On the generic part of the cohomology of compact unitary Shimura varieties

Abstract

The goal of this paper is to show that the cohomology of compact unitary Shimura varieties is concentrated in the middle degree and torsion-free, after localizing at a maximal ideal of the Hecke algebra satisfying a suitable genericity assumption. Along the way, we establish various foundational results on the geometry of the Hodge-Tate period map. In particular, we compare the fibres of the Hodge-Tate period map with Igusa varieties.

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      author = {Deligne, Pierre},
      title = {Variétés de {S}himura: interprétation modulaire, et techniques de construction de modèles canoniques},
      booktitle = {Automorphic Forms, Representations and {$L$}-Functions, {P}art 2},
      VENUE={{P}roc. {S}ympos. {P}ure {M}ath., {O}regon {S}tate {U}niv., {C}orvallis, {OR},
      1977},
      series = {Proc. Sympos. Pure Math.},
      VOLUME={XXXIII},
      pages = {247--289},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1979},
      mrclass = {10D20 (14D20 14G25 14K15)},
      mrnumber = {0546620},
      mrreviewer = {J. S. Milne},
      zblnumber = {0437.14012},
      }
  • [delignehodge] Go to document P. Deligne, J. S. Milne, A. Ogus, and K. Shih, Hodge Cycles, Motives, and Shimura Varieties, Springer-Verlag, New York, 1982, vol. 900.
    @BOOK{delignehodge,
      author = {Deligne, Pierre and Milne, James S. and Ogus, Arthur and Shih, Kuang-yen},
      title = {Hodge {C}ycles, {M}otives, and {S}himura {V}arieties},
      series = {Lecture Notes in Math.},
      volume = {900},
      publisher = {Springer-Verlag, New York},
      year = {1982},
      pages = {ii+414},
      isbn = {3-540-11174-3},
      mrclass = {14Kxx (10D25 12A67 14A20 14F30 14K22)},
      mrnumber = {0654325},
      zblnumber = {0465.00010},
      doi = {10.1007/978-3-540-38955-2},
     }
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    @ARTICLE{emerton-gee,
      author = {Emerton, Matthew and Gee, Toby},
      title = {{$p$}-adic {H}odge-theoretic properties of étale cohomology with {${\rm mod}\, p$} coefficients, and the cohomology of {S}himura varieties},
      journal = {Algebra Number Theory},
      fjournal = {Algebra \& Number Theory},
      volume = {9},
      year = {2015},
      number = {5},
      pages = {1035--1088},
      issn = {1937-0652},
      mrclass = {11F33 (14Gxx)},
      mrnumber = {3365999},
      doi = {10.2140/ant.2015.9.1035},
      url = {http://dx.doi.org/10.2140/ant.2015.9.1035},
      zblnumber = {1321.11050},
      }
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    @incollection{fargues,
      author = {Fargues, Laurent},
      title = {Quelques résultats et conjectures concernant la courbe},
      booktitle={De la Géométrie Algébrique aux Formes Automorphes (I)},
      NOTE={une collection d'articles en l'honneur du soixantime anniversaire de Gérard Laumon},
      PUBLISHER={Math. Soc. France, Paris},
      SERIES = {Astérisque},
      VOLUME = {369},
      year = {2015},
      pages = {325--374},
      issn = {0303-1179},
      isbn = {978-2-85629-805-3},
      mrclass = {14L05 (14G22)},
      mrnumber = {3379639},
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      }
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    @INCOLLECTION{farguesfontaine,
      author = {Fargues, Laurent and Fontaine, Jean-Marc},
      title = {Vector bundles on curves and {$p$}-adic {H}odge theory},
      booktitle = {Automorphic {F}orms and {G}alois {R}epresentations. {V}ol. 2},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {415},
      pages = {17--104},
      publisher = {Cambridge Univ. Press, Cambridge},
      year = {2014},
      mrclass = {11F80 (11S20)},
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      }
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      author = {Franke, Jens},
      title = {Harmonic analysis in weighted {$L_2$}-spaces},
      journal = {Ann. Sci. École Norm. Sup. (4)},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      volume = {31},
      year = {1998},
      number = {2},
      pages = {181--279},
      issn = {0012-9593},
      mrclass = {11F75 (11F70 22E41)},
      mrnumber = {1603257},
      doi = {10.1016/S0012-9593(98)80015-3},
      url = {http://dx.doi.org/10.1016/S0012-9593(98)80015-3},
      zblnumber = {0938.11026},
      }
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      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {150},
      year = {2014},
      number = {2},
      pages = {191--228},
      issn = {0010-437X},
      mrclass = {11F33 (11F55 11F70)},
      mrnumber = {3177267},
      mrreviewer = {Luis Alberto LomelÂ-},
      doi = {10.1112/S0010437X13007355},
      url = {http://dx.doi.org/10.1112/S0010437X13007355},
      zblnumber = {1309.11038},
      }
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      journal = {Duke Math. J.},
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      year = {2015},
      number = {15},
      pages = {2809--2895},
      issn = {0012-7094},
      mrclass = {14G35 (14L05 20G25)},
      mrnumber = {3430453},
      mrreviewer = {Eva Viehmann},
      doi = {10.1215/00127094-3328137},
      url = {http://dx.doi.org/10.1215/00127094-3328137},
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      author = {Harris, Michael and Taylor, Richard},
      title = {The {G}eometry and {C}ohomology of some {S}imple {S}himura {V}arieties},
      series = {Annals of Mathematics Studies},
      volume = {151},
      note = {With an appendix by Vladimir G. Berkovich},
      publisher = {Princeton University Press, Princeton, NJ},
      year = {2001},
      pages = {viii+276},
      isbn = {0-691-09090-4},
      mrclass = {11G18 (11F70 11S37 14G35 22E45)},
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      year = {1996},
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      isbn = {3-528-06794-2},
      mrclass = {14G22 (14F20)},
