
      ISBN = {0-8218-3633-1},
      MRCLASS = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
      MRNUMBER = {2005h:11005},
      MRREVIEWER = {K. Soundararajan},
      ZBLNUMBER = {1059.11001},
      }
  • [7] H. Iwaniec and P. Sarnak, "Perspectives on the analytic theory of $L$-functions," Geom. Funct. Anal., pp. 705-741, 2000.
    @article {7, MRKEY = {1826269},
      AUTHOR = {Iwaniec, Henryk and Sarnak, P.},
      TITLE = {Perspectives on the analytic theory of {$L$}-functions},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      YEAR = {2000},
      NOTE = {Special Volume, Part II},
      PAGES = {705--741},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {11M26 (11F67 11F70 11F72 11G40)},
      MRNUMBER = {2002b:11117},
      MRREVIEWER = {Henry H. Kim},
      ZBLNUMBER = {0996.11036},
      }
  • [8] Go to document E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006.
    @article {8, 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},
      }
  • [9] Go to document W. 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 {9, MRKEY = {1361757},
      AUTHOR = {Luo, Wenzhi 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},
      }
  • [10] Go to document W. Luo and P. Sarnak, "Mass equidistribution for Hecke eigenforms," Comm. Pure Appl. Math., vol. 56, iss. 7, pp. 874-891, 2003.
    @article {10, MRKEY = {1990480},
      AUTHOR = {Luo, Wenzhi and Sarnak, Peter},
      TITLE = {Mass equidistribution for {H}ecke eigenforms},
      NOTE = {Dedicated to the memory of J{ü}rgen K. Moser},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {56},
      YEAR = {2003},
      NUMBER = {7},
      PAGES = {874--891},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {11F11 (11F72 11L05)},
      MRNUMBER = {2004e:11038},
      MRREVIEWER = {D. R. Heath-Brown},
      DOI = {10.1002/cpa.10078},
      ZBLNUMBER = {1044.11022},
      }
  • [11] Go to document W. Luo and P. Sarnak, "Quantum variance for Hecke eigenforms," Ann. Sci. École Norm. Sup., vol. 37, iss. 5, pp. 769-799, 2004.
    @article {11, MRKEY = {2103474},
      AUTHOR = {Luo, Wenzhi and Sarnak, Peter},
      TITLE = {Quantum variance for {H}ecke eigenforms},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {37},
      YEAR = {2004},
      NUMBER = {5},
      PAGES = {769--799},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11F72 (11F25 11F66)},
      MRNUMBER = {2005k:11101},
      MRREVIEWER = {Ze{é}v Rudnick},
      DOI = {10.1016/j.ansens.2004.08.001},
      ZBLNUMBER = {1121.11042},
      }
  • [12] Z. Rudnick, "On the asymptotic distribution of zeros of modular forms," Int. Math. Res. Not., vol. 2005, iss. 34, pp. 2059-2074, 2005.
    @article {12, MRKEY = {2181743},
      AUTHOR = {Rudnick, Ze{é}v},
      TITLE = {On the asymptotic distribution of zeros of modular forms},
      JOURNAL = {Int. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2005},
      NUMBER = {34},
      PAGES = {2059--2074},
      ISSN = {1073-7928},
      MRCLASS = {11F72 (11F11)},
      MRNUMBER = {2006k:11099},
      MRREVIEWER = {Gergely Harcos},
      VOLUME = {2005},
      ZBLNUMBER = {1162.11345},
      }
  • [13] 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 {13, 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},
      }
  • [14] Go to document P. Sarnak, "Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity," J. Funct. Anal., vol. 184, iss. 2, pp. 419-453, 2001.
    @article {14, MRKEY = {1851004},
      AUTHOR = {Sarnak, Peter},
      TITLE = {Estimates for {R}ankin-{S}elberg {$L$}-functions and quantum unique ergodicity},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {184},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {419--453},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {11F72 (11F66 37A45)},
      MRNUMBER = {2003c:11050},
      MRREVIEWER = {Henry H. Kim},
      DOI = {10.1006/jfan.2001.3783},
      ZBLNUMBER = {1006.11022},
      }
  • [15] P. Sarnak, "Arithmetic quantum chaos," in The Schur Lectures (1992), Ramat Gan: Bar-Ilan Univ., 1995, vol. 8, pp. 183-236.
    @incollection {15, MRKEY = {1321639},
      AUTHOR = {Sarnak, Peter},
      TITLE = {Arithmetic quantum chaos},
      BOOKTITLE = {The {S}chur Lectures (1992)},
      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},
      }
  • [16] Go to document G. Shimura, "On the holomorphy of certain Dirichlet series," Proc. London Math. Soc., vol. 31, iss. 1, pp. 79-98, 1975.
    @article {16, MRKEY = {0382176},
      AUTHOR = {Shimura, Goro},
      TITLE = {On the holomorphy of certain {D}irichlet series},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {31},
      YEAR = {1975},
      NUMBER = {1},
      PAGES = {79--98},
      ISSN = {0024-6115},
      MRCLASS = {10D15},
      MRNUMBER = {52 \#3064},
      MRREVIEWER = {I. I. Pjateckii-Sapiro},
      DOI = {10.1112/plms/s3-31.1.79},
      ZBLNUMBER = {0311.10029},
      }
  • [17] K. Soundararajan, "Weak subconvexity for central values of $L$-functions," Ann. of Math., vol. 172, pp. 1469-1498, 2010.
    @article{17,
      author={Soundararajan, K.},
      TITLE={Weak subconvexity for central values of $L$-functions},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1469--1498},
      }
  • [18] K. Soundararajan, "Quantum unique ergodicity for $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$," Ann. of Math., vol. 172, pp. 1529-1538, 2010.
    @article{18,
      author={Soundararajan, K.},
      TITLE={Quantum unique ergodicity for $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$},
      JOURNAL={Ann. of Math.},
      VOLUME={172},
      YEAR={2010},
      PAGES={1529--1538},
      }
  • [19] Go to document T. C. Watson, Rankin triple products and quantum chaos, ProQuest LLC, Ann Arbor, MI, 2002.
    @book {19, MRKEY = {2618394},
      AUTHOR = {Watson, Thomas Crawford},
      TITLE = {Rankin triple products and quantum chaos},
      NOTE = {Ph.D. Thesis, 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},
      }
  • [20] 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 {20, 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

Roman Holowinsky

Department of Mathematics
The Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210-1174
United States

Kannan Soundararajan

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