Abstract
We approach the holomorphic analogue to the Quantum Unique Ergodicity conjecture through an application of the Large Sieve. We deal with shifted convolution sums as in [Hol09], with various simplifications in our analysis due to the knowledge of the Ramanujan-Petersson conjecture in this holomorphic case.
-
[E-M-S]
P. Elliott, C. Moreno, and F. Shahidi, "On the absolute value of Ramanujan’s $\tau$-function," Math. Ann., vol. 266, iss. 4, pp. 507-511, 1984.
@article {E-M-S, MRKEY = {735531},
AUTHOR = {Elliott, P. and Moreno, C. and Shahidi, F.},
TITLE = {On the absolute value of {R}amanujan's {$\tau$}-function},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {266},
YEAR = {1984},
NUMBER = {4},
PAGES = {507--511},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {11F11 (11N37)},
MRNUMBER = {85f:11030},
MRREVIEWER = {T. M. Apostol},
DOI = {10.1007/BF01458543},
ZBLNUMBER = {0513.10024},
} -
[G-H-L]
D. Goldfeld, J. Hoffstein, and D. Lieman, "An effective zero-free region," Ann. of Math., vol. 140, pp. 177-181, 1994.
@article {G-H-L, MRKEY = {1289494},
AUTHOR = {Goldfeld, D. and Hoffstein, J. and Lieman, D. },
TITLE = {An effective zero-free region},
NOTE = {appendix of \cite{H-L}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {140},
YEAR = {1994},
PAGES = {177--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},
} -
[H-L]
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 {H-L, 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},
} -
[Ho]
R. Holowinsky, "A sieve method for shifted convolution sums," Duke Math. J., vol. 146, iss. 3, pp. 401-448, 2009.
@article {Ho, MRKEY = {2484279},
AUTHOR = {Holowinsky, Roman},
TITLE = {A sieve method for shifted convolution sums},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {146},
YEAR = {2009},
NUMBER = {3},
PAGES = {401--448},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11N36 (11F30 11M36)},
MRNUMBER = {2484279},
MRREVIEWER = {Don Redmond},
DOI = {10.1215/00127094-2009-002},
ZBLNUMBER = {05505895},
} -
[H-S] R. Holowinsky and K. Soundararajan, "Mass equidistribution of Hecke eigenforms," Annals of Math., vol. 172, pp. 1517-1528, 2010.
@article{H-S,
author = {Holowinsky, Roman and Soundararajan, K.},
TITLE={Mass equidistribution of Hecke eigenforms},
JOURNAL={Annals of Math.},
VOLUME={172},
YEAR={2010},
PAGES={1517--1528},
} -
[Iw] H. Iwaniec, Introduction to the Spectral Theory of Automorphic Forms, Madrid: Revista Matemática Iberoamericana, 1995.
@book {Iw, MRKEY = {1325466},
AUTHOR = {Iwaniec, Henryk},
TITLE = {Introduction to the Spectral Theory of Automorphic Forms},
SERIES = {Biblioteca de la Revista Matemática Iberoamericana},
PUBLISHER = {Revista Matemática Iberoamericana},
ADDRESS = {Madrid},
YEAR = {1995},
PAGES = {xiv+247},
MRCLASS = {11F72 (11-02 11F12 11F37 35Pxx 58G25)},
MRNUMBER = {96f:11078},
MRREVIEWER = {Solomon Friedberg},
ZBLNUMBER = {0847.11028},
} -
[I-K] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004.
@book {I-K, MRKEY = {2061214},
AUTHOR = {Iwaniec, Henryk and Kowalski, Emmanuel},
TITLE = {Analytic Number Theory},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
NUMBER = {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 = {1071.11001},
} -
[I-S] H. Iwaniec and P. Sarnak, "Perspectives on the analytic theory of $L$-functions," Geom. Funct. Anal., iss. Special Volume, Part II, pp. 705-741, 2000.