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      mrreviewer = {Lorenzo Ramero},
      doi = {10.1007/978-3-663-09991-8},
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      }
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      author = {Illusie, Luc},
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      BOOKTITLE = {Périodes $p$-adiques},
      VENUE={Bures-sur-Yvette, 1988},
      SERIES = {Astérisque},
      VOLUME = {223},
      PUBLISHER={Math. Soc. France, Paris},
      year = {1994},
      pages = {9--57},
      issn = {0303-1179},
      mrclass = {14F20 (11G25 14D07 14G20)},
      mrnumber = {1293970},
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      }
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      journal = {Amer. J. Math.},
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      number = {4},
      pages = {777--815},
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      mrclass = {10D40 (12A67 22E55)},
      mrnumber = {0623137},
      mrreviewer = {Freydoon Shahidi},
      doi = {10.2307/2374050},
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    @INCOLLECTION{katz,
      author = {Katz, Nicholas M.},
      title = {Slope filtration of {$F$}-crystals},
      booktitle = {Journées de {G}éométrie {A}lgébrique de {R}ennes, {V}ol. {I}},
      VENUE={{R}ennes, 1978},
      series = {Astérisque},
      volume = {63},
      pages = {113--163},
      publisher = {Soc. Math. France, Paris},
      year = {1979},
      mrclass = {14F30 (14G20 14L05)},
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      }
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      journal = {J. Amer. Math. Soc.},
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      volume = {30},
      year = {2017},
      number = {3},
      pages = {819--914},
      issn = {0894-0347},
      mrclass = {11G18 (11G10 14G35)},
      mrnumber = {3630089},
      doi = {10.1090/jams/867},
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      mrclass = {11G18 (14G35)},
      mrnumber = {2669706},
      mrreviewer = {Jeffrey D. Achter},
      doi = {10.1090/S0894-0347-10-00667-3},
      url = {http://dx.doi.org/10.1090/S0894-0347-10-00667-3},
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      author = {Kedlaya, Kiran S. and Liu, Ruochuan},
      title = {Relative {$p$}-adic {H}odge Theory: Foundations},
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      VOLUME= {371},
      year = {2015},
      note = {239 pp.},
      issn = {0303-1179},
      isbn = {978-2-85629-807-7},
      mrclass = {14C30 (14G22)},
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      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
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      year = {1984},
      number = {3},
      pages = {611--650},
      issn = {0012-7094},
      mrclass = {11R39 (11F70 11F72 22E55)},
      mrnumber = {0757954},
      mrreviewer = {Jean-Pierre Labesse},
      doi = {10.1215/S0012-7094-84-05129-9},
      url = {http://dx.doi.org/10.1215/S0012-7094-84-05129-9},
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    @ARTICLE{kottwitz,
      author = {Kottwitz, Robert E.},
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      journal = {Compositio Math.},
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      volume = {56},
      year = {1985},
      number = {2},
      pages = {201--220},
      issn = {0010-437X},
      mrclass = {14L25 (14F30 14L15 20G25)},
      mrnumber = {0809866},
      mrreviewer = {K. F. Lai},
      url = {http://www.numdam.org/item?id=CM_1985__56_2_201_0},
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    @INCOLLECTION{kottwitz-lambda-adic,
      author = {Kottwitz, Robert E.},
      title = {Shimura varieties and {$\lambda$}-adic representations},
      booktitle = {Automorphic forms, {S}himura varieties, and {$L$}-functions, {V}ol. {I} ({A}nn {A}rbor, {MI},
      1988)},
      series = {Perspect. Math.},
      volume = {10},
      pages = {161--209},
      publisher = {Academic Press, Boston, MA},
      year = {1990},
      mrclass = {11G18 (11F70 11R39)},
      mrnumber = {1044820},
      zblnumber = {0743.14019},
      }
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    @ARTICLE{Kottwitz-lambda,
      author = {Kottwitz, Robert E.},
      title = {On the {$\lambda$}-adic representations associated to some simple {S}himura varieties},
      journal = {Invent. Math.},
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      volume = {108},
      year = {1992},
      number = {3},
      pages = {653--665},
      issn = {0020-9910},
      mrclass = {11G18 (11F70 11F85 14G35 22E55)},
      mrnumber = {1163241},
      mrreviewer = {Min Ho Lee},
      doi = {10.1007/BF02100620},
      url = {http://dx.doi.org/10.1007/BF02100620},
      zblnumber = {0765.22011},
      }
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      title = {Points on some {S}himura varieties over finite fields},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {5},
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      mrclass = {11G18 (11G25 14G10 14G35)},
      mrnumber = {1124982},
      mrreviewer = {Min Ho Lee},
      doi = {10.2307/2152772},
      url = {http://dx.doi.org/10.2307/2152772},
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      booktitle = {Motives ({S}eattle, {WA},
      1991)},
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      year = {1994},
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      mrclass = {22E46 (19K99 22E41)},
      mrnumber = {1135534},
      mrreviewer = {Jeffrey Fox},
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      mrclass = {10D40 (12A67 22E50)},
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Authors

Ana Caraiani

Mathematisches Institut der Universität Bonn, Bonn, Germany

Current address:

Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom Peter Scholze

Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany