Abstract
This paper contains a proof of a conjecture of Breuil and Schneider on the existence of an invariant norm on any locally algebraic representation of $\mathrm{GL}(n)$, with integral central character, whose smooth part is given by a generalized Steinberg representation. In fact, we prove the analogue for any connected reductive group $G$. This is done by passing to a global setting, using the trace formula for an $\mathbb{R}$-anisotropic model of $G$. The ultimate norm comes from classical $p$-adic modular forms.
-
[AT] E. Artin and J. Tate, Class Field Theory, AMS Chelsea Publishing, Providence, RI, 2009.
@book {AT, MRKEY = {2467155},
AUTHOR = {Artin, Emil and Tate, John},
TITLE = {Class Field Theory},
NOTE = {Reprinted with corrections from the 1967 original},
PUBLISHER = {AMS Chelsea Publishing, Providence, RI},
YEAR = {2009},
PAGES = {viii+194},
ISBN = {978-0-8218-4426-7},
MRCLASS = {11-01 (11R37)},
MRNUMBER = {2467155},
ZBLNUMBER = {1179.11040},
} -
[B] T. Barnet-Lamb, T. Gee, D. Gereghty, and R. Taylor, Local-global compatibility for $l=p$. I, II.
@misc{B,
author={Barnet-Lamb, T. and Gee, T. and Gereghty, D. and Taylor, R.},
TITLE={Local-global compatibility for $l=p$. {I},
{II} },
NOTE={\arxiv{1105.2240}},
ARXIV={1105.2242},
} -
[BH] A. Borel and G. Harder, "Existence of discrete cocompact subgroups of reductive groups over local fields," J. Reine Angew. Math., vol. 298, pp. 53-64, 1978.
@article {BH, MRKEY = {0483367},
AUTHOR = {Borel, A. and Harder, G.},
TITLE = {Existence of discrete cocompact subgroups of reductive groups over local fields},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {298},
YEAR = {1978},
PAGES = {53--64},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {22E50 (22E40)},
MRNUMBER = {0483367},
MRREVIEWER = {Gopal Prasad},
ZBLNUMBER = {0385.14014},
} -
[Br] C. Breuil, "The emerging $p$-adic Langlands programme," in Proceedings of the International Congress of Mathematicians. Volume II, New Delhi, 2010, pp. 203-230.
@inproceedings {Br, MRKEY = {2827792},
AUTHOR = {Breuil, Christophe},
TITLE = {The emerging {$p$}-adic {L}anglands programme},
BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians. {V}olume {II}},
PAGES = {203--230},
PUBLISHER = {Hindustan Book Agency},
ADDRESS = {New Delhi},
YEAR = {2010},
MRCLASS = {22E50 (11F70 11F80)},
MRNUMBER = {2827792},
MRREVIEWER = {Benjamin Schraen},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {05971096},
} -
[BS]
C. Breuil and P. Schneider, "First steps towards $p$-adic Langlands functoriality," J. Reine Angew. Math., vol. 610, pp. 149-180, 2007.@article {BS, MRKEY = {2359853},
AUTHOR = {Breuil, Christophe and Schneider, Peter},
TITLE = {First steps towards {$p$}-adic {L}anglands functoriality},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
VOLUME = {610},
YEAR = {2007},
PAGES = {149--180},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11S37 (22E50)},
MRNUMBER = {2359853},
MRREVIEWER = {Jacques Tilouine},
DOI = {10.1515/CRELLE.2007.070},
ZBLNUMBER = {1180.11036},
} -
[C] L. Clozel, "On limit multiplicities of discrete series representations in spaces of automorphic forms," Invent. Math., vol. 83, iss. 2, pp. 265-284, 1986.
@article {C, MRKEY = {0818353},
AUTHOR = {Clozel, Laurent},
TITLE = {On limit multiplicities of discrete series representations in spaces of automorphic forms},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {83},
YEAR = {1986},
NUMBER = {2},
PAGES = {265--284},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {22E45 (11F70 22E55)},
MRNUMBER = {0818353},
MRREVIEWER = {Marie-France Vign{é}ras},
DOI = {10.1007/BF01388963},
ZBLNUMBER = {0582.22012},
} -
[Ca] A. Caraiani, Monodromy and local-global compatibility for $l=p$.
