On torsion in the cohomology of locally symmetric varieties

Abstract

The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\mathrm{GL}_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic $0$ cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\mathrm{GL}_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties.
Thus, we prove new results on the $p$-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at $p$, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from $p$ (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

  • [StacksProject] Go to document The Stacks Project.
    @misc{StacksProject, title = {The {S}tacks {P}roject},
      url = {http://stacks.math.columbia.edu},
      key = {St},
      }
  • [AbbesMokrane] 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{AbbesMokrane, 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},
      }
  • [AndreattaGasbarri] F. Andreatta and C. Gasbarri, "The canonical subgroup for families of abelian varieties," Compos. Math., vol. 143, iss. 3, pp. 566-602, 2007.
    @article{AndreattaGasbarri, mrkey = {2330440},
      author = {Andreatta, F. and Gasbarri, C.},
      title = {The canonical subgroup for families of abelian varieties},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {143},
      year = {2007},
      number = {3},
      pages = {566--602},
      issn = {0010-437X},
      mrclass = {11G10 (11F33 14K10)},
      mrnumber = {2330440},
      mrreviewer = {Alessandra Bertapelle},
      zblnumber = {1219.11087},
      }
  • [ArthurBook] J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Providence, RI: Amer. Math. Soc., 2013.
    @book{ArthurBook, mrkey = {3135650},
      author = {Arthur, James},
      title = {The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups},
      series = {Amer. Math. Soc. Colloq. Publ.},
      number = {61},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2013},
      pages = {xviii+590},
      isbn = {978-0-8218-4990-3},
      mrclass = {22E55 (11F66 11R37 20G25 22E50)},
      mrnumber = {3135650},
      zblnumber = {06231010},
      }
  • [Ash1] Go to document A. Ash, "Galois representations and cohomology of ${ GL}(n,\Bbb Z)$," in Séminaire de Théorie des Nombres, Paris, 1989–90, Boston: Birkhäuser, 1992, vol. 102, pp. 9-22.
    @incollection{Ash1, mrkey = {1476725},
      author = {Ash, Avner},
      title = {Galois representations and cohomology of {${\rm GL}(n,\bold Z)$}},
      booktitle = {Séminaire de {T}héorie des {N}ombres, {P}aris, 1989--90},
      series = {Progr. Math.},
      volume = {102},
      pages = {9--22},
      publisher = {Birkhäuser},
      address = {Boston},
      year = {1992},
      mrclass = {11F80 (11F70 11F75 11R39)},
      mrnumber = {1476725},
      mrreviewer = {Ian Kiming},
      doi = {10.1007/978-1-4757-4269-5_2},
      zblnumber = {0754.11016},
     }
  • [Ash2] Go to document A. Ash, "Galois representations attached to mod $p$ cohomology of ${ GL}(n,{\bf Z})$," Duke Math. J., vol. 65, iss. 2, pp. 235-255, 1992.
    @article{Ash2, mrkey = {1150586},
      author = {Ash, Avner},
      title = {Galois representations attached to mod {$p$} cohomology of {${\rm GL}(n,{\bf Z})$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {65},
      year = {1992},
      number = {2},
      pages = {235--255},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F80 (11F33 11F75 11R39)},
      mrnumber = {1150586},
      mrreviewer = {Dipendra Prasad},
      doi = {10.1215/S0012-7094-92-06510-0},
      zblnumber = {0774.11024},
      }
  • [ADP] Go to document A. Ash, D. Doud, and D. Pollack, "Galois representations with conjectural connections to arithmetic cohomology," Duke Math. J., vol. 112, iss. 3, pp. 521-579, 2002.
    @article{ADP, mrkey = {1896473},
      author = {Ash, Avner and Doud, Darrin and Pollack, David},
      title = {Galois representations with conjectural connections to arithmetic cohomology},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {112},
      year = {2002},
      number = {3},
      pages = {521--579},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F75 (11F60 11F80 11R39)},
      mrnumber = {1896473},
      mrreviewer = {Gebhard B{ö}ckle},
      doi = {10.1215/S0012-9074-02-11235-6},
      zblnumber = {1023.11025},
     }
  • [AS] Go to document A. Ash and W. Sinnott, "An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod $p$ cohomology of ${ GL}(n,{\bf Z})$," Duke Math. J., vol. 105, iss. 1, pp. 1-24, 2000.
    @article{AS, mrkey = {1788040},
      author = {Ash, Avner and Sinnott, Warren},
      title = {An analogue of {S}erre's conjecture for {G}alois representations and {H}ecke eigenclasses in the mod {$p$} cohomology of {${\rm GL}(n,{\bf Z})$}},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {105},
      year = {2000},
      number = {1},
      pages = {1--24},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F80 (11F60 11F75 11R39)},
      mrnumber = {1788040},
      mrreviewer = {Gebhard B{ö}ckle},
      doi = {10.1215/S0012-7094-00-10511-X},
      zblnumber = {1015.11018},
      }
  • [BailyBorel] Go to document W. L. Baily Jr. and A. Borel, "Compactification of arithmetic quotients of bounded symmetric domains," Ann. of Math., vol. 84, pp. 442-528, 1966.
    @article{BailyBorel, mrkey = {0216035},
      author = {Baily, Jr., W. L. and Borel, A.},
      title = {Compactification of arithmetic quotients of bounded symmetric domains},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {84},
      year = {1966},
      pages = {442--528},
      issn = {0003-486X},
      mrclass = {32.65},
      mrnumber = {0216035},
      mrreviewer = {A. Kor{á}nyi},
      doi = {10.2307/1970457},
      zblnumber = {0154.08602},
      }
  • [BLGHT] Go to document T. Barnet-Lamb, D. Geraghty, M. Harris, and R. Taylor, "A family of Calabi-Yau varieties and potential automorphy II," Publ. Res. Inst. Math. Sci., vol. 47, iss. 1, pp. 29-98, 2011.
