On the \(p\)-adic cohomology of the Lubin-Tate tower. (English. French summary) Zbl 1419.14031

In this article, the author constructs a functor from the category of admissible \(\bmod p\) representations of \(\mathrm{GL}_n(F)\), where \(F/\mathbb{Q}_p\) is a finite extension, to the category of admissible \(\mathrm{Gal}_F\times D^\times\)-representations, where \(\mathrm{Gal}_F\) is the absolute Galois group of \(F\) and \(D/F\) is the central division algebra of invariant \(1/n\). His construction provides further evidence for the existence of mod \(p\) and \(p\)-adic local Langlands and Jacquet-Langlands correspondences. One advantage over a previously established candidate for such correspondences (see [A. Caraiani et al., Camb. J. Math. 4, No. 2, 197–287 (2016; Zbl 1403.11073)]) is that this functor is constructed by purely local means.
To describe the results in slightly more detail, fix a complete and algebraically closed extension \(C \supset F\). For a smooth admissible representation \(\pi\) of \(\mathrm{GL}_n(F)\) on a \(\mathbb{F}_p\)-vector space, the author constructs an étale sheaf \(\mathcal{F}_{\pi}\) on the adic space \(\mathbb{P}_C^{n-1}\) using the Lubin-Tate tower and the Gross-Hopkins period map. It is then shown that for all \(i\geq 0\) the cohomology groups \[ H_{\text{ét}}^i(\mathbb{P}_C^{n-1},\mathcal{F}_\pi) \] are admissible \(D^\times\)-representations, which carry a continuous action of the absolute Galois group \(\mathrm{Gal}_F\). They vanish in degree \(i > 2(n-1)\) and are independent of the choice of the algebraically closed and complete extension \(C\) of \(F\).
The proof of this result follows a strategy developed in an earlier work of the author to prove finiteness of \(\mathbb{F}_p\)-cohomology of proper (smooth) rigid analytic spaces, (Zbl 1297.14023). It depends crucially on properness of \(\mathbb{P}^{n-1}_C\), which is the image of the Gross-Hopkins period map.
The author then proves a local-global compatibility result for \(n=2\). To describe it, fix a totally real field \(F\) and a prime \(\mathfrak{p}\) dividing \(p\) such that \(F_{\mathfrak{p}}\) is the \(p\)-adic field considered above. Let \(D_0\) be a quaternion algebra over \(F\), which is split at \(\mathfrak{p}\) and is ramified at all infinite places. Let \(D\) be the quaternion algebra over \(F\), which is split at a fixed infinite place \(\infty_F\), ramified at \(\mathfrak{p}\) and isomorphic to \(D_0\) at all other places. Fix a compact open subgroup \(U^{\mathfrak{p}} \subset D_0^\times(\mathbb{A}_{F,f}^{\mathfrak{p}})\cong D^{\times}(\mathbb{A}_{F,f}^{\mathfrak{p}})\). Let \(\pi = \varinjlim_K S(K U^{\mathfrak{p}},\mathbb{Q}_p/\mathbb{Z}_p)\) be the direct limit of spaces of algebraic automorphic forms, where \(K\) runs through the compact open subgroups of \( \mathrm{GL}_2(F_{\mathfrak{p}})\cong D_{0,\mathfrak{p}}\). Consider the cohomology groups \(H^1(\mathrm{Sh}_{K^\prime U^{\mathfrak{p}},C},\mathbb{Q}_p/\mathbb{Z}_p)\), of the Shimura curve \(\mathrm{Sh}_{K^\prime U^{\mathfrak{p}}}/F\) for \(D/F\), for varying \(K^\prime\subset D_{\mathfrak{p}}^\times\). The author shows that there is a canonical \(\mathrm{Gal}_{F_{\mathfrak{p}}}\times D^\times_{\mathfrak{p}}\)-equivariant isomorphism \[ H^1_{\text{ét}}(\mathbb{P}^1_{C},\mathcal{F}_\pi) \cong \varinjlim_{K^\prime} H^1(\mathrm{Sh}_{K^\prime U^{\mathfrak{p}},C},\mathbb{Q}_p/\mathbb{Z}_p). \] He also shows that for \(n=2\) his functor is compatible with the patching construction of Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin [loc. cit.]. In this context, a new perspective on the patching construction using ultraproducts is explained.
