Abstract
We extend to an arbitrary number field the best known bounds towards Ramanujan for the group $\mathrm{GL}_n$, $n=2,3,4$. In particular, we present a technique which overcomes the analytic obstacles posed by the presence of an infinite group of units.
-
[BH]
V. Blomer and G. Harcos, "Twisted $L$-functions over number fields and Hilbert’s eleventh problem," Geom. Funct. Anal., vol. 20, iss. 1, pp. 1-52, 2010.@article {BH, MRKEY = {2647133},
AUTHOR = {Blomer, Valentin and Harcos, Gergely},
TITLE = {Twisted {$L$}-functions over number fields and {H}ilbert's eleventh problem},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {20},
YEAR = {2010},
NUMBER = {1},
PAGES = {1--52},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {11F41 (11E45 11F70 11F72 11M41)},
MRNUMBER = {2647133},
MRREVIEWER = {Roelof Wichert Bruggeman},
DOI = {10.1007/s00039-010-0063-x},
ZBLNUMBER = {05770217},
} -
[BM] R. W. Bruggeman and R. J. Miatello, "Sum formula for ${ SL}_2$ over a number field and Selberg type estimate for exceptional eigenvalues," Geom. Funct. Anal., vol. 8, iss. 4, pp. 627-655, 1998.
@article {BM, MRKEY = {1633975},
AUTHOR = {Bruggeman, R. W. and Miatello, R. J.},
TITLE = {Sum formula for {${\rm SL}\sb 2$} over a number field and {S}elberg type estimate for exceptional eigenvalues},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {8},
YEAR = {1998},
NUMBER = {4},
PAGES = {627--655},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {11F72 (11F55)},
MRNUMBER = {1633975},
MRREVIEWER = {Stefan K{ü}hnlein},
DOI = {10.1007/s000390050069},
} -
[BG] D. Bump and D. Ginzburg, "Symmetric square $L$-functions on ${ GL}(r)$," Ann. of Math., vol. 136, iss. 1, pp. 137-205, 1992.
@article {BG, MRKEY = {1173928},
AUTHOR = {Bump, Daniel and Ginzburg, David},
TITLE = {Symmetric square {$L$}-functions on {${\rm GL}(r)$}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {136},
YEAR = {1992},
NUMBER = {1},
PAGES = {137--205},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11F66 (11F27 11F70)},
MRNUMBER = {1173928},
MRREVIEWER = {Stephen Gelbart},
DOI = {10.2307/2946548},
ZBLNUMBER = {0753.11021},
} -
[De] P. Deligne, "La conjecture de Weil. I," Inst. Hautes Études Sci. Publ. Math., iss. 43, pp. 273-307, 1974.
@article {De, MRKEY = {0340258},
AUTHOR = {Deligne, Pierre},
TITLE = {La conjecture de {W}eil. {I}},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
NUMBER = {43},
YEAR = {1974},
PAGES = {273--307},
ISSN = {0073-8301},
MRCLASS = {14G13},
MRNUMBER = {0340258},
MRREVIEWER = {Nicholas M. Katz},
URL = {http://www.numdam.org/item?id=PMIHES_1974__43__273_0},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {0287.14001},
} -
[De1] P. Deligne, Cohomologie Étale, New York: Springer-Verlag, 1977, vol. 569.
@book {De1, MRKEY = {0463174},
AUTHOR = {Deligne, Pierre},
TITLE = {Cohomologie Étale},
NOTE={S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois-Marie SGA 4$\frac{1}{2}$},
SERIES = {Lecture Notes in Math},
VOLUME={569},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1977},
PAGES = {iv+312pp},
MRCLASS = {14F20},
MRNUMBER = {0463174},
MRREVIEWER = {J. S. Milne},
ZBLNUMBER = {0349.14008},
ZBLNUMBER = {0345.00010},
} -
[DI] W. Duke and H. Iwaniec, "Estimates for coefficients of $L$-functions. I," in Automorphic forms and analytic number theory (Montreal, PQ, 1989), Montreal, QC: Univ. Montréal, 1990, pp. 43-47.
@incollection {DI, MRKEY = {1111010},
AUTHOR = {Duke, W. and Iwaniec, H.},
TITLE = {Estimates for coefficients of {$L$}-functions. {I}},
BOOKTITLE = {Automorphic forms and analytic number theory ({M}ontreal, {PQ},
1989)},
PAGES = {43--47},
PUBLISHER = {Univ. Montréal},
ADDRESS = {Montreal, QC},
YEAR = {1990},
MRCLASS = {11F66 (11F25)},
MRNUMBER = {1111010},
MRREVIEWER = {Kazuyuki Hatada},
ZBLNUMBER = {0745.11030},
} -
[GS] S. Gelbart and F. Shahidi, "Boundedness of automorphic $L$-functions in vertical strips," J. Amer. Math. Soc., vol. 14, iss. 1, pp. 79-107, 2001.