    @article{BLGHT, mrkey = {2827723},
      author = {Barnet-Lamb, Tom and Geraghty, David and Harris, Michael and Taylor, Richard},
      title = {A family of {C}alabi-{Y}au varieties and potential automorphy {II}},
      journal = {Publ. Res. Inst. Math. Sci.},
      fjournal = {Publications of the Research Institute for Mathematical Sciences},
      volume = {47},
      year = {2011},
      number = {1},
      pages = {29--98},
      issn = {0034-5318},
      mrclass = {11F80 (11F11 11G18 14J32)},
      mrnumber = {2827723},
      mrreviewer = {Neil P. Dummigan},
      doi = {10.2977/PRIMS/31},
      zblnumber = {1264.11044},
      }
  • [BartenwerferHebbarkeit] Go to document W. Bartenwerfer, "Der erste Riemannsche Hebbarkeitssatz im nichtarchimedischen Fall," J. Reine Angew. Math., vol. 286/287, pp. 144-163, 1976.
    @article{BartenwerferHebbarkeit, mrkey = {0422680},
      author = {Bartenwerfer, Wolfgang},
      title = {Der erste {R}iemannsche {H}ebbarkeitssatz im nichtarchimedischen {F}all},
      journal = {J. Reine Angew. Math.},
      fjournal = {Journal für die Reine und Angewandte Mathematik},
      volume = {286/287},
      year = {1976},
      pages = {144--163},
      issn = {0075-4102},
      mrclass = {32D20 (32K05)},
      mrnumber = {0422680},
      mrreviewer = {C. Banica},
      doi = {10.1515/crll.1976.286-287.144},
      zblnumber = {0332.32019},
      }
  • [BergeronVenkatesh] Go to document N. Bergeron and A. Venkatesh, "The asymptotic growth of torsion homology for arithmetic groups," J. Inst. Math. Jussieu, vol. 12, iss. 2, pp. 391-447, 2013.
    @article{BergeronVenkatesh, mrkey = {3028790},
      author = {Bergeron, Nicolas and Venkatesh, Akshay},
      title = {The asymptotic growth of torsion homology for arithmetic groups},
      journal = {J. Inst. Math. Jussieu},
      fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      volume = {12},
      year = {2013},
      number = {2},
      pages = {391--447},
      issn = {1474-7480},
      mrclass = {57M25 (22E47)},
      mrnumber = {3028790},
      mrreviewer = {Paul E. Gunnells},
      doi = {10.1017/S1474748012000667},
      zblnumber = {1266.22013},
      }
  • [BorelSerre] Go to document A. Borel and J. -P. Serre, "Corners and arithmetic groups," Comment. Math. Helv., vol. 48, pp. 436-491, 1973.
    @article{BorelSerre, mrkey = {0387495},
      author = {Borel, A. and Serre, J.-P.},
      title = {Corners and arithmetic groups},
      note = {Avec un appendice: {A}rrondissement des vari{é}t{é}s {à} coins, par {A. D}ouady et {L. H}{é}rault},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici},
      volume = {48},
      year = {1973},
      pages = {436--491},
      issn = {0010-2571},
      mrclass = {22E40},
      mrnumber = {0387495},
      mrreviewer = {M. S. Raghunathan},
      doi = {10.1007/BF02566134},
      zblnumber = {0274.22011},
      }
  • [BoschGuentzerRemmert] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Berlin: Springer-Verlag, 1984, vol. 261.
    @book{BoschGuentzerRemmert, mrkey = {0746961},
      author = {Bosch, S. and G{ü}ntzer, U. and Remmert, R.},
      title = {Non-{A}rchimedean Analysis. A Systematic Approach to Rigid Analytic Geometry},
      series = {Grundlehren Math. Wissen.},
      volume = {261},
      mrnumber = {0746961},
      address = {Berlin},
      publisher = {Springer-Verlag},
      year = {1984},
      zblnumber = {0539.14017},
      }
  • [BoschLuetkebohmert1] Go to document S. Bosch and W. Lütkebohmert, "Formal and rigid geometry. I. Rigid spaces," Math. Ann., vol. 295, iss. 2, pp. 291-317, 1993.
    @article{BoschLuetkebohmert1, mrkey = {1202394},
      author = {Bosch, Siegfried and L{ü}tkebohmert, Werner},
      title = {Formal and rigid geometry. {I}. {R}igid spaces},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {295},
      year = {1993},
      number = {2},
      pages = {291--317},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {11G25 (14D15 14G20)},
      mrnumber = {1202394},
      mrreviewer = {Qing Liu},
      doi = {10.1007/BF01444889},
      zblnumber = {0808.14017},
     }
  • [BuzzardGee] K. Buzzard and T. Gee, The conjectural connections between automorphic representations and Galois representations.
    @misc{BuzzardGee,
      author = {Buzzard, K. and Gee, T.},
      title = {The conjectural connections between automorphic representations and {G}alois representations},
      arxiv = {1009.0785},
      }
  • [CalegariEmerton] F. Calegari and M. Emerton, "Completed cohomology—a survey," in Non-Abelian Fundamental Groups and Iwasawa Theory, Cambridge: Cambridge Univ. Press, 2012, vol. 393, pp. 239-257.
