Deformation spaces of one-dimensional formal modules and their cohomology. (English) Zbl 1140.22017

Strauch, Matthias

Adv. Math. 217, No. 3, 889-951 (2008).

From the abstract: Let \(\mathcal M _m\) be the formal scheme which represents the functor of deformations of a one-dimensional formal module over \(\bar {\mathbb F}_p\) equipped with a level-\(m\)-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the \(\ell \)-adic étale cohomology of the generic fibre \(M_m\) of \(\mathcal M _m\) realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper the author shows without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces \(M_m\).

Reviewer: Samuel James Patterson (Göttingen)

MSC:

22E50	Representations of Lie and linear algebraic groups over local fields
20G25	Linear algebraic groups over local fields and their integers
14G35	Modular and Shimura varieties
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

local Langlands correspondence; Jacquet-Langlands correspondence; Lubin-Tate spaces; Drinfeld level structures; Lefschetz trace formula; rigid analytic geometry

References:

[1]	Artin, M., Algebraic approximation of structures over complete local rings, Inst. hautes études sci. publ. math., 36, 23-58, (1969) · Zbl 0181.48802
[2]	Artin, M., Algebraization of formal moduli: I, (), 21-71 · Zbl 0205.50402
[3]	Badulescu, A.I., Orthogonalité des caractères pour \(\operatorname{GL}_n\) sur un corps local de caractéristique non nulle, Manuscripta math., 101, 1, 49-70, (2000) · Zbl 0957.20027
[4]	Berkovich, V.G., Spectral theory and analytic geometry over non-Archimedean fields, Math. surveys monogr., vol. 33, (1990), American Mathematical Society Providence, RI · Zbl 0715.14013
[5]	Berkovich, V.G., Étale cohomology for non-Archimedean analytic spaces, Inst. hautes études sci. publ. math., 78, 5-161, (1993), (1994) · Zbl 0804.32019
[6]	Berkovich, V.G., Vanishing cycles for formal schemes, Invent. math., 115, 3, 539-571, (1994) · Zbl 0791.14008
[7]	Berkovich, V.G., Vanishing cycles for formal schemes. II, Invent. math., 125, 2, 367-390, (1996) · Zbl 0852.14002
[8]	P. Berthelot, Cohomologie rigide et cohomologie rigde à support propre, Première partie, Prépublication IRMAR 96-03, 1996
[9]	Boyer, P., Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. math., 138, 3, 573-629, (1999) · Zbl 1161.11408
[10]	Bombosch, U., Der Zariski-Riemann-raum und die étale kohomologie rigider Räume, () · Zbl 0904.32029
[11]	Bourbaki, N., Commutative algebra, chapters 1-7, Elements math., (1989), Springer-Verlag Berlin · Zbl 0666.13001
[12]	Bushnell, C.J., Induced representations of locally profinite groups, J. algebra, 134, 1, 104-114, (1990) · Zbl 0727.22004
[13]	Bushnell, C.J.; Kutzko, P.C., The admissible dual of \(\operatorname{GL}(N)\) via compact open subgroups, Ann. of math. stud., vol. 129, (1993), Princeton University Press Princeton, NJ · Zbl 0787.22016
[14]	Carayol, H., Sur la mauvaise réduction des courbes de Shimura, Compos. math., 59, 2, 151-230, (1986) · Zbl 0607.14021
[15]	Carayol, H., Sur LES représentations ℓ-adiques associées aux formes modulaires de Hilbert, Ann. sci. école norm. sup. (4), 19, 3, 409-468, (1986) · Zbl 0616.10025