@article {GS, MRKEY = {1800349},
AUTHOR = {Gelbart, Stephen and Shahidi, Freydoon},
TITLE = {Boundedness of automorphic {$L$}-functions in vertical strips},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {14},
YEAR = {2001},
NUMBER = {1},
PAGES = {79--107},
ISSN = {0894-0347},
MRCLASS = {11F70 (11F66 11R39 22E46)},
MRNUMBER = {1800349},
MRREVIEWER = {Henry H. Kim},
DOI = {10.1090/S0894-0347-00-00351-9},
ZBLNUMBER = {1050.11053},
} -
[HT] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2001, vol. 151.
@book {HT, MRKEY = {1876802},
AUTHOR = {Harris, Michael and Taylor, Richard},
TITLE = {The Geometry and Cohomology of Some Simple {S}himura Varieties},
SERIES = {Ann. of Math. Studies},
VOLUME = {151},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {2001},
PAGES = {viii+276},
ISBN = {0-691-09090-4},
MRCLASS = {11G18 (11F70 11S37 14G35 22E45)},
MRNUMBER = {1876802},
MRREVIEWER = {James Milne},
ZBLNUMBER = {1036.11027},
} -
[Iw] H. Iwaniec, "The lowest eigenvalue for congruence groups," in Topics in Geometry, Boston, MA: Birkhäuser, 1996, vol. 20, pp. 203-212.
@incollection {Iw, MRKEY = {1390315},
AUTHOR = {Iwaniec, Henryk},
TITLE = {The lowest eigenvalue for congruence groups},
BOOKTITLE = {Topics in Geometry},
SERIES = {Progr. Nonlinear Differential Equations Appl.},
VOLUME = {20},
PAGES = {203--212},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1996},
MRCLASS = {11F72 (11F66)},
MRNUMBER = {1390315},
MRREVIEWER = {{Ö}zlem Imamo{\=g}lu},
ZBLNUMBER = {0884.11024},
} -
[JS] H. Jacquet and J. A. Shalika, "On Euler products and the classification of automorphic representations. I," Amer. J. Math., vol. 103, iss. 3, pp. 499-558, 1981.
@article {JS, MRKEY = {0618323},
AUTHOR = {Jacquet, H. and Shalika, J. A.},
TITLE = {On {E}uler products and the classification of automorphic representations. {I}},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {103},
YEAR = {1981},
NUMBER = {3},
PAGES = {499--558},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {10D40 (12A67 22E55)},
MRNUMBER = {0618323},
MRREVIEWER = {Freydoon Shahidi},
DOI = {10.2307/2374103},
ZBLNUMBER = {0473.12008},
} -
[K1] H. H. Kim, "Langlands-Shahidi method and poles of automorphic $L$-functions: application to exterior square $L$-functions," Canad. J. Math., vol. 51, iss. 4, pp. 835-849, 1999.
@article {K1, MRKEY = {1701344},
AUTHOR = {Kim, Henry H.},
TITLE = {Langlands-{S}hahidi method and poles of automorphic {$L$}-functions: application to exterior square {$L$}-functions},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
VOLUME = {51},
YEAR = {1999},
NUMBER = {4},
PAGES = {835--849},
ISSN = {0008-414X},
CODEN = {CJMAAB},
MRCLASS = {11F70 (11F66 22E55)},
MRNUMBER = {1701344},
MRREVIEWER = {Goran Mui{ć}},
DOI = {10.4153/CJM-1999-036-0},
ZBLNUMBER = {0934.11024},
} -
[KS] H. H. Kim and P. Sarnak, "Refined estimates towards the Ramanujan and Selberg conjectures," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 139-183, 2003.
@article {KS, MRKEY = {1937203},
AUTHOR = {Kim, Henry H. and Sarnak, P.},
TITLE = {Refined estimates towards the {R}amanujan and {S}elberg conjectures},
NOTE = {appendix to H. Kim, 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 = {1937203},
MRREVIEWER = {Mahdi Asgari},
DOI = {10.1090/S0894-0347-02-00410-1},
ZBLNUMBER = {1018.11024},
} -
[KSh] H. H. Kim and F. Shahidi, "Cuspidality of symmetric powers with applications," Duke Math. J., vol. 112, iss. 1, pp. 177-197, 2002.