    @incollection{CalegariEmerton, mrkey = {2905536},
      author = {Calegari, Frank and Emerton, Matthew},
      title = {Completed cohomology---a survey},
      booktitle = {Non-Abelian Fundamental Groups and {I}wasawa Theory},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {393},
      pages = {239--257},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {2012},
      mrclass = {11F85 (11F70 11F75)},
      mrnumber = {2905536},
      mrreviewer = {Neven Grbac},
      zblnumber = {1288.11056},
      }
  • [CalegariGeragthy] Go to document F. Calegari and D. Geraghty, Modularity lifting theorems beyond the Taylor-Wiles method, 2012.
    @misc{CalegariGeragthy,
      author = {Calegari, F. and Geraghty, D.},
      title = {Modularity lifting theorems beyond the {T}aylor-{W}iles method},
      url = {https://www2.bc.edu/david-geraghty/files/merge.pdf},
      year = {2012},
      }
  • [CaraianiLeHung] A. Caraiani and B. V. LeHung, On the image of complex conjugation in certain Galois representations, 2014.
    @misc{CaraianiLeHung,
      author = {Caraiani, A. and LeHung, B. V.},
      title = {On the image of complex conjugation in certain {G}alois representations},
      year = {2014},
      arxiv = {1409.2158},
      }
  • [ChenevierDet] G. Chenevier, "The $p$-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings," in Automorphic Forms and Galois Representations: Volume 1, Cambridge: Cambridge University Press, 2014, vol. 414.
    @incollection{ChenevierDet,
      author = {Chenevier, Ga{ë}tan},
      title = {The $p$-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings},
      BOOKTITLE = {Automorphic Forms and Galois Representations: Volume 1},
      EDITORS = {Fred Diamond, Payman L. Kassaei, and Minhyong Kim},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {414},
      YEAR = {2014},
      PUBLISHER = {Cambridge University Press},
      ADDRESS = {Cambridge},
     }
  • [ChenevierHarris] Go to document G. Chenevier and M. Harris, "Construction of automorphic Galois representations, II," Camb. J. Math., vol. 1, iss. 1, pp. 53-73, 2013.
    @article{ChenevierHarris, mrkey = {3272052},
      author = {Chenevier, Ga{ë}tan and Harris, Michael},
      title = {Construction of automorphic {G}alois representations, {II}},
      journal = {Camb. J. Math.},
      fjournal = {Cambridge Journal of Mathematics},
      volume = {1},
      year = {2013},
      number = {1},
      pages = {53--73},
      issn = {2168-0930},
      mrclass = {11F80},
      mrnumber = {3272052},
      mrreviewer = {Krzysztof Klosin},
      doi = {10.4310/CJM.2013.v1.n1.a2},
      zblnumber = {06266535},
      }
  • [ClozelMotifs] L. Clozel, "Motifs et formes automorphes: applications du principe de fonctorialité," in Automorphic Forms, Shimura Varieties, and $L$-Functions, Vol. I, Boston: Academic Press, 1990, vol. 10, pp. 77-159.
    @incollection{ClozelMotifs, mrkey = {1044819},
      author = {Clozel, Laurent},
      title = {Motifs et formes automorphes: applications du principe de fonctorialité},
      booktitle = {Automorphic Forms, {S}himura Varieties, and {$L$}-Functions, {V}ol. {I}},
      venue = {{A}nn {A}rbor, {MI},
      1988},
      series = {Perspect. Math.},
      volume = {10},
      pages = {77--159},
      publisher = {Academic Press},
      address = {Boston},
      year = {1990},
      mrclass = {11F70 (11F75 11F80 11R39 22E55)},
      mrnumber = {1044819},
      mrreviewer = {Ernst-Wilhelm Zink},
      zblnumber = {0705.11029},
      }
  • [Clozel] Go to document L. Clozel, "Représentations galoisiennes associées aux représentations automorphes autoduales de ${ GL}(n)$," Inst. Hautes Études Sci. Publ. Math., vol. 73, pp. 97-145, 1991.
    @article{Clozel, mrkey = {1114211},
      author = {Clozel, Laurent},
      title = {Représentations galoisiennes associées aux représentations automorphes autoduales de {${\rm GL}(n)$}},
      journal = {Inst. Hautes Études Sci. Publ. Math.},
      fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      volume = {73},
      year = {1991},
      pages = {97--145},
      issn = {0073-8301},
      coden = {PMIHA6},
      mrclass = {11F70 (11F80 11R39 22E55)},
      mrnumber = {1114211},
      mrreviewer = {David Manderscheid},
      zblnumber = {0739.11020},
      doi = {10.1007/BF02699257},
      zblnumber = {0739.11020},
      }
  • [Conrad] Go to document B. Conrad, Higher-level canonical subgroups in Abelian varieties, 2006.
    @misc{Conrad,
      author = {Conrad, B.},
      title = {Higher-level canonical subgroups in {A}belian varieties},
      year = {2006},
      url = {http://math.stanford.edu/~conrad/papers/subgppaper.pdf},
      }
  • [DeligneRapoport] P. Deligne and M. Rapoport, "Les schémas de modules de courbes elliptiques," in Modular Functions of One Variable, II, New York: Springer-Verlag, 1973, vol. 349, pp. 143-316.
    @incollection{DeligneRapoport, mrkey = {0337993},
      author = {Deligne, P. and Rapoport, M.},
      title = {Les schémas de modules de courbes elliptiques},
      booktitle = {Modular Functions of One Variable, {II}},
      venue = {{P}roc. {I}nternat. {S}ummer {S}chool, {U}niv. {A}ntwerp, {A}ntwerp, 1972},
      pages = {143--316},
      series = {Lecture Notes in Math.},
      volume = {349},
      publisher = {Springer-Verlag},
      year = {1973},
      mrclass = {14K10 (10D05)},
      mrnumber = {0337993},
      mrreviewer = {T. Oda},
      address = {New York},
      zblnumber = {0281.14010},
      }
  • [DeligneShimura] P. Deligne, "Travaux de Shimura," in Séminaire Bourbaki, 23ème Année (1970/71), Exp. No. 389, New York: Springer-Verlag, 1971, vol. 244, pp. 123-165.