[16]	Carayol, H., Nonabelian Lubin-Tate theory, (), 15-39 · Zbl 0704.11049
[17]	Casselman, W., Introduction of the theory of admissible representations of p-adic reductive groups, unpublished manuscript, draft May 1, 1995, available at
[18]	Chai, C.-L., The group action on the closed fiber of the Lubin-Tate moduli space, Duke math. J., 82, 3, 725-754, (1996) · Zbl 0864.14028
[19]	Corwin, L., A construction of the supercuspidal representations of \(\operatorname{GL}_n(F)\), fp-adic, Trans. amer. math. soc., 337, 1, 1-58, (1993) · Zbl 0789.22032
[20]	Deligne, P., Théorèmes de finitude en cohomologie ℓ-adique, () · Zbl 0349.14013
[21]	Deligne, P.; Kazhdan, D.; Vignéras, M.-F., Représentations des algèbres centrales simples p-adiques, (), 33-117 · Zbl 0583.22009
[22]	Drinfeld, V.G., Elliptic modules, Math. USSR-sb., 23, 4, 561-592, (1974), English translation: · Zbl 0321.14014
[23]	Drinfeld, V.G., Coverings of p-adic symmetric domains, Funct. anal. appl., 10, 107-115, (1976) · Zbl 0346.14010
[24]	Endler, O., Valuation theory, Universitext, (1972), Springer-Verlag New York-Heidelberg · Zbl 0257.12111
[25]	Faltings, G., The trace formula and Drinfeld’s upper halfplane, Duke math. J., 76, 2, 467-481, (1994) · Zbl 0855.11061
[26]	Faltings, G., Group schemes with strict \(\mathcal{O}\)-action, Mosc. math. J., 2, 2, 249-279, (2002) · Zbl 1013.11079
[27]	L. Fargues, Manuscript on generalized canonical subgroups, 2004
[28]	Fargues, L., Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales, Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales, Astérisque, 291, 1-199, (2004) · Zbl 1196.11087
[29]	Fujiwara, K., Theory of tubular neighborhood in étale topology, Duke math. J., 80, 1, 15-57, (1995) · Zbl 0872.14014
[30]	Fujiwara, K., Rigid geometry, Lefschetz-verdier trace formula and Deligne’s conjecture, Invent. math., 127, 3, 489-533, (1997) · Zbl 0920.14005
[31]	Freitag, E.; Kiehl, R., Étale cohomology and the Weil conjecture, Ergeb. math. grenzgeb. (3), vol. 13, (1988), Springer-Verlag Berlin · Zbl 0643.14012
[32]	Harris, M., Supercuspidal representations in the cohomology of Drinfeld upper half spaces; elaboration of Carayol’s program, Invent. math., 129, 1, 75-119, (1997) · Zbl 0886.11029
[33]	Harris, M., The local Langlands conjecture for \(\operatorname{GL}(n)\) over a p-adic field, n<p, Invent. math., 134, 1, 177-210, (1998) · Zbl 0921.11060
[34]	Harris, M.; Taylor, R., The geometry and cohomology of some simple Shimura varieties, Ann. of math. stud., vol. 151, (2001), Princeton University Press Princeton, NJ · Zbl 1036.11027
[35]	Hausberger, Th., Uniformisation des variétés de laumon-Rapoport-stuhler et conjecture de Drinfeld-carayol, Ann. inst. Fourier (Grenoble), 55, 4, 1285-1371, (2005) · Zbl 1138.11329
[36]	Henniart, G., Correspondance de Langlands-Kazhdan explicite dans le cas non ramifié, Math. nachr., 158, 7-26, (1992) · Zbl 0777.11048
[37]	Henniart, G., Correspondance de Jacquet-Langlands explicite. I. le cas modéré de degré premier, (), 85-114 · Zbl 0961.11018
[38]	Hopkins, M.J.; Gross, B.H., Equivariant vector bundles on the Lubin-Tate moduli space, (), 23-88 · Zbl 0807.14037