@article {KSh, MRKEY = {1890650},
AUTHOR = {Kim, Henry H. and Shahidi, Freydoon},
TITLE = {Cuspidality of symmetric powers with applications},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {112},
YEAR = {2002},
NUMBER = {1},
PAGES = {177--197},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F70 (11F30 11R42)},
MRNUMBER = {1890650},
MRREVIEWER = {A. Raghuram},
DOI = {10.1215/S0012-9074-02-11215-0},
ZBLNUMBER = {1074.11027},
} -
[LRS1] W. Luo, Z. Rudnick, and P. Sarnak, "On Selberg’s eigenvalue conjecture," Geom. Funct. Anal., vol. 5, iss. 2, pp. 387-401, 1995.
@article {LRS1, MRKEY = {1334872},
AUTHOR = {Luo, Wenzhi and Rudnick, Ze{é}v 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 = {1334872},
MRREVIEWER = {Jens Bolte},
DOI = {10.1007/BF01895672},
ZBLNUMBER = {0844.11038},
} -
[LRS2] W. Luo, Z. Rudnick, and P. Sarnak, "On the generalized Ramanujan conjecture for ${ GL}(n)$," in Automorphic Forms, Automorphic Representations, and Arithmetic, Providence, RI, 1999, pp. 301-310.
@inproceedings {LRS2, MRKEY = {1703764},
AUTHOR = {Luo, Wenzhi and Rudnick, Ze{é}v and Sarnak, Peter},
TITLE = {On the generalized {R}amanujan conjecture for {${\rm GL}(n)$}},
BOOKTITLE = {Automorphic Forms, Automorphic Representations, and Arithmetic},
VENUE={{F}ort {W}orth, {TX},
1996},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {66},
PAGES = {301--310},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {11F70 (11F72)},
MRNUMBER = {1703764},
MRREVIEWER = {Stefan K{ü}hnlein},
ZBLNUMBER = {0965.11023},
} -
[MS] W. Müller and B. Speh, "Absolute convergence of the spectral side of the Arthur trace formula for ${ GL}_n$," Geom. Funct. Anal., vol. 14, iss. 1, pp. 58-93, 2004.
@article {MS, MRKEY = {2053600},
AUTHOR = {M{ü}ller, W. and Speh, B.},
TITLE = {Absolute convergence of the spectral side of the {A}rthur trace formula for {${\rm GL}\sb n$}},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {14},
YEAR = {2004},
NUMBER = {1},
PAGES = {58--93},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {22E55 (11F70 11F72)},
MRNUMBER = {2053600},
MRREVIEWER = {Werner Hoffmann},
DOI = {10.1007/s00039-004-0452-0},
} -
[N] W. Narkiewicz, "Units in residue classes," Arch. Math. $($Basel$)$, vol. 51, iss. 3, pp. 238-241, 1988.
@article {N, MRKEY = {0960401},
AUTHOR = {Narkiewicz, W.},
TITLE = {Units in residue classes},
JOURNAL = {Arch. Math. $($Basel$)$},
FJOURNAL = {Archiv der Mathematik},
VOLUME = {51},
YEAR = {1988},
NUMBER = {3},
PAGES = {238--241},
ISSN = {0003-889X},
CODEN = {ACVMAL},
MRCLASS = {11R04 (11N69 11R27)},
MRNUMBER = {0960401},
MRREVIEWER = {M. Ram Murty},
DOI = {10.1007/BF01207477},
ZBLNUMBER = {0641.12004},
} -
[Maki] M. Nakasuji, Generalized Ramanujan conjecture over general imaginary quadratic fields.
@misc{Maki,
author={Nakasuji, M.},
TITLE={Generalized {R}amanujan conjecture over general imaginary quadratic fields},
NOTE={{\it Forum Math.},
to appear},
} -
[Ra] S. Ramanujan, "On certain Arithmetical Functions," Trans. Cambridge Phil. Soc., vol. 22, pp. 159-184, 1916.
@article{Ra,
author={Ramanujan, S.},
TITLE={On certain Arithmetical Functions},
JOURNAL={Trans. Cambridge Phil. Soc.},
VOLUME={22},
YEAR={1916},
PAGES={159--184},
} -
[Ro] D. E. Rohrlich, "Nonvanishing of $L$-functions for ${ GL}(2)$," Invent. Math., vol. 97, iss. 2, pp. 381-403, 1989.