The paper contains an appendix by M. Rapoport on the question of surjectivity of period maps from Rapoport-Zink spaces to partial flag varieties. Two variants of this question are studied. It is first investigated in which situations the period map is surjective on classical points, i.e., for which \(\mathrm{PD}\)-triples the period domain is the whole partial flag variety. It is then proved that surjectivity on all adic points (not just on classical points) occurs essentially only in the Lubin-Tate case.


14G22 Rigid analytic geometry
11S37 Langlands-Weil conjectures, nonabelian class field theory
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
Full Text: arXiv Link


[1] 1. Chenevier, Gaëtan and Harris, Michael Construction of automorphic Galois representations, II Camb. J. Math.1 (2013) 53-73 Math Reviews MR3272052 · Zbl 1310.11062
[2] 3. Chojecki, Przemysław On modp non-abelian Lubin-Tate theory for GL_2(Q_p)Compos. Math.151 (2015) 1433-1461 Math Reviews MR3383163 · Zbl 1331.11039
[3] 4. Drinfeld, V. G. Coverings of p-adic symmetric domains Funkcional. Anal. i Priložen.10 (1976) 29-40 Math Reviews MR0422290 (54 #10281)
[4] 5. Shin, Sug Woo Galois representations arising from some compact Shimura varieties Ann. of Math.173 (2011) 1645-1741 Math Reviews MR2800722 · Zbl 1269.11053
[5] 6. Scholze, Peter On torsion in the cohomology of locally symmetric varieties Ann. of Math.182 (2015) 945-1066 Math Reviews MR3418533 · Zbl 1345.14031
[6] 7. Taylor, Richard On Galois representations associated to Hilbert modular forms Invent. math.98 (1989) 265-280 Math Reviews MR1016264 (90m:11176)
[7] 8. Hopkins, M. J. and Gross, B. H. Equivariant vector bundles on the Lubin-Tate moduli space in Topology and representation theory (Evanston, IL, 1992)Contemp. Math.158 (1994) 23-88 Math Reviews MR1263712 (95b:14033) · Zbl 0807.14037
[8] 9. Paškūnas, Vytautas The image of Colmez’s Montreal functor Publ. Math. IHÉS118 (2013) 1-191 Math Reviews MR3150248
[9] 10. Colmez, Pierre Représentations de GL_2(Q_p) et (,)-modules Astérisque330 (2010) 281-509 Math Reviews MR2642409 (2011j:11224)
[10] 11. Breuil, Christophe The emerging p-adic Langlands programme in Proceedings of the International Congress of Mathematicians. Volume II(2010) 203-230 Math Reviews MR2827792 (2012k:22024)
[11] 12. Breuil, Christophe and Paškūnas, Vytautas Towards a modulo p Langlands correspondence for GL_2Mem. Amer. Math. Soc., vol. 216, 2012 Math Reviews MR2931521
[12] 13. Caraiani, A. and Emerton, Matthew and Gee, Toby and Geraghty, D. and Paškūnas, Vytautas and Shin, Sug Woo Patching and the p-adic local Langlands correspondence Cambridge Journal of Math.4 (2016) 197-287 · Zbl 1403.11073
[13] 14. Scholze, Peter Perfectoid Spaces Publ. Math. de l’IHES116 (2012) 245-313
[14] 15. Scholze, Peter p-adic Hodge theory for rigid-analytic varieties Forum of Mathematics, Pi1 (2013) · Zbl 1297.14023
[15] 16. Scholze, Peter and Weinstein, Jared Moduli of p-divisible groups Camb. J. Math.1 (2013) 145-237 Math Reviews MR3272049 · Zbl 1349.14149
[16] 18. Elkik, Renée Solutions d’équations à coefficients dans un anneau hensélien Ann. Sci. École Norm. Sup.6 (1973) 553-603 Math Reviews MR0345966 (49 #10692) · Zbl 0327.14001
[17] 19. Fargues, Laurent and Genestier, Alain and Lafforgue, Vincent L’isomorphisme entre les tours de Lubin-Tate et de DrinfeldProgress in Math., vol. 262, Birkhäuser, 2008 Math Reviews MR2441311 (2009k:14088)
[18] 20. Faltings, Gerd A relation between two moduli spaces studied by V. G. Drinfeld in Algebraic number theory and algebraic geometryContemp. Math.300 (2002) 115-129 Math Reviews MR1936369 (2003g:14062)