[39]	Huber, R., Bewertungsspektrum und rigide geometrie, Regensburger mathematische schriften, vol. 23, (1993), Universität Regensburg Fachbereich Mathematik, Regensburg · Zbl 0806.13001
[40]	Huber, R., Continuous valuations, Math. Z., 212, 3, 455-477, (1993) · Zbl 0788.13010
[41]	Huber, R., A generalization of formal schemes and rigid analytic varieties, Math. Z., 217, 4, 513-551, (1994) · Zbl 0814.14024
[42]	Huber, R., Étale cohomology of rigid analytic varieties and adic spaces, Aspects math., vol. E30, (1996), Friedr. Vieweg & Sohn Braunschweig · Zbl 0868.14010
[43]	Huber, R., A finiteness result for direct image sheaves on the étale site of rigid analytic varieties, J. algebraic geom., 7, 2, 359-403, (1998) · Zbl 1013.14007
[44]	Huber, R., Swan representations associated with rigid analytic curves, J. reine angew. math., 537, 165-234, (2001) · Zbl 0979.14012
[45]	Illusie, L.; Szpiro, Lucien, Déformations de groupes de Barsotti-Tate (d’après A. Grothendieck), Séminaire sur LES pinceaux arithmétiques: la conjecture de Mordell, Astérisque, 127, 151-198, (1985) · Zbl 1182.14050
[46]	de Jong, A.J.; Put, M.v.d., Étale cohomology of rigid analytic spaces, Doc. math., 1, 01, 1-56, (1996) · Zbl 0922.14012
[47]	Kazhdan, D., Cuspidal geometry of p-adic groups, J. anal. math., 47, 1-36, (1986) · Zbl 0634.22009
[48]	Lemaire, B., Intégrabilité locale des caractères-distributions de \(\operatorname{GL}_N(F)\) oùF est un corps local non-archimédien de caractéristique quelconque, Compos. math., 100, 1, 41-75, (1996) · Zbl 0856.22024
[49]	Lubin, J., Finite subgroups and isogenies of one-parameter formal Lie groups, Ann. of math. (2), 85, 296-302, (1967) · Zbl 0166.02803
[50]	Lubin, J.; Tate, J.T., Formal moduli for one-parameter formal Lie groups, Bull. soc. math. France, 94, 49-59, (1966) · Zbl 0156.04105
[51]	Messing, W., The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture notes in math., vol. 264, (1972), Springer-Verlag Berlin-New York · Zbl 0243.14013
[52]	Milne, J.S., Étale cohomology, Princeton math. ser., vol. 33, (1980), Princeton University Press Princeton, NJ · Zbl 0433.14012
[53]	Put, M.v.d.; Schneider, P., Points and topologies in rigid geometry, Math. ann., 302, 1, 81-103, (1995) · Zbl 0867.11049
[54]	Rapoport, M.; Zink, Th., Period spaces for p-divisible groups, Ann. of math. stud., vol. 141, (1996), Princeton University Press Princeton, NJ · Zbl 0873.14039
[55]	Rogawski, J.D., Representations of \(\operatorname{GL}(n)\) and division algebras over a p-adic field, Duke math. J., 50, 1, 161-196, (1983) · Zbl 0523.22015
[56]	Sally, P.J., Some remarks on discrete series characters for reductive p-adic groups, (), 337-348
[57]	Cohomologie ℓ-adique et fonctions L, ()
[58]	Strauch, M., On the Jacquet-Langlands correspondence in the cohomology of the Lubin-Tate deformation tower. automorphic forms. I, Astérisque, 298, 391-410, (2005) · Zbl 1073.22010
[59]	Wewers, S., Swan conductors on the boundary of Lubin-Tate spaces, arXiv:
[60]	S. Wewers, Non-abelian Lubin-Tate theory via stable reduction, preprint, January 2006
[61]	Yoshida, T., On non-abelian Lubin-Tate theory via vanishing cycles, Ann. Inst. Fourier, in press · Zbl 1257.11103
[62]	Yu, J.-K., On the moduli of quasi-canonical liftings, Compos. math., 96, 3, 293-321, (1995) · Zbl 0866.14029