    @incollection{DeligneShimura, mrkey = {0498581},
      author = {Deligne, Pierre},
      title = {Travaux de {S}himura},
      booktitle = {Séminaire {B}ourbaki, 23ème Année (1970/71), {E}xp. {N}o. 389},
      pages = {123--165},
      series = {Lecture Notes in Math.},
      volume = {244},
      publisher = {Springer-Verlag},
      year = {1971},
      mrclass = {14G25 (10D25 14K15)},
      mrnumber = {0498581},
      mrreviewer = {Author's review},
      address = {New York},
      zblnumber = {0225.14007},
      }
  • [FaltingsFieldOfDefinition] G. Faltings, "Arithmetic varieties and rigidity," in Seminar on Number Theory, Paris 1982–83, Birkhäuser, Boston, 1984, pp. 63-77.
    @incollection{FaltingsFieldOfDefinition, mrkey = {0791585},
      author = {Faltings, G.},
      title = {Arithmetic varieties and rigidity},
      booktitle = {Seminar on Number Theory, {P}aris 1982--83},
      series = {Progr. Math.},
      number = {51},
      pages = {63--77},
      publisher = {Birkhäuser, Boston},
      year = {1984},
      mrclass = {11F46 (14G25 32N10)},
      mrnumber = {0791585},
      mrreviewer = {James Milne},
      zblnumber = {0549.14010},
      }
  • [FaltingsTwoTowers] Go to document G. Faltings, "A relation between two moduli spaces studied by V. G. Drinfeld," in Algebraic Number Theory and Algebraic Geometry, Providence, RI: Amer. Math. Soc., 2002, vol. 300, pp. 115-129.
    @incollection{FaltingsTwoTowers, mrkey = {1936369},
      author = {Faltings, Gerd},
      title = {A relation between two moduli spaces studied by {V}. {G}. {D}rinfeld},
      booktitle = {Algebraic Number Theory and Algebraic Geometry},
      series = {Contemp. Math.},
      volume = {300},
      pages = {115--129},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {2002},
      mrclass = {14L05 (14G22)},
      mrnumber = {1936369},
      mrreviewer = {Noriko Yui},
      doi = {10.1090/conm/300/05145},
      zblnumber = {1062.14059},
      }
  • [FaltingsChai] Go to document G. Faltings and C. Chai, Degeneration of Abelian Varieties, New York: Springer-Verlag, 1990, vol. 22.
    @book{FaltingsChai, mrkey = {1083353},
      author = {Faltings, Gerd and Chai, Ching-Li},
      title = {Degeneration of Abelian Varieties},
      series = {Ergeb. Math. Grenzgeb.},
      volume = {22},
      note = {with an appendix by David Mumford},
      publisher = {Springer-Verlag},
      year = {1990},
      pages = {xii+316},
      isbn = {3-540-52015-5},
      mrclass = {14K10 (11G10 14D20 14K25)},
      mrnumber = {1083353},
      mrreviewer = {Min Ho Lee},
      doi = {10.1007/978-3-662-02632-8},
      address = {New York},
      zblnumber = {0744.14031},
      }
  • [FarguesCanonical] L. Fargues, "La filtration canonique des points de torsion des groupes $p$-divisibles," Ann. Sci. Éc. Norm. Supér., vol. 44, iss. 6, pp. 905-961, 2011.
    @article{FarguesCanonical, mrkey = {2919687},
      author = {Fargues, Laurent},
      title = {La filtration canonique des points de torsion des groupes {$p$}-divisibles},
      note = {with collaboration of Yichao Tian},
      journal = {Ann. Sci. Éc. Norm. Supér.},
      fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      volume = {44},
      year = {2011},
      number = {6},
      pages = {905--961},
      issn = {0012-9593},
      mrclass = {14L05},
      mrnumber = {2919687},
      mrreviewer = {Eva Viehmann},
      zblnumber = {06026211},
      }
  • [FarguesTwoTowers] L. Fargues, A. Genestier, and V. Lafforgue, L’isomorphisme Entre les Tours de Lubin-Tate et de Drinfeld, Basel: Birkhäuser, 2008, vol. 262.
    @book{FarguesTwoTowers, mrkey = {2441311},
      author = {Fargues, Laurent and Genestier, Alain and Lafforgue, Vincent},
      title = {L'isomorphisme Entre les Tours de {L}ubin-{T}ate et de {D}rinfeld},
      series = {Progr. Math.},
      volume = {262},
      publisher = {Birkhäuser},
      address = {Basel},
      year = {2008},
      pages = {xxii+406},
      isbn = {978-3-7643-8455-5},
      mrclass = {14L05 (11R39 14-06 14G22 14G35)},
      mrnumber = {2441311},
      mrreviewer = {Eva Viehmann},
      zblnumber = {1136.14001},
      }
  • [Franke] 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{Franke, 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 = {1603257},
      doi = {10.1016/S0012-9593(98)80015-3},
      zblnumber = {0938.11026},
      }
  • [GabberRamero] O. Gabber and L. Ramero, Almost Ring Theory, New York: Springer-Verlag, 2003, vol. 1800.
