Bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms

Abstract

Let $G_\infty$ be a semisimple real Lie group with unitary dual $\widehat{G}_{\infty}$. We produce new upper bounds for the multiplicities with which representations $\pi \in \widehat{G}_{\infty}$ of cohomological type appear in certain spaces of cusp forms on $G_\infty$. The main new idea is to apply noncommutative Iwasawa theory to certain $p$-adic completions of the cohomology of locally symmetric spaces.

  • [BE] Go to document N. Boston and J. S. Ellenberg, "Pro-$p$ groups and towers of rational homology spheres," Geom. Topol., vol. 10, pp. 331-334, 2006.
    @article{BE, MRKEY = {2224459},
      AUTHOR = {Boston, Nigel and Ellenberg, Jordan S.},
      TITLE = {Pro-{$p$} groups and towers of rational homology spheres},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {10},
      YEAR = {2006},
      PAGES = {331--334},
      ISSN = {1465-3060},
      MRCLASS = {20E18 (22E40 57M27)},
      MRNUMBER = {2007f:20052},
      DOI = {10.2140/gt.2006.10.331},
      MRREVIEWER = {Martyn Quick},
      ZBLNUMBER = {1108.57002},
      }
  • [BW] A. Borel and N. R. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Princeton, NJ: Princeton Univ. Press, 1980.
    @book{BW, MRKEY = {554917},
      AUTHOR = {Borel, Armand and Wallach, Nolan R.},
      TITLE = {Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups},
      SERIES = {Ann. of Math. Studies},
      NUMBER = {94},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1980},
      PAGES = {xvii+388},
      ISBN = {0-691-08248-0; 0-691-08249-9},
      MRCLASS = {22E41 (22E40 22E45 57T15)},
      MRNUMBER = {83c:22018},
      ZBLNUMBER = {0443.22010},
      MRREVIEWER = {Roger Howe},
      }
  • [CD] Go to document F. Calegari and N. M. Dunfield, "Automorphic forms and rational homology 3-spheres," Geom. Topol., vol. 10, pp. 295-329, 2006.
    @article{CD, MRKEY = {2224458},
      AUTHOR = {Calegari, Frank and Dunfield, Nathan M.},
      TITLE = {Automorphic forms and rational homology 3-spheres},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {10},
      YEAR = {2006},
      PAGES = {295--329},
      ISSN = {1465-3060},
      MRCLASS = {57M27 (11F75 11F80)},
      MRNUMBER = {2007h:57013},
      DOI = {10.2140/gt.2006.10.295},
      ZBLNUMBER = {1103.57007},
      MRREVIEWER = {Alexander Zvonkin},
      }
  • [CSS] Go to document J. Coates, P. Schneider, and R. Sujatha, "Modules over Iwasawa algebras," J. Inst. Math. Jussieu, vol. 2, iss. 1, pp. 73-108, 2003.
    @article{CSS, MRKEY = {1955208},
      AUTHOR = {Coates, J. and Schneider, P. and Sujatha, R.},
      TITLE = {Modules over {I}wasawa algebras},
      JOURNAL = {J. Inst. Math. Jussieu},
      FJOURNAL = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institute de Mathématiques de Jussieu},
      VOLUME = {2},
      YEAR = {2003},
      NUMBER = {1},
      PAGES = {73--108},
      ISSN = {1474-7480},
      MRCLASS = {11R23 (11G05 16S99 22E50)},
      MRNUMBER = {2004b:11152},
      DOI = {10.1017/S1474748003000045},
      ZBLNUMBER = {1061.11060},
      MRREVIEWER = {Francesc Bars},
      }
  • [DW] Go to document D. L. de George and N. R. Wallach, "Limit formulas for multiplicities in $L^{2}(\Gamma \backslash G)$," Ann. of Math., vol. 107, iss. 1, pp. 133-150, 1978.
    @article{DW, MRKEY = {0492077},
      AUTHOR = {de George, David L. and Wallach, Nolan R.},
      TITLE = {Limit formulas for multiplicities in {$L\sp{2}(\Gamma \backslash G)$}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {107},
      YEAR = {1978},
      NUMBER = {1},
      PAGES = {133--150},
      ISSN = {0003-486X},
      MRCLASS = {22E45 (58G99)},
      MRNUMBER = {58 \#11231},
      DOI = {10.2307/1971140},
      MRREVIEWER = {J. J. Duistermaat},
      ZBLNUMBER = {0397.22007},
      }
  • [Emerton] Go to document M. Emerton, "On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms," Invent. Math., vol. 164, iss. 1, pp. 1-84, 2006.
    @article{Emerton, MRKEY = {2207783},
      AUTHOR = {Emerton, Matthew},
      TITLE = {On the interpolation of systems of eigenvalues attached to automorphic {H}ecke eigenforms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {164},
      YEAR = {2006},
      NUMBER = {1},
      PAGES = {1--84},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E55 (11F70 11F75 11F85)},
      MRNUMBER = {2007k:22018},
      DOI = {10.1007/s00222-005-0448-x},
      ZBLNUMBER = {1090.22008},
      MRREVIEWER = {Payman L. Kassaei},
      }
  • [Fr] Go to document J. Franke, "Harmonic analysis in weighted $L_2$-spaces," Ann. Sci. École Norm. Sup., vol. 31, iss. 2, pp. 181-279, 1998.
    @article{Fr, MRKEY = {1603257},
      AUTHOR = {Franke, Jens},
      TITLE = {Harmonic analysis in weighted {$L\sb 2$}-spaces},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {31},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {181--279},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {11F75 (11F70 22E41)},
      MRNUMBER = {2000f:11065},
      DOI = {10.1016/S0012-9593(98)80015-3},
      ZBLNUMBER = {0938.11026},
      }
  • [FrS] Go to document J. Franke and J. Schwermer, "A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups," Math. Ann., vol. 311, iss. 4, pp. 765-790, 1998.
