Classicité de formes modulaires surconvergentes

Pages 975-1014 from Volume 183 (2016), Issue 3 by Stéphane Bijakowski, Vincent Pilloni, Benoît Stroh

Abstract

Nous généralisons le critère de classicité de formes modulaires surconvergentes sur les courbes modulaires dû à Coleman à certaines variétés de Shimura PEL de type (A) et~(C) associées à des groupes réductifs non ramifiés sur $\mathbb{Q}_p$. Notre démonstration s’inspire de la méthode de prolongement analytique de Buzzard et Kassaei. \vspace*4pt
We generalize Coleman classicity criterion for overconvergent modular forms on modular curves to the case of some PEL Shimura variety of type (A) or (C) associated to a reductive group unramified over $\mathbb{Q}_p$. Our demonstration is inspired by the analytic continuation method of Buzzard and Kassaei.

  • [AIP] Go to document F. Andreatta, A. Iovita, and V. Pilloni, "$p$-adic families of Siegel modular cuspforms," Ann. of Math., vol. 181, iss. 2, pp. 623-697, 2015.
    @article{AIP, mrkey = {3275848},
      author = {Andreatta, Fabrizio and Iovita, Adrian and Pilloni, Vincent},
      title = {{$p$}-adic families of {S}iegel modular cuspforms},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {181},
      year = {2015},
      number = {2},
      pages = {623--697},
      issn = {0003-486X},
      mrclass = {11F46 (11F85)},
      mrnumber = {3275848},
      mrreviewer = {Ellen E. Eischen},
      doi = {10.4007/annals.2015.181.2.5},
      zblnumber = {06399445},
      }
  • [AM] Go to document A. Abbes and A. Mokrane, "Sous-groupes canoniques et cycles évanescents $p$-adiques pour les variétés abéliennes," Publ. Math. Inst. Hautes Études Sci., vol. 99, pp. 117-162, 2004.
    @article{AM, mrkey = {2075884},
      author = {Abbes, Ahmed and Mokrane, Abdellah},
      title = {Sous-groupes canoniques et cycles évanescents {$p$}-adiques pour les variétés abéliennes},
      journal = {Publ. Math. Inst. Hautes Études Sci.},
      fjournal = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      volume = {99},
      year = {2004},
      pages = {117--162},
      issn = {0073-8301},
      mrclass = {14K15 (11S40 14G22 14G35)},
      mrnumber = {2075884},
      mrreviewer = {Alessandra Bertapelle},
      doi = {10.1007/s10240-004-0022-x},
      zblnumber = {1062.14057},
      }
  • [Ber] Go to document P. Berthelot, Cohomologie rigide et cohomologie rigide à support propres, 1996.
    @misc{Ber,
      author = {Berthelot, P.},
      title = {Cohomologie rigide et cohomologie rigide à support propres},
      note = {première partie, prépublication},
      year = {1996},
      url = {https://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf},
      }
  • [Bo] Go to document S. Bosch, "Half a century of rigid analytic spaces," Pure Appl. Math. Q., vol. 5, iss. 4, Special Issue: In honor of John Tate. Part 1, pp. 1435-1467, 2009.
    @article{Bo, mrkey = {2560322},
      author = {Bosch, Siegfried},
      title = {Half a century of rigid analytic spaces},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {5},
      year = {2009},
      number = {4, Special Issue: In honor of John Tate. Part 1},
      pages = {1435--1467},
      issn = {1558-8599},
      mrclass = {14G22},
      mrnumber = {2560322},
      mrreviewer = {Hans Schoutens},
      doi = {10.4310/PAMQ.2009.v5.n4.a9},
      zblnumber = {1256.32023},
      }
  • [BL] Go to document S. Bosch and W. Lütkebohmert, "Formal and rigid geometry. II. Flattening techniques," Math. Ann., vol. 296, iss. 3, pp. 403-429, 1993.
    @article{BL, mrkey = {1225983},
      author = {Bosch, Siegfried and L{ü}tkebohmert, Werner},
      title = {Formal and rigid geometry. {II}. {F}lattening techniques},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {296},
      year = {1993},
      number = {3},
      pages = {403--429},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {11G25 (14A15 14G20 32P05)},
      mrnumber = {1225983},
      doi = {10.1007/BF01445112},
      zblnumber = {0808.14018},
      }
  • [Buz] Go to document K. Buzzard, "Analytic continuation of overconvergent eigenforms," J. Amer. Math. Soc., vol. 16, iss. 1, pp. 29-55, 2003.
    @article{Buz, mrkey = {1937198},
      author = {Buzzard, Kevin},
      title = {Analytic continuation of overconvergent eigenforms},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {16},
      year = {2003},
      number = {1},
