Abstract
We describe a general method to obtain weak subconvexity bounds for many classes of $L$-functions. We give several examples of our bound, and our work has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms
-
[1]
J. Bernstein and A. Reznikov, "Periods, subconvexity of $L$-functions and representation theory," J. Differential Geom., vol. 70, iss. 1, pp. 129-141, 2005.
@article {1, MRKEY = {2192063},
AUTHOR = {Bernstein, Joseph and Reznikov, Andre},
TITLE = {Periods, subconvexity of {$L$}-functions and representation theory},
JOURNAL = {J. Differential Geom.},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {70},
YEAR = {2005},
NUMBER = {1},
PAGES = {129--141},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {11F67 (11F70 22E45)},
MRNUMBER = {2007b:11063},
MRREVIEWER = {Dipendra Prasad},
URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1143572016},
ZBLNUMBER = {1097.11022},
} -
[2] S. Böcherer, P. Sarnak, and R. Schulze-Pillot, "Arithmetic and equidistribution of measures on the sphere," Comm. Math. Phys., vol. 242, iss. 1-2, pp. 67-80, 2003.
@article {2, MRKEY = {2018269},
AUTHOR = {B{ö}cherer, Siegfried and Sarnak, Peter and Schulze-Pillot, Rainer},
TITLE = {Arithmetic and equidistribution of measures on the sphere},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {242},
YEAR = {2003},
NUMBER = {1-2},
PAGES = {67--80},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {11F72 (11F66 58J50)},
MRNUMBER = {2005a:11067},
MRREVIEWER = {Ze{é}v Rudnick},
ZBLNUMBER = {1090.11033},
} -
[3]
F. Brumley, "Effective multiplicity one on ${ GL}_N$ and narrow zero-free regions for Rankin-Selberg $L$-functions," Amer. J. Math., vol. 128, iss. 6, pp. 1455-1474, 2006.
@article {3, MRKEY = {2275908},
AUTHOR = {Brumley, Farrell},
TITLE = {Effective multiplicity one on {${\rm GL}\sb N$} and narrow zero-free regions for {R}ankin-{S}elberg {$L$}-functions},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {128},
YEAR = {2006},
NUMBER = {6},
PAGES = {1455--1474},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {11F70 (11F67 11M20 11M36 22E55)},
MRNUMBER = {2007h:11062},
MRREVIEWER = {Daniel Bump},
URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v128/128.6brumley.pdf},
ZBLNUMBER = {1137.11058},
} -
[4]
J. Cogdell and P. Michel, "On the complex moments of symmetric power $L$-functions at $s=1$," Int. Math. Res. Not., vol. 2004, iss. 31, pp. 1561-1617, 2004.
@article {4, MRKEY = {2035301},
AUTHOR = {Cogdell, J. and Michel, P.},
TITLE = {On the complex moments of symmetric power {$L$}-functions at {$s=1$}},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2004},
NUMBER = {31},
PAGES = {1561--1617},
ISSN = {1073-7928},
MRCLASS = {11F67 (11F66)},
MRNUMBER = {2005f:11094},
MRREVIEWER = {Emmanuel P. Royer},
DOI = {10.1155/S1073792804132455},
VOLUME = {2004},
ZBLNUMBER = {1093.11032},
} -
[5] P. D. T. A. Elliott, "Extrapolating the mean-values of multiplicative functions," Nederl. Akad. Wetensch. Indag. Math., vol. 51, iss. 4, pp. 409-420, 1989.
@article {5, MRKEY = {1041494},
AUTHOR = {Elliott, P. D. T. A.},
TITLE = {Extrapolating the mean-values of multiplicative functions},
JOURNAL = {Nederl. Akad. Wetensch. Indag. Math.},
FJOURNAL = {Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae},
VOLUME = {51},
YEAR = {1989},
NUMBER = {4},
PAGES = {409--420},
ISSN = {0019-3577},
CODEN = {IMTHBJ},
MRCLASS = {11N37},
MRNUMBER = {91a:11043},
MRREVIEWER = {Z. Kry{ž}ius},
ZBLNUMBER = {0696.10041},
} -
[6]
P. B. Garrett, "Decomposition of Eisenstein series: Rankin triple products," Ann. of Math., vol. 125, iss. 2, pp. 209-235, 1987.
@article {6, MRKEY = {881269},
AUTHOR = {Garrett, Paul B.},
TITLE = {Decomposition of {E}isenstein series: {R}ankin triple products},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {125},
YEAR = {1987},
NUMBER = {2},
PAGES = {209--235},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11F66 (11F46 11F67)},
MRNUMBER = {88m:11033},
MRREVIEWER = {Siegfried B{ö}cherer},
DOI = {10.2307/1971310},
ZBLNUMBER = {0625.10020},
} -
[7]
A. Granville and K. Soundararajan, "The distribution of values of $L(1,\chi_d)$," Geom. Funct. Anal., vol. 13, iss. 5, pp. 992-1028, 2003.
