Nous exposons dans cette Note les résultats constituant la première partie d'un programme visant à généraliser les articles [5,7] et ainsi construire des correspondances de Langlands locales pour d'autres groupes que GLn (par exemple des groupes unitaires quasidéployés) dans la cohomologie ℓ-adique des espaces de Rapoport–Zink. La méthode consiste à comparer la cohomologie de ces objets locaux à celle d'objets globaux : les variétés de Shimura. Pour cela nous généralisons les suites spectrales établies dans [5] et [4]. Une partie de ces résultats est mentionnée dans [6].
In this Note we announce results concerning the first part of a programme intending to generalize the articles [5,7] and thus construct local Langlands correspondences for groups other than GLn (for example, quasisplit unitary groups) inside the ℓ adic cohomology of Rapoport–Zink spaces. The method consists in comparing the cohomology of these local objects with that of global objects: Shimura varieties. For this we generalize the spectral sequences constructed in [5] and [4]. A part of these results is quoted in [6].
Accepté le :
Publié le :
Laurent Fargues 1
@article{CRMATH_2002__334_9_739_0, author = {Laurent Fargues}, title = {Une suite spectrale de {Hochschild{\textendash}Serre} pour l'uniformisation de {Rapoport{\textendash}Zink}}, journal = {Comptes Rendus. Math\'ematique}, pages = {739--742}, publisher = {Elsevier}, volume = {334}, number = {9}, year = {2002}, doi = {10.1016/S1631-073X(02)02340-3}, language = {fr}, }
Laurent Fargues. Une suite spectrale de Hochschild–Serre pour l'uniformisation de Rapoport–Zink. Comptes Rendus. Mathématique, Volume 334 (2002) no. 9, pp. 739-742. doi : 10.1016/S1631-073X(02)02340-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02340-3/
[1] V.G. Berkovich, Étale cohomology for p-adic analytic spaces, Notes d'un exposé à Toulouse, Juin 1994
[2] Étale cohomology for non-archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math., Volume 78 (1993), pp. 5-161
[3] Vanishing cycles for formal schemes, Invent. Math., Volume 115 (1994) no. 3, pp. 539-571
[4] The trace formula and Drinfeld's upper halfplane, Duke Math. J., Volume 76 (1994) no. 2, pp. 467-481
[5] Supercuspidal representations in the cohomology of Drinfel'd upper half spaces; elaboration of Carayol's program, Invent. Math., Volume 129 (1997) no. 1, pp. 75-119
[6] Local Langlands correspondences and vanishing cycles on Shimura varieties, Proc. E.C.M., 2000
[7] M. Harris, R. Taylor, On the geometry and cohomology of some simple Shimura varieties, Ann. Math. Stud., à paraitre
[8] A comparison theorem for l-adic cohomology, Compositio Math., Volume 112 (1998) no. 2, pp. 217-235
[9] Isocrystals with additional structure, Compositio Math., Volume 56 (1985) no. 2, pp. 201-220
[10] Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444
[11] Isocrystals with additional structure. II, Compositio Math., Volume 109 (1997) no. 3, pp. 255-339
[12] Cohomologie, stabilisation et changement de base, Astérisque, Volume 257 (1999)
[13] The points on a Shimura variety modulo a prime of good reduction, The Zeta Functions of Picard Modular Surfaces, University Montréal, Montréal, PQ, 1992, pp. 151-253
[14] On the classification and specialization of F-isocrystals with additional structure, Compositio Math., Volume 103 (1996) no. 2, pp. 153-181
[15] Period Spaces for p-Divisible Groups, Ann. of Math. Stud., 141, Princeton University Press, Princeton, NJ, 1996
[16] Representation theory and sheaves on the Bruhat–Tits building, Inst. Hautes Études Sci. Publ. Math., Volume 85 (1997), pp. 97-191
[17] Th. Zink, On the slope filtration, Preprint
Cité par Sources :
Commentaires - Politique