[19] 21. Emerton, Matthew Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors Astérisque331 (2010) 403-459 Math Reviews MR2667883 (2011k:22014) · Zbl 1205.22014
[20] 22. Emerton, Matthew Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties Astérisque331 (2010) 355-402 Math Reviews MR2667882 (2011k:22013) · Zbl 1205.22013
[21] 23. Chenevier, Gaëtan The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings in Automorphic forms and Galois representations. Vol. 1London Math. Soc. Lecture Note Ser.414 (2014) 221-285 Math Reviews MR3444227 · Zbl 1350.11063
[22] 24. Carayol, Henri Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991)Contemp. Math.165 (1994) 213-237 Math Reviews MR1279611 (95i:11059)
[23] 25. Čerednik, I. V. Uniformization of algebraic curves by discrete arithmetic subgroups of PGL_2(k_w) with compact quotient spaces Mat. Sb. (N.S.)100(142) (1976) 59-88, 165 Math Reviews MR0491706 (58 #10909)
[24] 27. Carayol, Henri Sur les représentations l-adiques associées aux formes modulaires de Hilbert Ann. Sci. École Norm. Sup.19 (1986) 409-468 Math Reviews MR870690 (89c:11083)
[25] 28. Weinstein, Jared Semistable models for modular curves of arbitrary level Invent. math.205 (2016) 459-526 Math Reviews MR3529120 · Zbl 1357.14034
[26] 29. Strauch, Matthias Geometrically connected components of Lubin-Tate deformation spaces with level structures Pure Appl. Math. Q.4 (2008) 1215-1232 Math Reviews MR2441699 (2009e:11116) · Zbl 1193.11112
[27] 30. Abbes, Ahmed Éléments de géométrie rigide. Volume IProgress in Math., vol. 286, Birkhäuser, 2010 Math Reviews MR2815110 · Zbl 1223.14003
[28] 31. Andreatta, Fabrizio and Iovita, Adrian Comparison Isomorphisms for Formal Schemes http://www.mathstat.concordia.ca/faculty/iovita/paper##img##14.pdf · Zbl 1281.14013
[29] 32. Arthur, James The L2-Lefschetz numbers of Hecke operators Invent. math.97 (1989) 257-290 Math Reviews MR1001841
[30] 33. Arthur, James and Clozel, Laurent Simple algebras, base change, and the advanced theory of the trace formulaAnnals of Math. Studies, vol. 120, Princeton Univ. Press, 1989 Math Reviews MR1007299 · Zbl 0682.10022
[31] 34. Artin, M. Algebraization of formal moduli. I in Global Analysis (Papers in Honor of K. Kodaira)(1969) 21-71 Math Reviews MR0260746 (41 #5369)
[32] 35. Bartenwerfer, Wolfgang Die höheren metrischen Kohomologiegruppen affinoider Räume Math. Ann.241 (1979) 11-34 Math Reviews MR531147 (80h:32051)
[33] 36. Beilinson, Alexander p-adic periods and derived de Rham cohomology http://arxiv.org/abs/1102.1294
[34] 37. Berkovich, Vladimir G. Vanishing cycles for formal schemes Invent. math.115 (1994) 539-571 Math Reviews MR1262943 · Zbl 0791.14008
[35] 38. Berkovich, Vladimir G. Vanishing cycles for formal schemes. II Invent. math.125 (1996) 367-390 Math Reviews MR1395723 · Zbl 0852.14002
[36] 39. Bernstein, J. N. Le “centre” de Bernstein in Representations of reductive groups over a local fieldTravaux en Cours (1984) 1-32 Edited by P. Deligne Math Reviews MR771671
[37] 40. Bernstein, I. N. and Zelevinsky, A. V. Induced representations of reductive p-adic groups. I Ann. Sci. École Norm. Sup.10 (1977) 441-472 Math Reviews MR0579172 (58 #28310) · Zbl 0412.22015
[38] 41. Bosch, S. and Güntzer, U. and Remmert, R. Non-Archimedean analysisGrundl. math. Wiss., vol. 261, Springer, 1984 A systematic approach to rigid analytic geometry Math Reviews MR746961 (86b:32031) · Zbl 0539.14017
[39] 42. Brinon, Olivier Représentations p-adiques cristallines et de de Rham dans le cas relatif Mém. Soc. Math. Fr. (N.S.)112 (2008) 159 Math Reviews MR2484979 (2010a:14034) · Zbl 1170.14016
[40] 43. Bushnell, Colin J. and Kutzko, Philip C. The admissible dual of GL(N) via compact open subgroupsAnnals of Math. Studies, vol. 129, Princeton Univ. Press, 1993 Math Reviews MR1204652 · Zbl 0787.22016