    @book{GabberRamero, mrkey = {2004652},
      author = {Gabber, Ofer and Ramero, Lorenzo},
      title = {Almost Ring Theory},
      series = {Lecture Notes in Math.},
      volume = {1800},
      publisher = {Springer-Verlag},
      year = {2003},
      pages = {vi+307},
      isbn = {3-540-40594-1},
      mrclass = {13D10 (13B40 13D03 14G22 18D10)},
      mrnumber = {2004652},
      address = {New York},
      zblnumber = {1045.13002},
      }
  • [GorenKassaei] Go to document E. Z. Goren and P. L. Kassaei, "The canonical subgroup: a “subgroup-free” approach," Comment. Math. Helv., vol. 81, iss. 3, pp. 617-641, 2006.
    @article{GorenKassaei, mrkey = {2250856},
      author = {Goren, Eyal Z. and Kassaei, Payman L.},
      title = {The canonical subgroup: a ``subgroup-free'' approach},
      journal = {Comment. Math. Helv.},
      fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
      volume = {81},
      year = {2006},
      number = {3},
      pages = {617--641},
      issn = {0010-2571},
      mrclass = {11F85 (11F33 11G18 14G22 14G35)},
      mrnumber = {2250856},
      mrreviewer = {Anton Deitmar},
      doi = {10.4171/CMH/66},
      zblnumber = {1122.11037},
      }
  • [HLTT] M. Harris, K. -W. Lan, R. Taylor, and J. Thorne, Galois representations for regular algebraic cusp forms over CM fields.
    @misc{HLTT,
      author = {Harris, M. and Lan, K.-W. and Taylor, R. and Thorne, J.},
      title = {Galois representations for regular algebraic cusp forms over {CM} fields},
      note = {in preperation},
      }
  • [HarrisTaylor] M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2001, vol. 151.
    @book{HarrisTaylor, mrkey = {1876802},
      author = {Harris, Michael and Taylor, Richard},
      title = {The Geometry and Cohomology of Some Simple {S}himura Varieties},
      series = {Ann. of Math. Stud.},
      volume = {151},
      note = {with an appendix by Vladimir G. Berkovich},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {2001},
      pages = {viii+276},
      isbn = {0-691-09090-4},
      mrclass = {11G18 (11F70 11S37 14G35 22E45)},
      mrnumber = {1876802},
      mrreviewer = {James Milne},
      zblnumber = {1036.11027},
      }
  • [HuberContVal] Go to document R. Huber, "Continuous valuations," Math. Z., vol. 212, iss. 3, pp. 455-477, 1993.
    @article{HuberContVal, mrkey = {1207303},
      author = {Huber, R.},
      title = {Continuous valuations},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {212},
      year = {1993},
      number = {3},
      pages = {455--477},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {13F30 (12J17 13J07)},
      mrnumber = {1207303},
      mrreviewer = {N. Sankaran},
      doi = {10.1007/BF02571668},
      zblnumber = {0788.13010},
      }
  • [Huber] Go to document R. Huber, Étale Cohomology of Rigid Analytic Varieties and Adic Spaces, Braunschweig: Friedr. Vieweg & Sohn, 1996, vol. E30.
    @book{Huber, mrkey = {1734903},
      author = {Huber, Roland},
      title = {\'{E}tale Cohomology of Rigid Analytic Varieties and Adic Spaces},
      series = {Aspects Math.},
      volume = {E30},
      publisher = {Friedr. Vieweg \& Sohn},
      address = {Braunschweig},
      year = {1996},
      pages = {x+450},
      isbn = {3-528-06794-2},
      mrclass = {14G22 (14F20)},
      mrnumber = {1734903},
      mrreviewer = {Lorenzo Ramero},
      doi = {10.1007/978-3-663-09991-8},
      zblnumber = {0868.14010},
      }
  • [IllusieThesis1] L. Illusie, Complexe Cotangent et Déformations. I, New York: Springer-Verlag, 1971, vol. 239.
    @book{IllusieThesis1, mrkey = {0491680},
      author = {Illusie, Luc},
      title = {Complexe Cotangent et Déformations. {I}},
      series = {Lecture Notes in Math.},
      volume = {239},
      publisher = {Springer-Verlag},
      year = {1971},
      pages = {xv+355},
      mrclass = {14B10 (14D15 14F20 55D20)},
      mrnumber = {0491680},
      mrreviewer = {Larry Breen},
      address = {New York},
      zblnumber = {0224.13014},
      }
  • [IllusieThesis2] L. Illusie, Complexe Cotangent et Déformations. II, New York: Springer-Verlag, 1972, vol. 283.
    @book{IllusieThesis2, mrkey = {0491681},
      author = {Illusie, Luc},
      title = {Complexe Cotangent et Déformations. {II}},
      series = {Lecture Notes in Math.},
      volume = {283},
      publisher = {Springer-Verlag},
      year = {1972},
      pages = {vii+304},
      mrclass = {14B10 (14D15 14F20 55D20)},
      mrnumber = {0491681},
      mrreviewer = {Larry Breen},
      address = {New York},
      zblnumber = {0238.13017},
      }
  • [IllusieDeformationsBT] L. Illusie, "Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck)," in Seminar on Arithmetic Bundles: The Mordell Conjecture, , 1985, vol. 127, pp. 151-198.