@article {I-S, MRKEY = {1826269},
AUTHOR = {Iwaniec, H. and Sarnak, P.},
TITLE = {Perspectives on the analytic theory of {$L$}-functions},
NOTE = {GAFA 2000 (Tel Aviv, 1999)},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
YEAR = {2000},
NUMBER = {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},
} -
[Ki]
H. H. Kim, "Functoriality for the exterior square of ${ GL}_4$ and the symmetric fourth of ${ GL}_2$," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 139-183, 2003.
@article {Ki, MRKEY = {1937203},
AUTHOR = {Kim, Henry H.},
TITLE = {Functoriality for the exterior square of {${\rm GL}\sb 4$} and the symmetric fourth of {${\rm GL}\sb 2$}},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {16},
YEAR = {2003},
NUMBER = {1},
PAGES = {139--183},
ISSN = {0894-0347},
MRCLASS = {11F70 (11R39 22E46)},
MRNUMBER = {2003k:11083},
MRREVIEWER = {Mahdi Asgari},
DOI = {10.1090/S0894-0347-02-00410-1},
ZBLNUMBER = {1018.11024},
} -
[K-Sa]
H. H. Kim, "Functoriality for the exterior square of ${ GL}_4$ and the symmetric fourth of ${ GL}_2$," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 139-183, 2003.
@article {K-Sa, MRKEY = {1937203},
AUTHOR = {Kim, Henry H.},
TITLE = {Functoriality for the exterior square of {${\rm GL}\sb 4$} and the symmetric fourth of {${\rm GL}\sb 2$}},
NOTE = {With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {16},
YEAR = {2003},
NUMBER = {1},
PAGES = {139--183},
ISSN = {0894-0347},
MRCLASS = {11F70 (11R39 22E46)},
MRNUMBER = {2003k:11083},
MRREVIEWER = {Mahdi Asgari},
DOI = {10.1090/S0894-0347-02-00410-1},
ZBLNUMBER = {1018.11024},
} -
[K-Sh]
H. H. Kim and F. Shahidi, "Functorial products for $ GL_2\times GL_3$ and functorial symmetric cube for $ GL_2$," C. R. Acad. Sci. Paris Sér. I Math., vol. 331, iss. 8, pp. 599-604, 2000.
@article {K-Sh, MRKEY = {1799096},
AUTHOR = {Kim, Henry H. and Shahidi, Freydoon},
TITLE = {Functorial products for {$\rm GL\sb 2\times GL\sb 3$} and functorial symmetric cube for {$\rm GL\sb 2$}},
JOURNAL = {C. R. Acad. Sci. Paris Sér. {\rm I} Math.},
FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {331},
YEAR = {2000},
NUMBER = {8},
PAGES = {599--604},
ISSN = {0764-4442},
CODEN = {CASMEI},
MRCLASS = {11F70 (22E55)},
MRNUMBER = {2002i:11049},
DOI = {10.1016/S0764-4442(00)01677-3},
ZBLNUMBER = {1002.11044},
} -
[Li]
E. Lindenstrauss, "Invariant measures and arithmetic quantum unique ergodicity," Ann. of Math., vol. 163, iss. 1, pp. 165-219, 2006.
@article {Li, 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},
} -
[L-S]
W. Luo and P. Sarnak, "Mass equidistribution for Hecke eigenforms," Comm. Pure Appl. Math., vol. 56, iss. 7, pp. 874-891, 2003.
@article {L-S, 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},
} -
[M-V] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory. I. Classical Theory, Cambridge: Cambridge Univ. Press, 2007.
@book {M-V, MRKEY = {2378655},
AUTHOR = {Montgomery, Hugh L. and Vaughan, Robert C.},
TITLE = {Multiplicative Number Theory. {\rm I}. {C}lassical Theory},
SERIES = {Cambridge Stud. Adv. Math.},
NUMBER = {97},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {2007},
PAGES = {xviii+552},
ISBN = {978-0-521-84903-6; 0-521-84903-9},
MRCLASS = {11-02 (11-01 11M06 11M26 11M45 11N05 11N37)},
MRNUMBER = {2009b:11001},
MRREVIEWER = {Wolfgang Schwarz},
ZBLNUMBER={1142.11001},
} -
[N] M. Nair, "Multiplicative functions of polynomial values in short intervals," Acta Arith., vol. 62, iss. 3, pp. 257-269, 1992.