[41] 44. Buzzard, Kevin and Gee, Toby The conjectural connections between automorphic representations and Galois representations arXiv:1009.0785 · Zbl 1377.11067
[42] 45. Casselman, W. The Hasse-Weil ##img##-function of some moduli varieties of dimension greater than one in Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2Proc. Sympos. Pure Math., XXXIII (1979) 141-163 Math Reviews MR546615
[43] 46. Casselman, W. Characters and Jacquet modules Math. Ann.230 (1977) 101-105 Math Reviews MR0492083 · Zbl 0337.22019
[44] 47. Clozel, Laurent The fundamental lemma for stable base change Duke Math. J.61 (1990) 255-302 Math Reviews MR1068388
[45] 48. Colmez, Pierre Espaces de Banach de dimension finie J. Inst. Math. Jussieu1 (2002) 331-439 Math Reviews MR1956055 (2004b:11160) · Zbl 1044.11102
[46] 49. de Jong, A. J. Smoothness, semi-stability and alterations Publ. Math. IHÉS83 (1996) 51-93 Math Reviews MR1423020 (98e:14011) · Zbl 0916.14005
[47] 50. de Jong, A. J. Étale fundamental groups of non-Archimedean analytic spaces Compos. math.97 (1995) 89-118 Special issue in honour of Frans Oort Math Reviews MR1355119 (97c:32047)
[48] 51. de Jong, Johan and van der Put, Marius Étale cohomology of rigid analytic spaces Doc. Math.1 (1996) No. 01, 1-56 Math Reviews MR1386046 (98d:14024) · Zbl 0922.14012
[49] 52. Deligne, P. and Rapoport, M. Les schémas de modules de courbes elliptiques in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)Lecture Notes in Math.349 (1973) 143-316 Math Reviews MR0337993 · Zbl 0281.14010
[50] 53. Faltings, Gerd Group schemes with strict O-action Mosc. Math. J.2 (2002) 249-279 Dedicated to Yuri I. Manin on the occasion of his 65th birthday Math Reviews MR1944507 (2004i:14052)
[51] 54. Faltings, Gerd Almost étale extensions Astérisque279 (2002) 185-270 Math Reviews MR1922831 (2003m:14031)
[52] 55. Fargues, Laurent Motives and automorphic forms: the abelian case http://www.math.u-psud.fr/ ##img## fargues/Motifsabeliens.ps
[53] 56. Folkman, Jon The homology groups of a lattice J. Math. Mech.15 (1966) 631-636 Math Reviews MR0188116 · Zbl 0146.01602
[54] 57. Gabber, Ofer and Ramero, Lorenzo Almost ring theoryLecture Notes in Math., vol. 1800, Springer, 2003 Math Reviews MR2004652 (2004k:13027) · Zbl 1045.13002
[55] 58. Gabber, Ofer and Ramero, Lorenzo Foundations of almost ring theory http://math.univ-lille1.fr/##img##ramero/hodge.pdf
[56] 59. Haines, Thomas J. Introduction to Shimura varieties with bad reduction of parahoric type in Harmonic analysis, the trace formula, and Shimura varietiesClay Math. Proc.4 (2005) 583-642 Math Reviews MR2192017 · Zbl 1148.11028
[57] 60. Haines, Thomas J. The base change fundamental lemma for central elements in parahoric Hecke algebras Duke Math. J.149 (2009) 569-643 Math Reviews MR2553880 · Zbl 1194.22019
[58] 61. Haines, Thomas J. and Ngô, B. C. Nearby cycles for local models of some Shimura varieties Compos. math.133 (2002) 117-150 Math Reviews MR1923579 · Zbl 1009.11042
[59] 62. Haines, Thomas J. and Ngô, B. C. On the semi-simple local L-factors for some simple Shimura varieties in preparation
[60] 63. Haines, Thomas J. and Rapoport, M. Shimura varieties with_1(p)-level structure in preparation · Zbl 1337.11041
[61] 64. Harris, Michael and Taylor, Richard The geometry and cohomology of some simple Shimura varieties Annals of Math. Studies151 (2001) 276 Math Reviews MR1876802 · Zbl 1036.11027
[62] 65. Henniart, Guy Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique Invent. math.139 (2000) 439-455 Math Reviews MR1738446 (2001e:11052) · Zbl 1048.11092