    @incollection{IllusieDeformationsBT, mrkey = {0801922},
      author = {Illusie, Luc},
      title = {Déformations de groupes de {B}arsotti-{T}ate (d'après {A}. {G}rothendieck)},
      booktitle = {Seminar on Arithmetic Bundles: The Mordell Conjecture},
      venue = {Paris, 1983/84},
      series = {Astérisque},
      volume = {127},
      year = {1985},
      pages = {151--198},
      issn = {0303-1179},
      mrclass = {14L05 (11G07 11G10 11S31)},
      mrnumber = {0801922},
      zblnumber = {1182.14050},
      }
  • [deJongvanderPut] J. de Jong and M. van der Put, "Étale cohomology of rigid analytic spaces," Doc. Math., vol. 1, p. no. 01, 1-56, 1996.
    @article{deJongvanderPut, mrkey = {1386046},
      author = {de Jong, Johan and van der Put, Marius},
      title = {\'{E}tale cohomology of rigid analytic spaces},
      journal = {Doc. Math.},
      fjournal = {Documenta Mathematica},
      volume = {1},
      year = {1996},
      pages = {No. 01, 1--56},
      issn = {1431-0635},
      mrclass = {14F20 (32P05)},
      mrnumber = {1386046},
      mrreviewer = {Jean-Claude Douai},
      zblnumber = {0922.14012},
      }
  • [Katz] N. M. Katz, "$p$-adic properties of modular schemes and modular forms," in Modular Functions of One Variable, III, New York: Springer-Verlag, 1973, vol. 350, pp. 69-190.
    @incollection{Katz, mrkey = {0447119},
      author = {Katz, Nicholas M.},
      title = {{$p$}-adic properties of modular schemes and modular forms},
      booktitle = {Modular Functions of One Variable, {III}},
      venue = {{P}roc. {I}nternat. {S}ummer {S}chool, {U}niv. {A}ntwerp, {A}ntwerp, 1972},
      pages = {69--190},
      series = {Lecture Notes in Math.},
      volume = {350},
      publisher = {Springer-Verlag},
      year = {1973},
      mrclass = {10D15 (14D20)},
      mrnumber = {0447119},
      mrreviewer = {V. V. Sokurov},
      address = {New York},
      zblnumber = {0271.10033},
      }
  • [KedlayaLiu1] Go to document K. S. Kedlaya and R. Liu, Relative $p$-adic Hodge Theory, I: Foundations, , 2015, vol. 371.
    @book{KedlayaLiu1,
      author = {Kedlaya, K. S. and Liu, R.},
      title = {Relative {$p$}-adic {H}odge Theory, {I}: {F}oundations},
      fjournal = {Astérisque},
      series = {Astérisque},
      volume = {371},
      year = {2015},
      url = {http://smf4.emath.fr/en/Publications/Asterisque/2015/371/html/smf_ast_371.php},
      }
  • [Kottwitz] Go to document R. E. Kottwitz, "On the $\lambda$-adic representations associated to some simple Shimura varieties," Invent. Math., vol. 108, iss. 3, pp. 653-665, 1992.
    @article{Kottwitz, mrkey = {1163241},
      author = {Kottwitz, Robert E.},
      title = {On the {$\lambda$}-adic representations associated to some simple {S}himura varieties},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {108},
      year = {1992},
      number = {3},
      pages = {653--665},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {11G18 (11F70 11F85 14G35 22E55)},
      mrnumber = {1163241},
      mrreviewer = {Min Ho Lee},
      doi = {10.1007/BF02100620},
      zblnumber = {0765.22011},
      }
  • [Lubin] Go to document J. Lubin, "Canonical subgroups of formal groups," Trans. Amer. Math. Soc., vol. 251, pp. 103-127, 1979.
    @article{Lubin, mrkey = {0531971},
      author = {Lubin, Jonathan},
      title = {Canonical subgroups of formal groups},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {251},
      year = {1979},
      pages = {103--127},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {14L05},
      mrnumber = {0531971},
      mrreviewer = {V. V. Shokurov},
      doi = {10.2307/1998685},
      zblnumber = {0431.14014},
      }
  • [MoeglinWaldspurger] C. Moeglin and J. -L. Waldspurger, Stabilisation de la formule des traces tordue X: stabilisation spectrale, 2014.
    @misc{MoeglinWaldspurger,
      author = {Moeglin, C. and Waldspurger, J.-L.},
      title = {Stabilisation de la formule des traces tordue {X}: stabilisation spectrale},
      year = {2014},
      arxiv = {1412.2981},
      }
  • [Mok] Go to document C. P. Mok, "Endoscopic classification of representations of quasi-split unitary groups," Mem. Amer. Math. Soc., vol. 235, iss. 1108, p. vi, 2015.
    @article{Mok, mrkey = {3338302},
      author = {Mok, Chung Pang},
      title = {Endoscopic classification of representations of quasi-split unitary groups},
      journal = {Mem. Amer. Math. Soc.},
      fjournal = {Memoirs of the American Mathematical Society},
      volume = {235},
      year = {2015},
      number = {1108},
      pages = {vi+248},
      issn = {0065-9266},
      isbn = {978-1-4704-1041-4; 978-1-4704-2226-4},
      mrclass = {22E55 (11R42 22E50)},
      mrnumber = {3338302},
      doi = {10.1090/memo/1108},
      }
  • [MorelBook] Go to document S. Morel, On the Cohomology of Certain Noncompact Shimura Varieties, Princeton, NJ: Princeton Univ. Press, 2010, vol. 173.
    @book{MorelBook, mrkey = {2567740},
      author = {Morel, Sophie},
      title = {On the Cohomology of Certain Noncompact {S}himura Varieties},
      series = {Ann. of Math. Stud.},
      volume = {173},
      note = {with an appendix by Robert Kottwitz},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {2010},
      pages = {xii+217},
      isbn = {978-0-691-14293-7},
      mrclass = {11F75 (11F72 11G18 14G35)},
      mrnumber = {2567740},
      mrreviewer = {Eva Viehmann},
      doi = {10.1515/9781400835393},
      zblnumber = {1233.11069},
      }
  • [Pfaff] Go to document J. Pfaff, "Exponential growth of homological torsion for towers of congruence subgroups of Bianchi groups," Ann. Global Anal. Geom., vol. 45, iss. 4, pp. 267-285, 2014.
    @article{Pfaff, mrkey = {3180949},
      author = {Pfaff, Jonathan},
      title = {Exponential growth of homological torsion for towers of congruence subgroups of {B}ianchi groups},
      journal = {Ann. Global Anal. Geom.},
      fjournal = {Annals of Global Analysis and Geometry},
      volume = {45},
      year = {2014},
      number = {4},
      pages = {267--285},
      issn = {0232-704X},
      mrclass = {30F40 (11F75 20H10)},
      mrnumber = {3180949},
      mrreviewer = {Bernd O. Stratmann},
      doi = {10.1007/s10455-013-9400-2},
      zblnumber = {06298097},
      }
  • [PilloniStroh] V. Pilloni and B. Stroh, Cohomologie cohérente et représentations galoisiennes, 2015.
    @misc{PilloniStroh,
      author = {Pilloni, V. and Stroh, B.},
      title = {Cohomologie cohérente et représentations galoisiennes},
      year = {2015},
      }
  • [Rabinoff] Go to document J. Rabinoff, "Higher-level canonical subgroups for $p$-divisible groups," J. Inst. Math. Jussieu, vol. 11, iss. 2, pp. 363-419, 2012.
    @article{Rabinoff, mrkey = {2905309},
      author = {Rabinoff, Joseph},
      title = {Higher-level canonical subgroups for {$p$}-divisible groups},
      journal = {J. Inst. Math. Jussieu},
      fjournal = {Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l'Institut de Mathématiques de Jussieu},
      volume = {11},
      year = {2012},
      number = {2},
      pages = {363--419},
      issn = {1474-7480},
      mrclass = {14L05 (11G10 14T05)},
      mrnumber = {2905309},
      mrreviewer = {Eva Viehmann},
      doi = {10.1017/S1474748011000132},
      zblnumber = {1285.14047},
      }
  • [ScholzeSurvey] P. Scholze, "Perfectoid spaces: a survey," in Current Developments in Mathematics 2012, Somerville, MA: Int. Press, 2013, pp. 193-227.
    @incollection{ScholzeSurvey, mrkey = {3204346},
      author = {Scholze, Peter},
      title = {Perfectoid spaces: a survey},
      booktitle = {Current Developments in Mathematics 2012},
      pages = {193--227},
      publisher = {Int. Press},
      address = {Somerville, MA},
      year = {2013},
      mrclass = {14G22 (14C30 14G20 14L05)},
      mrnumber = {3204346},
      zblnumber = {06302780},
      }
  • [ScholzeWeinstein] Go to document P. Scholze and J. Weinstein, "Moduli of $p$-divisible groups," Camb. J. Math., vol. 1, iss. 2, pp. 145-237, 2013.
    @article{ScholzeWeinstein, mrkey = {3272049},
      author = {Scholze, Peter and Weinstein, Jared},
      title = {Moduli of {$p$}-divisible groups},
      journal = {Camb. J. Math.},
      fjournal = {Cambridge Journal of Mathematics},
      volume = {1},
      year = {2013},
      number = {2},
      pages = {145--237},
      issn = {2168-0930},
      mrclass = {14L05 (11G25)},
      mrnumber = {3272049},
      doi = {10.4310/CJM.2013.v1.n2.a1},
      zblnumber = {06292858},
      }
  • [ScholzePerfectoid] Go to document P. Scholze, "Perfectoid spaces," Publ. Math. Inst. Hautes Études Sci., vol. 116, pp. 245-313, 2012.
    @article{ScholzePerfectoid, mrkey = {3090258},
      author = {Scholze, Peter},
      title = {Perfectoid spaces},
      journal = {Publ. Math. Inst. Hautes Études Sci.},
      fjournal = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      volume = {116},
      year = {2012},
      pages = {245--313},
      issn = {0073-8301},
      mrclass = {14G99},
      mrnumber = {3090258},
      mrreviewer = {Jean-Marc Fontaine},
      doi = {10.1007/s10240-012-0042-x},
      zblnumber = {1263.14022},
      }
  • [ScholzePAdicHodge] Go to document P. Scholze, "$p$-adic Hodge theory for rigid-analytic varieties," Forum Math. Pi, vol. 1, p. 1, 2013.
    @article{ScholzePAdicHodge, mrkey = {3090230},
      author = {Scholze, Peter},
      title = {{$p$}-adic {H}odge theory for rigid-analytic varieties},
      journal = {Forum Math. Pi},
      fjournal = {Forum of Mathematics. Pi},
      volume = {1},
      year = {2013},
      pages = {e1, 77},
      issn = {2050-5086},
      mrclass = {14G22 (14C30 14F30 14G20 32J27 32P05)},
      mrnumber = {3090230},
      mrreviewer = {Hui June Zhu},
      doi = {10.1017/fmp.2013.1},
      zblnumber = {1297.14023},
      }
  • [ShinGoldringApp] S. W. Shin, On the cohomological base change for unitary similitude groups, 2012.
    @misc{ShinGoldringApp,
      author = {Shin, Sug Woo},
      title = {On the cohomological base change for unitary similitude groups},
      year = {2012},
      note = {appendix to {W. G}oldring, {G}alois representations associated to holomorphic limits of discrete series {I}: {U}nitary groups, to appear in {\it {C}ompositio {M}ath.}},
      }
  • [Shin] Go to document S. W. Shin, "Galois representations arising from some compact Shimura varieties," Ann. of Math., vol. 173, iss. 3, pp. 1645-1741, 2011.
