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

Kannan Soundararajan

doi:http://doi.org/10.4007/annals.2010.172.1529

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 ${\rm SL}_2(\mathbb{Z})\backslash \mathbb{H}$.

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[1] $Go to document$ P. Dusart, Autour de la fonction qui compte le nombre de nombre premiers, 1998.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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.

Show bibtex

@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}, }

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

Abstract

Authors