@article {N, MRKEY = {1197420},
AUTHOR = {Nair, Mohan},
TITLE = {Multiplicative functions of polynomial values in short intervals},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {62},
YEAR = {1992},
NUMBER = {3},
PAGES = {257--269},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11N32 (11N64)},
MRNUMBER = {94b:11093},
MRREVIEWER = {Michael Filaseta},
ZBLNUMBER = {0768.11038},
} -
[N-T]
M. Nair and G. Tenenbaum, "Short sums of certain arithmetic functions," Acta Math., vol. 180, iss. 1, pp. 119-144, 1998.
@article {N-T, MRKEY = {1618321},
AUTHOR = {Nair, Mohan and Tenenbaum, G{é}rald},
TITLE = {Short sums of certain arithmetic functions},
JOURNAL = {Acta Math.},
FJOURNAL = {Acta Mathematica},
VOLUME = {180},
YEAR = {1998},
NUMBER = {1},
PAGES = {119--144},
ISSN = {0001-5962},
CODEN = {ACMAA8},
MRCLASS = {11N37 (11N25 11N32)},
MRNUMBER = {99j:11110},
MRREVIEWER = {A. J. Hildebrand},
DOI = {10.1007/BF02392880},
ZBLNUMBER = {0917.11048},
} -
[R-S]
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 {R-S, 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},
} -
[Sa]
P. Sarnak, "Spectra of hyperbolic surfaces," Bull. Amer. Math. Soc., vol. 40, iss. 4, pp. 441-478, 2003.
@article {Sa, MRKEY = {1997348},
AUTHOR = {Sarnak, Peter},
TITLE = {Spectra of hyperbolic surfaces},
JOURNAL = {Bull. Amer. Math. Soc.},
FJOURNAL = {American Mathematical Society. Bulletin. New Series},
VOLUME = {40},
YEAR = {2003},
NUMBER = {4},
PAGES = {441--478},
ISSN = {0273-0979},
CODEN = {BAMOAD},
MRCLASS = {11N75 (11F03 11F72 11R42 30F45 35P30)},
MRNUMBER = {2004f:11107},
MRREVIEWER = {Stephen D. Miller},
DOI = {10.1090/S0273-0979-03-00991-1},
ZBLNUMBER = {1045.11033},
} -
[Se] A. Selberg, "On the estimation of Fourier coefficients of modular forms," in Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., 1965, vol. 8, pp. 1-15.
@incollection {Se, MRKEY = {0182610},
AUTHOR = {Selberg, Atle},
TITLE = {On the estimation of {F}ourier coefficients of modular forms},
BOOKTITLE = {Proc. {S}ympos. {P}ure {M}ath.},
VOLUME={8},
PAGES = {1--15},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1965},
MRCLASS = {10.20},
MRNUMBER = {32 \#93},
MRREVIEWER = {J. R. Smart},
ZBLNUMBER = {0142.33903},
} -
[So] K. Soundararajan, "Weak subconvexity for central values of $L$-functions," Ann. of Math., vol. 172, pp. 1469-1498, 2010.
@article{So,
author={Soundararajan, K.},
TITLE={Weak subconvexity for central values of $L$-functions},
JOURNAL={Ann. of Math.},
VOLUME={172},
YEAR={2010},
PAGES={1469--1498},
} -
[Ze]
S. Zelditch, "Uniform distribution of eigenfunctions on compact hyperbolic surfaces," Duke Math. J., vol. 55, iss. 4, pp. 919-941, 1987.
@article {Ze, MRKEY = {916129},
AUTHOR = {Zelditch, Steven},
TITLE = {Uniform distribution of eigenfunctions on compact hyperbolic surfaces},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {55},
YEAR = {1987},
NUMBER = {4},
PAGES = {919--941},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {58G25 (58G30)},
MRNUMBER = {89d:58129},
MRREVIEWER = {Alejandro Uribe},
DOI = {10.1215/S0012-7094-87-05546-3},
ZBLNUMBER = {0643.58029},
}