@misc{Ca,
author={Caraiani, A.},
TITLE={Monodromy and local-global compatibility for $l=p$},
ARXIV={1202.4683},
} -
[CD] L. Clozel and P. Delorme, "Sur le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs réels," C. R. Acad. Sci. Paris Sér. I Math., vol. 300, iss. 11, pp. 331-333, 1985.
@article {CD, MRKEY = {0786912},
AUTHOR = {Clozel, Laurent and Delorme, Patrick},
TITLE = {Sur le théorème de {P}aley-{W}iener invariant pour les groupes de {L}ie réductifs réels},
JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
FJOURNAL = {Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique},
VOLUME = {300},
YEAR = {1985},
NUMBER = {11},
PAGES = {331--333},
ISSN = {0249-6291},
CODEN = {CASMEI},
MRCLASS = {22E30 (43A80)},
MRNUMBER = {0786912},
ZBLNUMBER = {0593.22009},
} -
[Em] 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 {Em, 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 = {2207783},
MRREVIEWER = {Payman L. Kassaei},
DOI = {10.1007/s00222-005-0448-x},
ZBLNUMBER = {1090.22008},
} -
[Hu] Y. Hu, "Normes invariantes et existence de filtrations admissibles," J. Reine Angew. Math., vol. 634, pp. 107-141, 2009.
@article {Hu, MRKEY = {2560407},
AUTHOR = {Hu, Yongquan},
TITLE = {Normes invariantes et existence de filtrations admissibles},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
VOLUME = {634},
YEAR = {2009},
PAGES = {107--141},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {11S37 (11F80 22E50)},
MRNUMBER = {2560407},
MRREVIEWER = {Jeremy T. Teitelbaum},
DOI = {10.1515/CRELLE.2009.070},
ZBLNUMBER = {1214.11132},
} -
[JS] H. Jacquet and J. Shalika, "The Whittaker models of induced representations," Pacific J. Math., vol. 109, iss. 1, pp. 107-120, 1983.
@article {JS, MRKEY = {0716292},
AUTHOR = {Jacquet, Herv{é} and Shalika, Joseph},
TITLE = {The {W}hittaker models of induced representations},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {109},
YEAR = {1983},
NUMBER = {1},
PAGES = {107--120},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {22E50},
MRNUMBER = {0716292},
MRREVIEWER = {Stephen Gelbart},
URL = {http://projecteuclid.org/euclid.pjm/1102720206},
ZBLNUMBER = {0535.22017},
} -
[Ku] S. S. Kudla, "The local Langlands correspondence: the non-Archimedean case," in Motives, Providence, RI: Amer. Math. Soc., 1994, vol. 55, pp. 365-391.
@incollection{Ku, MRKEY = {1265559},
AUTHOR = {Kudla, Stephen S.},
TITLE = {The local {L}anglands correspondence: the non-{A}rchimedean case},
BOOKTITLE = {Motives},
VENUE={{S}eattle, {WA},
1991},
SERIES = {Proc. Sympos. Pure Math.},
VOLUME = {55},
PAGES = {365--391},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1994},
MRCLASS = {11F70 (11R39 11S37 22E50)},
MRNUMBER = {1265559},
MRREVIEWER = {Stephen Gelbart},
ZBLNUMBER = {0811.11072},
} -
[PR] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Boston, MA: Academic Press, 1994, vol. 139.
@book {PR, MRKEY = {1278263},
AUTHOR = {Platonov, Vladimir and Rapinchuk, Andrei},
TITLE = {Algebraic Groups and Number Theory},
SERIES = {Pure and Applied Mathematics},
VOLUME = {139},
NOTE = {translated from the 1991 Russian original by Rachel Rowen},
PUBLISHER = {Academic Press},
ADDRESS = {Boston, MA},
YEAR = {1994},
PAGES = {xii+614},
ISBN = {0-12-558180-7},
MRCLASS = {11E57 (11-02 20Gxx)},
MRNUMBER = {1278263},
ZBLNUMBER = {0841.20046},
} -
[Sc] P. Schneider, Nonarchimedean Functional Analysis, New York: Springer-Verlag, 2002.