@article {Ro, MRKEY = {1001846},
AUTHOR = {Rohrlich, David E.},
TITLE = {Nonvanishing of {$L$}-functions for {${\rm GL}(2)$}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {97},
YEAR = {1989},
NUMBER = {2},
PAGES = {381--403},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {11F67 (11F70 11M56 11N35 11R42)},
MRNUMBER = {1001846},
MRREVIEWER = {David Joyner},
DOI = {10.1007/BF01389047},
ZBLNUMBER = {0677.10020},
} -
[RS] 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 {RS, 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 = {1395406},
MRREVIEWER = {Yiannis Petridis},
DOI = {10.1215/S0012-7094-96-08115-6},
ZBLNUMBER = {0866.11050},
} -
[Sh81] F. Shahidi, "On certain $L$-functions," Amer. J. Math., vol. 103, iss. 2, pp. 297-355, 1981.
@article {Sh81, MRKEY = {0610479},
AUTHOR = {Shahidi, Freydoon},
TITLE = {On certain {$L$}-functions},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {103},
YEAR = {1981},
NUMBER = {2},
PAGES = {297--355},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {10D15 (10D40 22E45 22E55)},
MRNUMBER = {0610479},
MRREVIEWER = {Stephen Gelbart},
DOI = {10.2307/2374219},
ZBLNUMBER = {0467.12013},
} -
[Sh85] F. Shahidi, "Local coefficients as Artin factors for real groups," Duke Math. J., vol. 52, iss. 4, pp. 973-1007, 1985.
@article {Sh85, MRKEY = {0816396},
AUTHOR = {Shahidi, Freydoon},
TITLE = {Local coefficients as {A}rtin factors for real groups},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {52},
YEAR = {1985},
NUMBER = {4},
PAGES = {973--1007},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F67 (11F70 11R39 22E30)},
MRNUMBER = {0816396},
MRREVIEWER = {K. F. Lai},
DOI = {10.1215/S0012-7094-85-05252-4},
ZBLNUMBER = {0674.10027},
} -
[Sh0] F. Shahidi, "A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups," Ann. of Math., vol. 132, iss. 2, pp. 273-330, 1990.
@article {Sh0, MRKEY = {1070599},
AUTHOR = {Shahidi, Freydoon},
TITLE = {A proof of {L}anglands' conjecture on {P}lancherel measures; complementary series for {$p$}-adic groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {132},
YEAR = {1990},
NUMBER = {2},
PAGES = {273--330},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11R39 (11F70 11S37 22E35 22E55)},
MRNUMBER = {1070599},
MRREVIEWER = {Stephen Gelbart},
DOI = {10.2307/1971524},
} -
[Sh92] F. Shahidi, "Twisted endoscopy and reducibility of induced representations for $p$-adic groups," Duke Math. J., vol. 66, iss. 1, pp. 1-41, 1992.
@article {Sh92, MRKEY = {1159430},
AUTHOR = {Shahidi, Freydoon},
TITLE = {Twisted endoscopy and reducibility of induced representations for {$p$}-adic groups},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {66},
YEAR = {1992},
NUMBER = {1},
PAGES = {1--41},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {22E50 (11F70)},
MRNUMBER = {1159430},
MRREVIEWER = {David Joyner},
DOI = {10.1215/S0012-7094-92-06601-4},
ZBLNUMBER = {0785.22022},
} -
[Sh1] F. Shahidi, "On the Ramanujan conjecture and finiteness of poles for certain $L$-functions," Ann. of Math., vol. 127, iss. 3, pp. 547-584, 1988.
@article {Sh1, MRKEY = {0942520},
AUTHOR = {Shahidi, Freydoon},
TITLE = {On the {R}amanujan conjecture and finiteness of poles for certain {$L$}-functions},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {127},
YEAR = {1988},
NUMBER = {3},
PAGES = {547--584},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {11F70 (11F66 11F67 22E50 22E55)},
MRNUMBER = {0942520},
MRREVIEWER = {David Joyner},
DOI = {10.2307/2007005},
ZBLNUMBER = {0654.10029},
} -
[Ta] S. Takeda, The twisted symmetric square $L$-function of GL($r$).
@misc{Ta,
author={Takeda, S.},
TITLE={The twisted symmetric square $L$-function of {GL}($r$)},
NOTE={preprint},
ARXIV={1005.1979},
} -
[We] A. Weil, Basic Number Theory, New York: Springer-Verlag, 1967, vol. 144.
@book {We, MRKEY = {0234930},
AUTHOR = {Weil, Andr{é}},
TITLE = {Basic Number Theory},
SERIES = {Die Grundlehren der mathematischen Wissenschaften},
VOLUME={144},
PUBLISHER = {Springer-Verlag},
YEAR = {1967},
PAGES = {xviii+294},
MRCLASS = {10.65},
MRNUMBER = {0234930},
MRREVIEWER = {G. Whaples},
ADDRESS = {New York},
ZBLNUMBER = {0176.33601},
}
Print version