Abstract
We derive a refinement of the spectral expansion of Arthur’s trace formula. The expression is absolutely convergent with respect to the trace norm.
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[MR518111]
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@article {MR518111, MRKEY = {0518111},
AUTHOR = {Arthur, James},
TITLE = {A trace formula for reductive groups. {I}. {T}erms associated to classes in {$G({\bf Q})$}},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {45},
YEAR = {1978},
NUMBER = {4},
PAGES = {911--952},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {10D40 (22E55)},
MRNUMBER = {0518111},
MRREVIEWER = {Stephen Gelbart},
ZBLNUMBER = {0499.10032},
DOI = {10.1215/S0012-7094-78-04533-7},
} -
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@article {MR558260, MRKEY = {0558260},
AUTHOR = {Arthur, James},
TITLE = {A trace formula for reductive groups. {II}. {A}pplications of a truncation operator},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {40},
YEAR = {1980},
NUMBER = {1},
PAGES = {87--121},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {22E55 (10D40 20G05)},
MRNUMBER = {0558260},
MRREVIEWER = {Stephen Gelbart},
ZBLNUMBER = {0499.10033},
} -
[MR625344]
J. Arthur, "The trace formula in invariant form," Ann. of Math., vol. 114, iss. 1, pp. 1-74, 1981.
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AUTHOR = {Arthur, James},
TITLE = {The trace formula in invariant form},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
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YEAR = {1981},
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ISSN = {0003-486X},
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MRCLASS = {10D40 (22E55)},
MRNUMBER = {0625344},
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DOI = {10.2307/1971376},
ZBLNUMBER = {0495.22006},
} -
[MR681737]
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AUTHOR = {Arthur, James},
TITLE = {On a family of distributions obtained from {E}isenstein series. {I}. {A}pplication of the {P}aley-{W}iener theorem},
JOURNAL = {Amer. J. Math.},
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PAGES = {1243--1288},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {22E55 (11F20 11F70 11F72 46F99)},
MRNUMBER = {0681737},
DOI = {10.2307/2374061},
ZBLNUMBER = {0541.22010},
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[MR681738]
J. Arthur, "On a family of distributions obtained from Eisenstein series. II. Explicit formulas," Amer. J. Math., vol. 104, iss. 6, pp. 1289-1336, 1982.
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AUTHOR = {Arthur, James},
TITLE = {On a family of distributions obtained from {E}isenstein series. {II}. {E}xplicit formulas},
JOURNAL = {Amer. J. Math.},
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VOLUME = {104},
YEAR = {1982},
NUMBER = {6},
PAGES = {1289--1336},
ISSN = {0002-9327},
CODEN = {AJMAAN},
MRCLASS = {22E55 (11F70 11F72 46F99)},
MRNUMBER = {0681738},
MRREVIEWER = {Freydoon Shahidi},
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ZBLNUMBER = {0562.22004},
} -
[MR650368]
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FJOURNAL = {Duke Mathematical Journal},
VOLUME = {49},
YEAR = {1982},
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PAGES = {35--70},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {22E55 (10D40 22E50)},
MRNUMBER = {0650368},
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DOI = {10.1215/S0012-7094-82-04904-3},
} -
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@incollection {MR2192011, MRKEY = {2192011},
AUTHOR = {Arthur, James},
TITLE = {An introduction to the trace formula},
BOOKTITLE = {Harmonic Analysis, the Trace Formula, and {S}himura Varieties},
SERIES = {Clay Math. Proc.},
VOLUME = {4},
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PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2005},
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} -
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} -
[FL2] T. Finis and E. M. Lapid, "On the Arthur-Selberg trace formula for ${G}{L}(2)$," Groups Geom. Dyn., vol. 5, pp. 367-391, 2011.
@article{FL2,
author = {Finis, Tobias and Lapid, Erez M.},
TITLE={On the {A}rthur-{S}elberg trace formula for ${G}{L}(2)$},
JOURNAL={Groups Geom. Dyn.},
VOLUME={5},
YEAR={2011},
NOTE={special issue on the occasion of {F}ritz {G}runewald's 60th birthday},
PAGES={367--391},
} -
[FL3]
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author = {Finis, Tobias and Lapid, Erez M.},
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YEAR={2011},
PAGES={784--802},
DOI={10.1112/S0010437X11004891},
} -
[FLAn] T. Finis and E. M. Lapid, On the spectral side of Arthur’s trace formula — combinatorial setup.