    @article{Shin, mrkey = {2800722},
      author = {Shin, Sug Woo},
      title = {Galois representations arising from some compact {S}himura varieties},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {173},
      year = {2011},
      number = {3},
      pages = {1645--1741},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {11F80 (11G18)},
      mrnumber = {2800722},
      mrreviewer = {Peter Scholze},
      doi = {10.4007/annals.2011.173.3.9},
      zblnumber = {1269.11053},
      }
  • [TaylorPseudoRepr] Go to document R. Taylor, "Galois representations associated to Siegel modular forms of low weight," Duke Math. J., vol. 63, iss. 2, pp. 281-332, 1991.
    @article{TaylorPseudoRepr, mrkey = {1115109},
      author = {Taylor, Richard},
      title = {Galois representations associated to {S}iegel modular forms of low weight},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {63},
      year = {1991},
      number = {2},
      pages = {281--332},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {11F46 (11F80 11R39 14G35)},
      mrnumber = {1115109},
      mrreviewer = {Bas Edixhoven},
      doi = {10.1215/S0012-7094-91-06312-X},
      zblnumber = {0810.11033},
      }
  • [TaylorWiles] Go to document R. Taylor and A. Wiles, "Ring-theoretic properties of certain Hecke algebras," Ann. of Math., vol. 141, iss. 3, pp. 553-572, 1995.
    @article{TaylorWiles, mrkey = {1333036},
      author = {Taylor, Richard and Wiles, Andrew},
      title = {Ring-theoretic properties of certain {H}ecke algebras},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {141},
      year = {1995},
      number = {3},
      pages = {553--572},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {11G18 (11D41 11F11 11R34 13C40)},
      mrnumber = {1333036},
      mrreviewer = {Karl Rubin},
      doi = {10.2307/2118560},
      zblnumber = {0823.11030},
      }
  • [Temkin] Go to document M. Temkin, "On local properties of non-Archimedean analytic spaces," Math. Ann., vol. 318, iss. 3, pp. 585-607, 2000.
    @article{Temkin, mrkey = {1800770},
      author = {Temkin, M.},
      title = {On local properties of non-{A}rchimedean analytic spaces},
      journal = {Math. Ann.},
      fjournal = {Mathematische Annalen},
      volume = {318},
      year = {2000},
      number = {3},
      pages = {585--607},
      issn = {0025-5831},
      coden = {MAANA},
      mrclass = {14G22 (32P05)},
      mrnumber = {1800770},
      mrreviewer = {Lorenzo Ramero},
      doi = {10.1007/s002080000123},
      zblnumber = {0972.32019},
      }
  • [Tian] Go to document Y. Tian, "Canonical subgroups of Barsotti-Tate groups," Ann. of Math., vol. 172, iss. 2, pp. 955-988, 2010.
    @article{Tian, mrkey = {2680485},
      author = {Tian, Yichao},
      title = {Canonical subgroups of {B}arsotti-{T}ate groups},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {172},
      year = {2010},
      number = {2},
      pages = {955--988},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {14L15},
      mrnumber = {2680485},
      doi = {10.4007/annals.2010.172.955},
      zblnumber = {1203.14026},
      }
  • [Venjakob] Go to document O. Venjakob, "On the structure theory of the Iwasawa algebra of a $p$-adic Lie group," J. Eur. Math. Soc. $($JEMS$)$, vol. 4, iss. 3, pp. 271-311, 2002.
    @article{Venjakob, mrkey = {1924402},
      author = {Venjakob, Otmar},
      title = {On the structure theory of the {I}wasawa algebra of a {$p$}-adic {L}ie group},
      journal = {J. Eur. Math. Soc. $($JEMS$)$},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {4},
      year = {2002},
      number = {3},
      pages = {271--311},
      issn = {1435-9855},
      mrclass = {16S34 (22E35)},
      mrnumber = {1924402},
      doi = {10.1007/s100970100038},
      zblnumber = {1049.16016},
      }
  • [WilesPAdicInterpolation] Go to document A. Wiles, "On ordinary $\lambda$-adic representations associated to modular forms," Invent. Math., vol. 94, iss. 3, pp. 529-573, 1988.
    @article{WilesPAdicInterpolation, mrkey = {0969243},
      author = {Wiles, A.},
      title = {On ordinary {$\lambda$}-adic representations associated to modular forms},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {94},
      year = {1988},
      number = {3},
      pages = {529--573},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {11F41 (11F80 11R23 11R80)},
      mrnumber = {0969243},
      mrreviewer = {Sheldon Kamienny},
      doi = {10.1007/BF01394275},
      zblnumber = {0664.10013},
      }
  • [WilesFLT] Go to document A. Wiles, "Modular elliptic curves and Fermat’s last theorem," Ann. of Math., vol. 141, iss. 3, pp. 443-551, 1995.
    @article{WilesFLT, mrkey = {1333035},
      author = {Wiles, Andrew},
      title = {Modular elliptic curves and {F}ermat's last theorem},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {141},
      year = {1995},
      number = {3},
      pages = {443--551},
      issn = {0003-486X},
      coden = {ANMAAH},
      mrclass = {11G05 (11D41 11F11 11F80 11G18)},
      mrnumber = {1333035},
      mrreviewer = {Karl Rubin},
      doi = {10.2307/2118559},
      zblnumber = {0823.11029},
      }

Authors

Peter Scholze

Mathematisches Institut, Universität Bonn, Bonn, Germany