      pages = {29--55},
      issn = {0894-0347},
      mrclass = {11F33 (11F11 11F80 11G18)},
      mrnumber = {1937198},
      mrreviewer = {Gebhard B{ö}ckle},
      doi = {10.1090/S0894-0347-02-00405-8},
      zblnumber = {1076.11029},
      }
  • [Col] Go to document R. F. Coleman, "Classical and overconvergent modular forms," Invent. Math., vol. 124, iss. 1-3, pp. 215-241, 1996.
    @article{Col, mrkey = {1369416},
      author = {Coleman, Robert F.},
      title = {Classical and overconvergent modular forms},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {124},
      year = {1996},
      number = {1-3},
      pages = {215--241},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {11F85 (14G20)},
      mrnumber = {1369416},
      mrreviewer = {Fernando Q. Gouv{ê}a},
      doi = {10.1007/s002220050051},
      zblnumber = {0851.11030},
      }
  • [Fa] Go to document L. Fargues, "La filtration de Harder-Narasimhan des schémas en groupes finis et plats," J. Reine Angew. Math., vol. 645, pp. 1-39, 2010.
    @article{Fa, mrkey = {2673421},
      author = {Fargues, Laurent},
      title = {La filtration de {H}arder-{N}arasimhan des schémas en groupes finis et plats},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik. [Crelle's Journal]},
      volume = {645},
      year = {2010},
      pages = {1--39},
      issn = {0075-4102},
      coden = {JRMAA8},
      mrclass = {14L15},
      mrnumber = {2673421},
      mrreviewer = {Alan Koch},
      doi = {10.1515/CRELLE.2010.058},
      zblnumber = {1199.14015},
      }
  • [GrtzEL] Go to document U. Görtz, "On the flatness of models of certain Shimura varieties of PEL-type," Math. Ann., vol. 321, iss. 3, pp. 689-727, 2001.
    @article{GrtzEL, mrkey = {1871975},
      author = {G{ö}rtz, Ulrich},
      title = {On the flatness of models of certain {S}himura varieties of {PEL}-type},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {321},
      year = {2001},
      number = {3},
      pages = {689--727},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {14G35 (11G18)},
      mrnumber = {1871975},
      mrreviewer = {Conjeeveram S. Rajan},
      doi = {10.1007/s002080100250},
      zblnumber = {1073.14526},
      }
  • [GrtzPEL] Go to document U. Görtz, "On the flatness of local models for the symplectic group," Adv. Math., vol. 176, iss. 1, pp. 89-115, 2003.
    @article{GrtzPEL, mrkey = {1978342},
      author = {G{ö}rtz, Ulrich},
      title = {On the flatness of local models for the symplectic group},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {176},
      year = {2003},
      number = {1},
      pages = {89--115},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {14G35 (11G18)},
      mrnumber = {1978342},
      mrreviewer = {Rolf Berndt},
      doi = {10.1016/S0001-8708(02)00062-2},
      zblnumber = {1073.14526},
      }
  • [He] Go to document X. He, "Normality and Cohen-Macaulayness of local models of Shimura varieties," Duke Math. J., vol. 162, iss. 13, pp. 2509-2523, 2013.
    @article{He, mrkey = {3127807},
      author = {He, Xuhua},
      title = {Normality and {C}ohen-{M}acaulayness of local models of {S}himura varieties},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {162},
      year = {2013},
      number = {13},
      pages = {2509--2523},
      issn = {0012-7094},
      mrclass = {14G35 (13H10 14M15 14M27)},
      mrnumber = {3127807},
      mrreviewer = {Marc-Hubert Nicole},
      doi = {10.1215/00127094-2371864},
      zblnumber = {06236002},
      }
  • [Jo] Go to document C. Johansson, "Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C," Math. Ann., vol. 357, iss. 1, pp. 51-88, 2013.
    @article{Jo, mrkey = {3084343},
      author = {Johansson, Christian},
      title = {Classicality for small slope overconvergent automorphic forms on some compact {PEL} {S}himura varieties of type {C}},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {357},
      year = {2013},
      number = {1},
      pages = {51--88},
      issn = {0025-5831},
      mrclass = {11F33 (11G18 14F30 14G35)},
      mrnumber = {3084343},
      mrreviewer = {Alessandra Bertapelle},
      doi = {10.1007/s00208-013-0900-y},
      zblnumber = {06210505},
      }
  • [Kas] Go to document P. L. Kassaei, "A gluing lemma and overconvergent modular forms," Duke Math. J., vol. 132, iss. 3, pp. 509-529, 2006.
    @article{Kas, mrkey = {2219265},
      author = {Kassaei, Payman L.},
