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

Pages 649-766 from Volume 186 (2017), Issue 3 by Ana Caraiani, Peter Scholze

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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    @INCOLLECTION{deligne-varietes,
      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},
      }
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    @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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      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},
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      }
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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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      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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      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},
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      mrclass = {14F20 (11G25 14D07 14G20)},
      mrnumber = {1293970},
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      journal = {Amer. J. Math.},
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      pages = {777--815},
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      mrclass = {10D40 (12A67 22E55)},
      mrnumber = {0623137},
      mrreviewer = {Freydoon Shahidi},
      doi = {10.2307/2374050},
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      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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      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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      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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      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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      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
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      mrclass = {11G18 (11G25 14G10 14G35)},
      mrnumber = {1124982},
      mrreviewer = {Min Ho Lee},
      doi = {10.2307/2152772},
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      mrclass = {22E46 (19K99 22E41)},
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Keywords
, , ,
Mathematical Subject Classification
Primary: ,
DOI
https://doi.org/10.4007/annals.2017.186.3.1
MR
3702677
Zbl
06804005
Milestones
Received: 19 November 2015
Accepted: 5 April 2017
Published online: 12 September 2017

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