@article {7, MRKEY = {2024414},
AUTHOR = {Granville, A. and Soundararajan, K.},
TITLE = {The distribution of values of {$L(1,\chi\sb d)$}},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {13},
YEAR = {2003},
NUMBER = {5},
PAGES = {992--1028},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {11M20},
MRNUMBER = {2005d:11129},
MRREVIEWER = {Philippe G. Michel},
DOI = {10.1007/s00039-003-0438-3},
ZBLNUMBER = {1044.11080},
} -
[8]
A. Granville and K. Soundararajan, "Decay of mean values of multiplicative functions," Canad. J. Math., vol. 55, iss. 6, pp. 1191-1230, 2003.
@article {8, MRKEY = {2016245},
AUTHOR = {Granville, Andrew and Soundararajan, Kannan},
TITLE = {Decay of mean values of multiplicative functions},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
VOLUME = {55},
YEAR = {2003},
NUMBER = {6},
PAGES = {1191--1230},
ISSN = {0008-414X},
CODEN = {CJMAAB},
MRCLASS = {11N56 (11N37 11N60)},
MRNUMBER = {2005b:11157},
MRREVIEWER = {G. Greaves},
URL = {http://journals.cms.math.ca/ams/ams-redirect.php?Journal=CJM&Volume=55&FirstPage=1191},
ZBLNUMBER = {1047.11093},
} -
[9] A. Granville and K. Soundararajan, "Pretentious multiplicative functions and an inequality for the zeta-function," in Anatomy of Integers, Providence, RI: Amer. Math. Soc., 2008, vol. 46, pp. 191-197.
@incollection {9, MRKEY = {2437976},
AUTHOR = {Granville, Andrew and Soundararajan, Kannan},
TITLE = {Pretentious multiplicative functions and an inequality for the zeta-function},
BOOKTITLE = {Anatomy of Integers},
SERIES = {CRM Proc. Lecture Notes},
VOLUME = {46},
PAGES = {191--197},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2008},
MRCLASS = {11N25 (11L20 11L40 11M06 11N60)},
MRNUMBER = {2009k:11145},
MRREVIEWER = {Nigel Watt},
ZBLNUMBER = {1187.11032},
} -
[10]
A. Granville and K. Soundararajan, "Large character sums: pretentious characters and the Pólya-Vinogradov theorem," J. Amer. Math. Soc., vol. 20, iss. 2, pp. 357-384, 2007.
@article {10, MRKEY = {2276774},
AUTHOR = {Granville, Andrew and Soundararajan, Kannan},
TITLE = {Large character sums: pretentious characters and the {P}ólya-{V}inogradov theorem},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {20},
YEAR = {2007},
NUMBER = {2},
PAGES = {357--384},
ISSN = {0894-0347},
MRCLASS = {11L40},
MRNUMBER = {2007k:11128},
MRREVIEWER = {S. W. Graham},
DOI = {10.1090/S0894-0347-06-00536-4},
ZBLNUMBER = {05120647},
} -
[11] G. Halász, "On the distribution of additive and the mean values of multiplicative arithmetic functions," Studia Sci. Math. Hungar., vol. 6, pp. 211-233, 1971.
@article {11, MRKEY = {0319930},
AUTHOR = {Hal{á}sz, G{á}bor},
TITLE = {On the distribution of additive and the mean values of multiplicative arithmetic functions},
JOURNAL = {Studia Sci. Math. Hungar.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica. A Quarterly of the Hungarian Academy of Sciences},
VOLUME = {6},
YEAR = {1971},
PAGES = {211--233},
ISSN = {0081-6906},
MRCLASS = {10K20},
MRNUMBER = {47 \#8471},
MRREVIEWER = {J. Knopfmacher},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0226.10046},
} -
[12] G. Halász, "On the distribution of additive arithmetic functions," Acta Arith., vol. 27, pp. 143-152, 1975.
@article {12, MRKEY = {0369292},
AUTHOR = {Hal{á}sz, G{á}bor},
TITLE = {On the distribution of additive arithmetic functions},
JOURNAL = {Acta Arith.},
FJOURNAL = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
VOLUME = {27},
YEAR = {1975},
PAGES = {143--152},
ISSN = {0065-1036},
MRCLASS = {10H25},
MRNUMBER = {51 \#5527},
MRREVIEWER = {S. L. Segal},
ZBLNUMBER = {0256.10028},
} -
[13]
G. Harcos, "Uniform approximate functional equation for principal $L$-functions," Int. Math. Res. Not., vol. 2002, iss. 18, pp. 923-932, 2002.
@article {13, MRKEY = {1902296},
AUTHOR = {Harcos, Gergely},
TITLE = {Uniform approximate functional equation for principal {$L$}-functions},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2002},
NUMBER = {18},
PAGES = {923--932},
ISSN = {1073-7928},
MRCLASS = {11F66 (11M41)},
MRNUMBER = {2003d:11074},
MRREVIEWER = {K. Soundararajan},
DOI = {10.1155/S1073792802111184},
VOLUME = {2002},
ZBLNUMBER = {0998.11026},
} -
[14]
D. R. Heath-Brown, "Convexity bounds for $L$-functions," Acta Arith., vol. 136, iss. 4, pp. 391-395, 2009.
@article {14, MRKEY = {2476604},
AUTHOR = {Heath-Brown, D. R.},
TITLE = {Convexity bounds for {$L$}-functions},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {136},
YEAR = {2009},
NUMBER = {4},
PAGES = {391--395},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11M41},
MRNUMBER = {2010b:11115},
MRREVIEWER = {W. A. Z{ú}{ñ}iga Galindo},
DOI = {10.4064/aa136-4-6},
ZBLNUMBER = {1169.11038},
} -
[15] A. Hildebrand, "A note on Burgess’ character sum estimate," C. R. Math. Rep. Acad. Sci. Canada, vol. 8, iss. 1, pp. 35-37, 1986.
@article {15, MRKEY = {827113},
AUTHOR = {Hildebrand, Adolf},
TITLE = {A note on {B}urgess' character sum estimate},
JOURNAL = {C. R. Math. Rep. Acad. Sci. Canada},
FJOURNAL = {La Société Royale du Canada. L'Academie des Sciences. Comptes Rendus Mathématiques. (Mathematical Reports)},
VOLUME = {8},
YEAR = {1986},
NUMBER = {1},
PAGES = {35--37},
ISSN = {0706-1994},
MRCLASS = {11L40},
MRNUMBER = {87e:11095},
MRREVIEWER = {Ming Gao Lu},
} -
[16]
A. Hildebrand, "On Wirsing’s mean value theorem for multiplicative functions," Bull. London Math. Soc., vol. 18, iss. 2, pp. 147-152, 1986.
@article {16, MRKEY = {818818},
AUTHOR = {Hildebrand, Adolf},
TITLE = {On {W}irsing's mean value theorem for multiplicative functions},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {The Bulletin of the London Mathematical Society},
VOLUME = {18},
YEAR = {1986},
NUMBER = {2},
PAGES = {147--152},
ISSN = {0024-6093},
CODEN = {LMSBBT},
MRCLASS = {11N37},
MRNUMBER = {87f:11075},
MRREVIEWER = {G. Greaves},
DOI = {10.1112/blms/18.2.147},
ZBLNUMBER = {0564.10048},
} -
[17] R. Holowinsky, "Sieving for mass equidistribution," Ann. of Math., vol. 172, pp. 1499-1516, 2010.
@article{17,
author={Holowinsky, R.},
TITLE={Sieving for mass equidistribution},
JOURNAL={Ann. of Math.},
VOLUME={172},
YEAR={2010},
PAGES={1499--1516},
} -
[18] R. Holowinsky and K. Soundararajan, "Mass equidistribution of Hecke eigenforms," Ann. of Math., vol. 172, p. 1517-1528, %%%%, 2010.
@article{18,
author={Holowinsky, R. and Soundararajan, K.},
TITLE={Mass equidistribution of Hecke eigenforms},
JOURNAL={Ann. of Math.},
VOLUME={172},
YEAR={2010},
PAGES={1517--1528},
}%%%% -
[19] H. Iwaniec, Spectral Methods of Automorphic Forms, Second ed., Providence, RI: Amer. Math. Soc., 2002, vol. 53.
@book {19, 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},
} -
[20] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
@book {20, 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},
} -
[21] H. Iwaniec and P. Sarnak, "Perspectives on the analytic theory of $L$-functions," Geom. Funct. Anal., vol. Special Volume, Part II, pp. 705-741, 2000.
@article {21, 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},
VOLUME = {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},
} -
[22] X. Li, Bounds for $\operatorname{GL}(3)\times \operatorname{GL}( 2)$ $L$-functions and $\operatorname{GL}(3)$ $L$-functions.
@misc{22,
author={Li, X.},
TITLE={Bounds for $\operatorname{GL}(3)\times \operatorname{GL}( 2)$ $L$-functions and $\operatorname{GL}(3)$ $L$-functions},
NOTE={{\it Ann. of Math.},
to appear},
} -
[23]
E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006.
@article {23, 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},
} -
[24]
W. Luo, "Values of symmetric square $L$-functions at $1$," J. Reine Angew. Math., vol. 506, pp. 215-235, 1999.
@article {24, MRKEY = {1665705},
AUTHOR = {Luo, Wenzhi},
TITLE = {Values of symmetric square {$L$}-functions at {$1$}},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {506},
YEAR = {1999},
PAGES = {215--235},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11F67 (11F30)},
MRNUMBER = {2001d:11055},
DOI = {10.1515/crll.1999.007},
ZBLNUMBER = {0969.11018},
} -
[25]
W. Luo and P. Sarnak, "Mass equidistribution for Hecke eigenforms," Comm. Pure Appl. Math., vol. 56, iss. 7, pp. 874-891, 2003.
@article {25, MRKEY = {1990480},
AUTHOR = {Luo, Wenzhi and Sarnak, Peter},
TITLE = {Mass equidistribution for {H}ecke eigenforms},
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},
} -
[26]
W. Luo, Z. Rudnick, and P. Sarnak, "On Selberg’s eigenvalue conjecture," Geom. Funct. Anal., vol. 5, iss. 2, pp. 387-401, 1995.
@article {26, MRKEY = {1334872},
AUTHOR = {Luo, Wenzhi and Rudnick, Z. and Sarnak, Peter},
TITLE = {On {S}elberg's eigenvalue conjecture},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {5},
YEAR = {1995},
NUMBER = {2},
PAGES = {387--401},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {11F72 (11F70)},
MRNUMBER = {96h:11045},
MRREVIEWER = {Jens Bolte},
DOI = {10.1007/BF01895672},
ZBLNUMBER = {0844.11038},
} -
[27] P. Michel, "Analytic number theory and families of automorphic $L$-functions," in Automorphic Forms and Applications, Providence, RI: Amer. Math. Soc., 2007, vol. 12, pp. 181-295.
@incollection {27, MRKEY = {2331346},
AUTHOR = {Michel, Philippe},
TITLE = {Analytic number theory and families of automorphic {$L$}-functions},
BOOKTITLE = {Automorphic Forms and Applications},
SERIES = {IAS/Park City Math. Ser.},
VOLUME = {12},
PAGES = {181--295},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2007},
MRCLASS = {11F66 (11M26 11M41)},
MRNUMBER = {2008m:11104},
MRREVIEWER = {Emmanuel P. Royer},
ZBLNUMBER = {1168.11016},
} -
[28]
G. Molteni, "Upper and lower bounds at $s=1$ for certain Dirichlet series with Euler product," Duke Math. J., vol. 111, iss. 1, pp. 133-158, 2002.
@article {28, MRKEY = {1876443},
AUTHOR = {Molteni, Giuseppe},
TITLE = {Upper and lower bounds at {$s=1$} for certain {D}irichlet series with {E}uler product},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {111},
YEAR = {2002},
NUMBER = {1},
PAGES = {133--158},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11M41 (11F66 11F70)},
MRNUMBER = {2002m:11084},
MRREVIEWER = {Ettore Carletti},
DOI = {10.1215/S0012-7094-02-11114-4},
ZBLNUMBER = {1100.11028},
} -
[29]
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 {29, 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},
} -
[30]
Z. Rudnick and P. Sarnak, "Zeros of principal $L$-functions and random matrix theory," Duke Math. J., vol. 81, iss. 2, pp. 269-322, 1996.
@article {30, MRKEY = {1395406},
AUTHOR = {Rudnick, Ze{é}v and Sarnak, Peter},
TITLE = {Zeros of principal {$L$}-functions and random matrix theory},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {81},
YEAR = {1996},
NUMBER = {2},
PAGES = {269--322},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11M41 (11F70)},
MRNUMBER = {97f:11074},
MRREVIEWER = {Yiannis Petridis},
DOI = {10.1215/S0012-7094-96-08115-6},
ZBLNUMBER = {0866.11050},
} -
[31]
P. Sarnak, "Estimates for Rankin-Selberg $L$-functions and quantum unique ergodicity," J. Funct. Anal., vol. 184, iss. 2, pp. 419-453, 2001.
@article {31, 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},
} -
[32] K. Soundararajan, "Quantum unique ergodicity for $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}$," Ann. of Math., vol. 172, pp. 1529-1538, 2010.
@article{32,
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},
} -
[33] A. Venkatesh, "Sparse equidistribution problems, period bounds, and subconvexity," Ann. of Math., vol. 172, pp. 989-1094, 2010.
@article{33,
author={Venkatesh, A.},
TITLE={Sparse equidistribution problems, period bounds, and subconvexity},
JOURNAL={Ann. of Math.},
VOLUME={172},
YEAR={2010},
PAGES={989--1094},
} -
@book {34, 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},
}