@misc{FLAn,
author = {Finis, Tobias and Lapid, Erez M.},
TITLE={On the spectral side of {A}rthur's trace formula -- combinatorial setup},
NOTE={this volume},
} -
[MR2550527]
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AUTHOR = {Finis, Tobias and Lapid, Erez M. and M{ü}ller, Werner},
TITLE = {The spectral side of {A}rthur's trace formula},
JOURNAL = {Proc. Natl. Acad. Sci. USA},
FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
VOLUME = {106},
YEAR = {2009},
NUMBER = {37},
PAGES = {15563--15566},
ISSN = {1091-6490},
CODEN = {PNASFB},
MRCLASS = {11F72 (22E55)},
MRNUMBER = {2550527},
MRREVIEWER = {{\Dbar}{\cftil{o}} Ng{\d{o}}c Di{\cfudot{e}}p},
DOI = {10.1073/pnas.0905408106},
ZBLNUMBER = {1203.11042},
} -
[MR1622604]
L. Ji, "The trace class conjecture for arithmetic groups," J. Differential Geom., vol. 48, iss. 1, pp. 165-203, 1998.
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AUTHOR = {Ji, Lizhen},
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VOLUME = {48},
YEAR = {1998},
NUMBER = {1},
PAGES = {165--203},
ISSN = {0022-040X},
CODEN = {JDGEAS},
MRCLASS = {11F72 (22E40)},
MRNUMBER = {1622604},
MRREVIEWER = {Werner Hoffmann},
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} -
[MR1625060]
H. Jacquet, E. Lapid, and J. Rogawski, "Periods of automorphic forms," J. Amer. Math. Soc., vol. 12, iss. 1, pp. 173-240, 1999.
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MRNUMBER = {1625060},
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[MR0579181]
R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, New York: Springer-Verlag, 1976, vol. 544.
@book {MR0579181, MRKEY = {0579181},
AUTHOR = {Langlands, Robert P.},
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ADDRESS = {New York},
YEAR = {1976},
PAGES = {v+337},
MRCLASS = {22E55 (10C15 10D20 32N05)},
MRNUMBER = {0579181},
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ZBLNUMBER = {0332.10018},
DOI = {1007/BFb0079929},
} -
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@inproceedings{MR1159111,
author = {Langlands, Robert P.},
TITLE = {Rank-one residues of {E}isenstein series},
BOOKTITLE={Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday, Part {\rm II}},
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SERIES={Israel Math. Conf. Proc.},
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MRNUMBER={1159111},
ZBLNUMBER = {0711.11022},
} -
[LSh] E. Lapid, "On Arthur’s asymptotic inner product formula of truncated Eisenstein series," in On Certain ${L}$-Functions: A Volume in Honor of Freydoon Shahidi on the Occasion of his 60 th Birthday, Providence, RI: Amer. Math. Soc., 2011, vol. 13, pp. 309-331.
@incollection{LSh,
author={Lapid, Erez},
TITLE={On {A}rthur's asymptotic inner product formula of truncated {E}isenstein series},
BOOKTITLE={On {C}ertain ${L}$-Functions: A Volume in Honor of Freydoon Shahidi on the Occasion of his 60{\rm th} Birthday},
SERIES={Clay Math. Proc.},
VOLUME={13},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
PAGES={309--331},
YEAR={2011},
} -
[MR2541128]
E. Lapid and W. Müller, "Spectral asymptotics for arithmetic quotients of ${ SL}(n,\Bbb R)/{ SO}(n)$," Duke Math. J., vol. 149, iss. 1, pp. 117-155, 2009.
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AUTHOR = {Lapid, Erez and M{ü}ller, Werner},
TITLE = {Spectral asymptotics for arithmetic quotients of {${\rm SL}(n,\Bbb R)/{\rm SO}(n)$}},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {149},
YEAR = {2009},
NUMBER = {1},
PAGES = {117--155},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11F70 (11F72 22E55)},
MRNUMBER = {2541128},
MRREVIEWER = {{\Dbar}{\cftil{o}} Ng{\d{o}}c Di{\cfudot{e}}p},
DOI = {10.1215/00127094-2009-037},
ZBLNUMBER = {pre05588173},
} -
[MR2053600]
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.
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AUTHOR = {M{ü}ller, W. and Speh, B.},
TITLE = {Absolute convergence of the spectral side of the {A}rthur trace formula for {${\rm GL}\sb n$}},
NOTE = {with an appendix by {E. M. L}apid},
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},
ZBLNUMBER = {1083.11031},
} -
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VOLUME = {130},
YEAR = {1989},
PAGES = {473--529},
MRCLASS = {11F72 (11F70 22E55)},
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W. Müller, "The trace class conjecture in the theory of automorphic forms. II," Geom. Funct. Anal., vol. 8, iss. 2, pp. 315-355, 1998.
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ZBLNUMBER = {1073.11514},
} -
[MR1800065]
W. Müller, "On the singularities of residual intertwining operators," Geom. Funct. Anal., vol. 10, iss. 5, pp. 1118-1170, 2000.
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YEAR = {2000},
NUMBER = {5},
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ZBLNUMBER = {0964.22016},
} -
[MR1935546]
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TITLE = {On the spectral side of the {A}rthur trace formula},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {12},
YEAR = {2002},
NUMBER = {4},
PAGES = {669--722},
ISSN = {1016-443X},
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DOI = {10.1007/s00039-002-8263-7},
ZBLNUMBER = {1169.11313},
} -
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AUTHOR = {Ziegler, G{ü}nter M.},
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VOLUME = {152},
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}