    @article{FrS, MRKEY = {1637980},
      AUTHOR = {Franke, Jens and Schwermer, Joachim},
      TITLE = {A decomposition of spaces of automorphic forms, and the {E}isenstein cohomology of arithmetic groups},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {311},
      YEAR = {1998},
      NUMBER = {4},
      PAGES = {765--790},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11F75 (11F55)},
      MRNUMBER = {99k:11077},
      DOI = {10.1007/s002080050208},
      ZBLNUMBER = {0924.11042},
      MRREVIEWER = {Chandrashekhar Khare},
      }
  • [Harrisorig] Go to document M. Harris, "$p$-adic representations arising from descent on abelian varieties," Compositio Math., vol. 39, iss. 2, pp. 177-245, 1979.
    @article{Harrisorig, MRKEY = {546966},
      AUTHOR = {Harris, Michael},
      TITLE = {{$p$}-adic representations arising from descent on abelian varieties},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {39},
      YEAR = {1979},
      NUMBER = {2},
      PAGES = {177--245},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14K15 (14G25)},
      MRNUMBER = {80j:14035},
      URL = {http://www.numdam.org/item?id=CM_1979__39_2_177_0},
      ZBLNUMBER = {0417.14034},
      MRREVIEWER = {J. S. Milne},
      }
  • [Harris] M. Harris, "Correction to the article \citeHarrisorig," Compositio Math., vol. 121, iss. 1, pp. 105-108, 2000.
    @article{Harris, MRKEY = {MR1753112},
      AUTHOR = {Harris, Michael},
      TITLE = {Correction to the article \cite{Harrisorig}},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {121},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {105--108},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {11G10 (14G25 14K15)},
      MRNUMBER = {2001b:11050},
      ZBLNUMBER={1060.14524},
      }
  • [L] Go to document M. Lazard, "Sur les groupes nilpotents et les anneaux de Lie," Ann. Sci. Ecole Norm. Sup., vol. 71, pp. 101-190, 1954.
    @article{L, MRKEY = {0088496},
      AUTHOR = {Lazard, Michel},
      TITLE = {Sur les groupes nilpotents et les anneaux de {L}ie},
      JOURNAL = {Ann. Sci. Ecole Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Troisième Série},
      VOLUME = {71},
      YEAR = {1954},
      PAGES = {101--190},
      ISSN = {0012-9593},
      MRCLASS = {20.0X},
      MRNUMBER = {19,529b},
      URL = {http://www.numdam.org/item?id=ASENS_1954_3_71_2_101_0},
      ZBLNUMBER = {0055.25103},
      MRREVIEWER = {B. H. Neumann},
      }
  • [RS] Go to document J. Rohlfs and B. Speh, "On limit multiplicities of representations with cohomology in the cuspidal spectrum," Duke Math. J., vol. 55, iss. 1, pp. 199-211, 1987.
    @article{RS, MRKEY = {883670},
      AUTHOR = {Rohlfs, J{ü}rgen and Speh, Birgit},
      TITLE = {On limit multiplicities of representations with cohomology in the cuspidal spectrum},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {55},
      YEAR = {1987},
      NUMBER = {1},
      PAGES = {199--211},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E46 (22E70)},
      MRNUMBER = {88k:22010},
      DOI = {10.1215/S0012-7094-87-05511-6},
      ZBLNUMBER = {0626.22008},
      MRREVIEWER = {Roberto J. Miatello},
      }
  • [SX] Go to document P. Sarnak and X. X. Xue, "Bounds for multiplicities of automorphic representations," Duke Math. J., vol. 64, iss. 1, pp. 207-227, 1991.
    @article{SX, MRKEY = {1131400},
      AUTHOR = {Sarnak, Peter and Xue, Xiao Xi},
      TITLE = {Bounds for multiplicities of automorphic representations},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {64},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {207--227},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {22E45 (11F70 11F72 22E40)},
      MRNUMBER = {92h:22026},
      DOI = {10.1215/S0012-7094-91-06410-0},
      ZBLNUMBER = {0741.22010},
      MRREVIEWER = {David Joyner},
      }
  • [Savin] Go to document G. Savin, "Limit multiplicities of cusp forms," Invent. Math., vol. 95, iss. 1, pp. 149-159, 1989.
    @article{Savin, MRKEY = {969416},
      AUTHOR = {Savin, Gordan},
      TITLE = {Limit multiplicities of cusp forms},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {95},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {149--159},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {22E40 (11F70 22E45)},
      MRNUMBER = {90c:22035},
      DOI = {10.1007/BF01394147},
      ZBLNUMBER = {0673.22003},
      MRREVIEWER = {Roberto J. Miatello},
      }
  • [ST] Go to document P. Schneider and J. Teitelbaum, "Banach space representations and Iwasawa theory," Israel J. Math., vol. 127, pp. 359-380, 2002.
    @article{ST, MRKEY = {1900706},
      AUTHOR = {Schneider, P. and Teitelbaum, J.},
      TITLE = {Banach space representations and {I}wasawa theory},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {127},
      YEAR = {2002},
      PAGES = {359--380},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {22E50},
      MRNUMBER = {2003c:22026},
      DOI = {10.1007/BF02784538},
      ZBLNUMBER = {1006.46053},
      MRREVIEWER = {Dubravka Ban},
      }

Authors

Frank Calegari

Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208-2730
United States

Matthew Emerton

Department of Mathematics
Northwestern University
2033 Sheridan Road
Evanston, IL 60208-2730
United States