@book {Sc, MRKEY = {1869547},
AUTHOR = {Schneider, Peter},
TITLE = {Nonarchimedean Functional Analysis},
SERIES = {Springer Monogr. Math.},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2002},
PAGES = {vi+156},
ISBN = {3-540-42533-0},
MRCLASS = {46S10 (46-01 47S10)},
MRNUMBER = {1869547},
MRREVIEWER = {Dinam{é}rico P. Pombo, Jr.},
ZBLNUMBER = {0998.46044},
} -
[Si] A. J. Silberger, "Isogeny restrictions of irreducible admissible representations are finite direct sums of irreducible admissible representations," Proc. Amer. Math. Soc., vol. 73, iss. 2, pp. 263-264, 1979.
@article {Si, MRKEY = {0516475},
AUTHOR = {Silberger, Allan J.},
TITLE = {Isogeny restrictions of irreducible admissible representations are finite direct sums of irreducible admissible representations},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {73},
YEAR = {1979},
NUMBER = {2},
PAGES = {263--264},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {22E50},
MRNUMBER = {0516475},
MRREVIEWER = {G. I. Ol{\cprime}shanski{\u\i}},
DOI = {10.2307/2042302},
ZBLNUMBER = {0368.22009},
} -
[So] C. Sorensen, Eigenvarieties and invariant norms: Towards $p$-adic Langlands for ${U}(n)$, 2012.
@misc{So,
author={Sorensen, C.},
TITLE={Eigenvarieties and invariant norms: Towards $p$-adic {L}anglands for ${U}(n)$},
YEAR={2012},
ARXIV={1208.0703},
} -
[ST] P. Schneider and J. Teitelbaum, "Banach-Hecke algebras and $p$-adic Galois representations," Doc. Math., iss. Extra Vol., pp. 631-684, 2006.
@article {ST, MRKEY = {2290601},
AUTHOR = {Schneider, Peter and Teitelbaum, J.},
TITLE = {Banach-{H}ecke algebras and {$p$}-adic {G}alois representations},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
YEAR = {2006},
NUMBER = {Extra Vol.},
PAGES = {631--684},
ISSN = {1431-0635},
MRCLASS = {11S37 (11F80 22E50)},
MRNUMBER = {2290601},
MRREVIEWER = {Laurent N. Berger},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1140.11026},
} -
[Ta] M. Tadić, "Notes on representations of non-archimedean ${ SL}(n)$," Pacific J. Math., vol. 152, iss. 2, pp. 375-396, 1992.
@article {Ta, MRKEY = {1141803},
AUTHOR = {Tadi{ć},
Marko},
TITLE = {Notes on representations of non-{a}rchimedean {${\rm SL}(n)$}},
JOURNAL = {Pacific J. Math.},
FJOURNAL = {Pacific Journal of Mathematics},
VOLUME = {152},
YEAR = {1992},
NUMBER = {2},
PAGES = {375--396},
ISSN = {0030-8730},
CODEN = {PJMAAI},
MRCLASS = {22E50},
MRNUMBER = {1141803},
MRREVIEWER = {David Manderscheid},
URL = {http://projecteuclid.org/euclid.pjm/1102636173},
ZBLNUMBER = {0724.22017},
} -
[Vi] M. Vignéras, Représentations $l$-Modulaires d’un Groupe Réductif $p$-adique avec $l\ne p$, Boston, MA: Birkhäuser, 1996, vol. 137.
@book {Vi, MRKEY = {1395151},
AUTHOR = {Vign{é}ras, Marie-France},
TITLE = {Représentations {$l$}-Modulaires d'un Groupe Réductif {$p$}-adique avec {$l\ne p$}},
SERIES = {Progr. Math.},
VOLUME = {137},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston, MA},
YEAR = {1996},
PAGES = {xviii and 233},
ISBN = {0-8176-3929-2},
MRCLASS = {22E35 (20G25 22-02 22E50)},
MRNUMBER = {1395151},
MRREVIEWER = {Chon Hu Cheng},
ZBLNUMBER = {0859.22001},
ZBLNUMBER = {0846.22004},
}
Full Article