[63] 66. Henniart, Guy La conjecture de Langlands locale numérique pour GL(n)Ann. Sci. École Norm. Sup.21 (1988) 497-544 Math Reviews MR982332 (90f:11094) · Zbl 0666.12013
[64] 67. Henniart, Guy Caractérisation de la correspondance de Langlands locale par les facteurs ##img## de paires Invent. math.113 (1993) 339-350 Math Reviews MR1228128 (96e:11078)
[65] 68. Dat, Jean François Espaces symétriques de Drinfeld et correspondance de Langlands locale Ann. Sci. École Norm. Sup.39 (2006) 1-74 Math Reviews MR2224658 · Zbl 1141.22004
[66] 69. Berkovich, Vladimir G. Vanishing cycles for formal schemes Invent. math.115 (1994) 539-571 Math Reviews MR1262943 · Zbl 0791.14008
[67] 70. Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces Publ. Math. IHÉS78 (1993) 5-161 Math Reviews MR1259429 · Zbl 0804.32019
[68] 71. Huber, Roland Étale cohomology of rigid analytic varieties and adic spacesAspects of Mathematics, E30, Friedr. Vieweg Sohn, 1996 Math Reviews MR1734903 (2001c:14046)
[69] 72. Huber, Roland Continuous valuations Math. Z.212 (1993) 455-477 Math Reviews MR1207303 (94e:13041) · Zbl 0788.13010
[70] 73. Huber, Roland A generalization of formal schemes and rigid analytic varieties Math. Z.217 (1994) 513-551 Math Reviews MR1306024 (95k:14001) · Zbl 0814.14024
[71] 74. Huber, Roland A finiteness result for direct image sheaves on the étale site of rigid analytic varieties J. Algebraic Geom.7 (1998) 359-403 Math Reviews MR1620118 (99h:14021) · Zbl 1013.14007
[72] 75. Iovita, Adrian and Spiess, Michael Logarithmic differential forms on p-adic symmetric spaces Duke Math. J.110 (2001) 253-278 Math Reviews MR1865241 (2002j:11055)
[73] 76. Jacquet, H. and Piatetski-Shapiro, I. I. and Shalika, J. Conducteur des représentations du groupe linéaire Math. Ann.256 (1981) 199-214 Math Reviews MR620708 (83c:22025) · Zbl 0443.22013
[74] 77. Katz, Nicholas M. and Mazur, Barry Arithmetic moduli of elliptic curvesAnnals of Math. Studies, vol. 108, Princeton Univ. Press, 1985 Math Reviews MR772569
[75] 78. Kazhdan, David Cuspidal geometry of p-adic groups J. Analyse Math.47 (1986) 1-36 Math Reviews MR874042 · Zbl 0634.22009
[76] 79. Kiehl, Reinhardt Der Endlichkeitssatz für eigentliche Abbildungen in der nichtarchimedischen Funktionentheorie Invent. math.2 (1967) 191-214 Math Reviews MR0210948 (35 #1833) · Zbl 0202.20101
[77] 80. Köpf, Ursula Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen Schr. Math. Inst. Univ. MünsterHeft 7 (1974) 72 Math Reviews MR0422671 (54 #10657) · Zbl 0275.14006
[78] 81. Kottwitz, Robert E. On the ##img##-adic representations associated to some simple Shimura varieties Invent. math.108 (1992) 653-665 Math Reviews MR1163241 · Zbl 0765.22011
[79] 82. Kottwitz, Robert E. Isomorphism classes of elliptic curves within an isogeny class over a finite field, unpublished notes
[80] 83. Kottwitz, Robert E. Points on some Shimura varieties over finite fields J. Amer. Math. Soc.5 (1992) 373-444 Math Reviews MR1124982 · Zbl 0796.14014
[81] 84. Kottwitz, Robert E. Shimura varieties and twisted orbital integrals Math. Ann.269 (1984) 287-300 Math Reviews MR761308 · Zbl 0533.14009
[82] 85. Langlands, Robert P. Base change for GL(2)Annals of Math. Studies, vol. 96, Princeton Univ. Press, 1980 Math Reviews MR574808 · Zbl 0444.22007
[83] 86. Langlands, Robert P. Representations of abelian algebraic groups Pacific J. Math.Special Issue (1997) 231-250 Olga Taussky-Todd: in memoriam Math Reviews MR1610871 (99b:11125)
[84] 87. Lütkebohmert, W. From Tate’s elliptic curve to abeloid varieties Pure Appl. Math. Q.5 (2009) 1385-1427 Math Reviews MR2560320 (2011c:14073) · Zbl 1185.14001
[85] 88. Mantovan, Elena On certain unitary group Shimura varieties Astérisque291 (2004) 201-331 Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales Math Reviews MR2074715 (2005g:11110c) · Zbl 1050.11002