      title = {A gluing lemma and overconvergent modular forms},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {132},
      year = {2006},
      number = {3},
      pages = {509--529},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F33 (11G18)},
      mrnumber = {2219265},
      mrreviewer = {Thomas A. Weston},
      doi = {10.1215/S0012-7094-06-13234-9},
      zblnumber = {1112.11020},
      }
  • [Kiehl] Go to document R. Kiehl, "Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie," Invent. Math., vol. 2, pp. 191-214, 1967.
    @article{Kiehl, mrkey = {0210948},
      author = {Kiehl, Reinhardt},
      title = {Der {E}ndlichkeitssatz für eigentliche {A}bbildungen in der nichtarchimedischen {F}unktionentheorie},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {2},
      year = {1967},
      pages = {191--214},
      issn = {0020-9910},
      mrclass = {32.50 (12.70)},
      mrnumber = {0210948},
      mrreviewer = {A. F. Monna},
      doi = {10.1007/BF01425513},
      zblnumber = {0202.20101},
      }
  • [Ko] Go to document R. E. Kottwitz, "Points on some Shimura varieties over finite fields," J. Amer. Math. Soc., vol. 5, iss. 2, pp. 373-444, 1992.
    @article{Ko, mrkey = {1124982},
      author = {Kottwitz, Robert E.},
      title = {Points on some {S}himura varieties over finite fields},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {5},
      year = {1992},
      number = {2},
      pages = {373--444},
      issn = {0894-0347},
      mrclass = {11G18 (11G25 14G10 14G35)},
      mrnumber = {1124982},
      mrreviewer = {Min Ho Lee},
      doi = {10.2307/2152772},
      zblnumber = {0796.14014},
      }
  • [LanBook] Go to document K. Lan, Arithmetic Compactifications of PEL-type Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2013, vol. 36.
    @book{LanBook, mrkey = {3186092},
      author = {Lan, Kai-Wen},
      title = {Arithmetic Compactifications of PEL-type {S}himura Varieties},
      series = {London Math. Soc. Monogr. Ser.},
      volume = {36},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {2013},
      pages = {xxvi+561},
      isbn = {978-0-691-15654-5},
      mrclass = {14G35 (11G18 14D23 14M27)},
      mrnumber = {3186092},
      mrreviewer = {Rolf Berndt},
      doi = {10.1515/9781400846016},
      zblnumber = {1284.14004},
      }
  • [LanKoecher] K. Lan, Higher Koecher’s principle.
    @misc{LanKoecher,
      author = {Lan, Kai-Wen},
      title = {Higher {K}oecher's principle},
      note = {{\it Math. Res. Lett.},
      prépublication},
      sortyear={2017},
      }
  • [LanRam] Go to document K. Lan, "Compactifications of PEL-type Shimura varieties in ramified characteristics," Forum Math. Sigma, vol. 4, p. 98, 2016.
    @article{LanRam,
      author = {Lan, Kai-Wen},
      title = {Compactifications of {PEL}-type {S}himura varieties in ramified characteristics},
      journal = {Forum Math. Sigma},
      fjournal = {Forum of Mathematics. Sigma},
      volume = {4},
      year = {2016},
      pages = {98 pp.},
      doi = {10.1017/fms.2015.31},
      }
  • [Lan] K. Lan, Integral models of toroidal compactifications with projective cone decompositions, 2015.
    @misc{Lan,
      author = {Lan, Kai-Wen},
      title = {Integral models of toroidal compactifications with projective cone decompositions},
      note = {prépublication},
      year = {2015},
      }
  • [PiPro] Go to document V. Pilloni, "Prolongement analytique sur les variétés de Siegel," Duke Math. J., vol. 157, iss. 1, pp. 167-222, 2011.
    @article{PiPro, mrkey = {2783930},
      author = {Pilloni, Vincent},
      title = {Prolongement analytique sur les variétés de {S}iegel},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {157},
      year = {2011},
      number = {1},
      pages = {167--222},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F46 (11G18 14G22 14G35)},
      mrnumber = {2783930},
      mrreviewer = {Fabrizio Andreatta},
      doi = {10.1215/00127094-2011-004},
      zblnumber = {1315.11033},
     }
  • [PSHilb] V. Pilloni and B. Stroh, Surconvergence et classicité : le cas Hilbert, 2011.
    @misc{PSHilb,
      author = {Pilloni, Vincent and Stroh, B.},
      title = {Surconvergence et classicité : le cas {H}ilbert},
      note = {prépublication},
      year = {2011},
      }
  • [Sa] Go to document S. Sasaki, "Analytic continuation of overconvergent Hilbert eigenforms in the totally split case," Compos. Math., vol. 146, iss. 3, pp. 541-560, 2010.