[86] 89. Mantovan, Elena On the cohomology of certain PEL-type Shimura varieties Duke Math. J.129 (2005) 573-610 Math Reviews MR2169874 (2006g:11122) · Zbl 1112.11033
[87] 90. Mieda, Yoichi On l-independence for the étale cohomology of rigid spaces over local fields Compos. Math.143 (2007) 393-422 Math Reviews MR2309992 (2008d:14030) · Zbl 1132.14016
[88] 91. Rapoport, M. and Zink, Th. Period spaces for p-divisible groupsAnnals of Math. Studies, vol. 141, Princeton Univ. Press, Princeton, NJ, 1996 Math Reviews MR1393439 (97f:14023) · Zbl 0873.14039
[89] 92. Schneider, Peter The cohomology of local systems on p-adically uniformized varieties Math. Ann.293 (1992) 623-650 Math Reviews MR1176024 (93k:14032) · Zbl 0774.14022
[90] 93. Schneider, Peter and Zink, E.-W. K-types for the tempered components of a p-adic general linear group J. reine angew. Math.517 (1999) 161-208 Math Reviews MR1728541 · Zbl 0934.22021
[91] 94. Scholze, Peter The Langlands-Kottwitz approach for the modular curve (2010) · Zbl 1309.14019
[92] 96. Scholze, Peter The Langlands-Kottwitz approach and deformation spaces of p-divisible groups (2010) in preparation · Zbl 1383.11082
[93] 98. Shin, Sug Woo Counting points on Igusa varieties Duke Math. J.146 (2009) 509-568 Math Reviews MR2484281 · Zbl 1218.11061
[94] 99. Strauch, Matthias On the Jacquet-Langlands correspondence in the cohomology of the Lubin-Tate deformation tower Astérisque298 (2005) 391-410 Automorphic forms. I Math Reviews MR2141708
[95] 100. Tate, J. T. p-divisiblegroups.in Proc. Conf. Local Fields (Driebergen, 1966)(1967) 158-183 Math Reviews MR0231827 (38 #155)
[96] 101. Temkin, M. On local properties of non-Archimedean analytic spaces Math. Ann.318 (2000) 585-607 Math Reviews MR1800770 (2001m:14037) · Zbl 0972.32019
[97] 102. Thomason, R. Absolute cohomological purity Bull. Soc. Math. France112 (1984) 397-406 Math Reviews MR794741 · Zbl 0584.14007
[98] 103. Bushnell, Colin J. and Henniart, Guy The local Langlands conjecture for GL(2)Grundl. math. Wiss., vol. 335, Springer, Berlin, 2006 Math Reviews MR2234120 (2007m:22013) · Zbl 1100.11041
[99] 104. Kisin, Mark The Fontaine-Mazur conjecture for GL_2J. Amer. Math. Soc.22 (2009) 641-690 Math Reviews MR2505297 (2010j:11084)
[100] 105. Taylor, Richard and Wiles, Andrew Ring-theoretic properties of certain Hecke algebras Ann. of Math.141 (1995) 553-572 Math Reviews MR1333036 (96d:11072) · Zbl 0823.11030
[101] 106. Taylor, Richard On the meromorphic continuation of degree two L-functions Doc. Math.Extra Vol. (2006) 729-779 Math Reviews MR2290604 (2008c:11154)
[102] 107. Varshavsky, Yakov p-adic uniformization of unitary Shimura varieties. II J. Differential Geom.49 (1998) 75-113 Math Reviews MR1642109 (2000e:14030) · Zbl 0993.14008
[103] 108. Zelevinsky, A. V. Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n)Ann. Sci. École Norm. Sup.13 (1980) 165-210 Math Reviews MR584084 (83g:22012) · Zbl 0441.22014
[104] 109. Théorie des topos et cohomologie étale des schémas. Tome 3Lecture Notes in Math., vol. 305, Springer, 1973 Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat Math Reviews MR0354654
[105] 110. Cohomologie étaleLecture Notes in Math., vol. 569, Springer, 1977 Math Reviews MR0463174
[106] 111. Groupes de monodromie en géométrie algébrique. ILecture Notes in Math., vol. 288, Springer, 1972 Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim Math Reviews MR0354656
[107] 112. Groupes de monodromie en géométrie algébrique. IILecture Notes in Math., vol. 340, Springer, 1973 Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz Math Reviews MR0354657
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.