    @article{Sa, mrkey = {2644926},
      author = {Sasaki, Shu},
      title = {Analytic continuation of overconvergent {H}ilbert eigenforms in the totally split case},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {146},
      year = {2010},
      number = {3},
      pages = {541--560},
      issn = {0010-437X},
      mrclass = {11F33 (11F41 11G18 14G35)},
      mrnumber = {2644926},
      mrreviewer = {Fabrizio Andreatta},
      doi = {10.1112/S0010437X09004552},
      zblnumber = {1206.11058},
      }
  • [SGA1] Go to document Revêtements Étales et Groupe Fondamental, Grothendieck, A., Ed., New York: Springer-Verlag, 1971, vol. 224.
    @book{SGA1, mrkey = {0354651},
      title = {Revêtements Étales et Groupe Fondamental},
      note = {S{é}minaire de G{é}om{é}trie Alg{é}brique du Bois Marie 1960--1961 (SGA 1), augment{é} de deux expos{é}s de M. Raynaud},
      editor = {Grothendieck, Alexandre},
      series = {Lecture Notes in Math.},
      volume = {224},
      publisher = {Springer-Verlag},
      year = {1971},
      pages = {xxii+447},
      mrclass = {14-06 (14E20)},
      mrnumber = {0354651},
      address = {New York},
      zblnumber = {0234.14002},
      doi = {10.1007/BFb0058656},
      }
  • [St] Go to document B. Stroh, "Compactification de variétés de Siegel aux places de mauvaise réduction," Bull. Soc. Math. France, vol. 138, iss. 2, pp. 259-315, 2010.
    @article{St, mrkey = {2679041},
      author = {Stroh, Beno{\^ı}t},
      title = {Compactification de variétés de {S}iegel aux places de mauvaise réduction},
      journal = {Bull. Soc. Math. France},
      fjournal = {Bulletin de la Société Mathématique de France},
      volume = {138},
      year = {2010},
      number = {2},
      pages = {259--315},
      issn = {0037-9484},
      coden = {BSMFAA},
      mrclass = {14K10 (14G35 14M27)},
      mrnumber = {2679041},
      mrreviewer = {Rolf Berndt},
      zblnumber = {1203.14048},
      url = {http://smf4.emath.fr/Publications/Bulletin/138/html/smf_bull_138_259-315.php},
      }
  • [HilbertTian] Go to document Y. Tian, "Classicality of overconvergent Hilbert modular forms: case of quadratic inert degree," Rend. Semin. Mat. Univ. Padova, vol. 132, pp. 133-229, 2014.
    @article{HilbertTian,
      author={Tian, Y.},
      TITLE={Classicality of overconvergent {H}ilbert modular forms: case of quadratic inert degree},
      journal = {Rend. Semin. Mat. Univ. Padova},
      fjournal = {Rendiconti del Seminario Matematico della Università di Padova},
      volume = {132},
      year = {2014},
      pages = {133--229},
      mrnumber = {3276831},
      zblnumber = {1311.11036},
      doi = {10.4171/RSMUP/132-10},
      }
  • [T-X] Y. Tian and L. Xiao, $p$-adic cohomology and classicality of overconvergent Hilbert modular forms, 2013.
    @misc{T-X,
      author={Tian, Y. and Xiao, L.},
      TITLE={$p$-adic cohomology and classicality of overconvergent {H}ilbert modular forms},
      ARXIV = {1308.0779},
      NOTE={{\em Astérisque},
      to appear},
      YEAR={2013},
     }
  • [Wed] Go to document T. Wedhorn, "Ordinariness in good reductions of Shimura varieties of PEL-type," Ann. Sci. École Norm. Sup., vol. 32, iss. 5, pp. 575-618, 1999.
    @article{Wed, mrkey = {1710754},
      author = {Wedhorn, Torsten},
      title = {Ordinariness in good reductions of {S}himura varieties of {PEL}-type},
      journal = {Ann. Sci. École Norm. Sup.},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      volume = {32},
      year = {1999},
      number = {5},
      pages = {575--618},
      issn = {0012-9593},
      coden = {ASENAH},
      mrclass = {11G18 (14G35 14L05)},
      mrnumber = {1710754},
      mrreviewer = {Ben Moonen},
      doi = {10.1016/S0012-9593(01)80001-X},
      zblnumber = {0983.14024},
      }

Keywords
, , , ,
Mathematical Subject Classification
Primary: , , , , ,
DOI
https://doi.org/10.4007/annals.2016.183.3.5
MR
3488741
zbMATH
06589412
Milestones
Received: 16 September 2014
Revised: 29 June 2015
Accepted: 30 September 2015
Published online: 19 April 2016

Authors

Stéphane Bijakowski

Université Paris 13, LAGA, Villetaneuse, France

Current address:

Imperial College, London, UK Vincent Pilloni

CNRS, ENS Lyon, UMPA, Lyon, France

Benoît Stroh

CNRS, Université Paris 13, LAGA, Villetaneuse, France