no. 6
Un cas PEL de la conjecture de Kottwitz
Nguyen, Kieu Hieu1
1 University of Münster (WWU) Einsteinstrasse, 62. 48149 Münster (Germany)
Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2239-2304.

La conjecture de Kottwitz décrit la cohomologie des espaces de Rapoport–Zink basiques à l’aide des correspondances de Langlands locales. Dans cet article, par voie globale via l’étude de la géométrie de certaines variétés de Shimura de type Kottwitz, on prouve cette conjecture pour des espaces de Rapoport–Zink de type PEL unitaires non ramifiés simples basiques de signature (1,n-1).

The Kottwitz conjecture describes the cohomology of basic Rapoport–Zink spaces using local Langlands correspondences. In this paper, via geometrical studies of some Kottwitz-type Shimura varieties, we prove this conjecture for basic simple unramified unitary PEL type Rapoport–Zink spaces of signature (1,n-1).

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3577
Classification : 11F70, 11F80, 11F85, 11G18, 20C08
Mot clés : Espaces de Rapoport–Zink, Conjecture de Kottwitz, Formes automorphes.
Keywords: Rapoport–Zink spaces, Kottwitz conjecture, Automorphic forms.

Nguyen, Kieu Hieu 1

1 University of Münster (WWU) Einsteinstrasse, 62. 48149 Münster (Germany)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Nguyen, Kieu Hieu. Un cas PEL de la conjecture de Kottwitz. Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2239-2304. doi : 10.5802/aif.3577. https://aif.centre-mersenne.org/articles/10.5802/aif.3577/

[1] Arthur, James Classifying automorphic representations, Current developments in mathematics 2012, International Press, Somerville, MA, 2013, pp. 1-58 | MR | Zbl

[2] Arthur, James The endoscopic classification of representations. Orthogonal and symplectic groups, American Mathematical Society Colloquium Publications, 61, American Mathematical Society, Providence, RI, 2013, xviii+590 pages | DOI | MR | Zbl

[3] Borel, A.; Casselman, W. L 2 -cohomology of locally symmetric manifolds of finite volume, Duke Math. J., Volume 50 (1983) no. 3, pp. 625-647 | DOI | MR | Zbl

[4] Borel, Armand Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974), pp. 235-272 | DOI | MR | Zbl

[5] Boyer, P. Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math., Volume 138 (1999) no. 3, pp. 573-629 | DOI | MR | Zbl

[6] Boyer, Pascal Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math., Volume 177 (2009) no. 2, pp. 239-280 | DOI | MR | Zbl

[7] Clozel, Laurent On limit multiplicities of discrete series representations in spaces of automorphic forms, Invent. Math., Volume 83 (1986) no. 2, pp. 265-284 | DOI | MR | Zbl

[8] On the stabilization of the trace formula (Clozel, Laurent; Harris, Michael; Labesse, Jean-Pierre; Ngô, Bao-Châu, eds.), Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, 1, International Press, Somerville, MA, 2011, xiv+527 pages | MR | Zbl

[9] Faltings, Gerd A relation between two moduli spaces studied by V. G. Drinfeld, Algebraic number theory and algebraic geometry (Contemp. Math.), Volume 300, American Mathematical Society, Providence, RI, 2002, pp. 115-129 | DOI | MR | Zbl

[10] Fargues, Laurent 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), Société Mathématique de France, 2004 no. 291, pp. 1-199 | MR | Zbl

[11] Fargues, Laurent; Genestier, Alain; Lafforgue, Vincent L’isomorphisme entre les tours de Lubin–Tate et de Drinfeld, Progress in Mathematics, 262, Birkhäuser Verlag, Basel, 2008, xxii+406 pages | MR | Zbl

[12] Guerberoff, L.; Lin, J. Galois Equivariance of critical values of L-Functions for Unitary groups, 2016

[13] Harris, Michael; Labesse, Jean-Pierre Conditional base change for unitary groups, Asian J. Math., Volume 8 (2004) no. 4, pp. 653-683 | DOI | MR | Zbl

[14] Harris, Michael; Taylor, Richard The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, 151, Princeton University Press, Princeton, NJ, 2001, viii+276 pages (With an appendix by Vladimir G. Berkovich) | MR | Zbl

[15] Henniart, Guy Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math., Volume 139 (2000) no. 2, pp. 439-455 | DOI | MR | Zbl

[16] Jacquet, H.; Shalika, J. A. On Euler products and the classification of automorphic representations. II, Amer. J. Math., Volume 103 (1981), pp. 777-815 | DOI | Zbl

[17] Kaletha, Tasho; Minguez, Alberto; Shin, Sug Woo; White, Paul-James Endoscopic Classification of Representations : Inner Forms of Unitary Groups (2014) (https://arxiv.org/abs/1409.3731)

[18] Kottwitz, Robert E. Stable trace formula : cuspidal tempered terms, Duke Math. J., Volume 51 (1984) no. 3, pp. 611-650 | DOI | MR | Zbl

[19] Kottwitz, Robert E. Points on some Shimura varieties over finite fields, J. Amer. Math. Soc., Volume 5 (1992) no. 2, pp. 373-444 | DOI | MR | Zbl

[20] Kottwitz, Robert E. Isocrystals with additional structure. II, Compos. Math., Volume 109 (1997) no. 3, pp. 255-339 | MR | Zbl

[21] Kottwitz, Robert E. B(G) for all local and global fields (2014) (https://arxiv.org/abs/1401.5728)

[22] Lan, Kai-Wen Compactifications of PEL-type Shimura varieties in ramified characteristics, Forum Math. Sigma, Volume 4 (2016), e1, 98 pages | DOI | MR | Zbl

[23] Lan, Kai-Wen; Stroh, Benoît Nearby cycles of automorphic étale sheaves, Compos. Math., Volume 154 (2018) no. 1, pp. 80-119 | DOI | MR | Zbl

[24] Langlands, R. P. Automorphic representations, Shimura varieties, and motives. Ein Märchen, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math., XXXIII), American Mathematical Society, Providence, R.I. (1979), pp. 205-246 | MR | Zbl

[25] Langlands, R. P. On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups (Math. Surveys Monogr.), Volume 31, American Mathematical Society, Providence, RI, 1989, pp. 101-170 | DOI | MR | Zbl

[26] Looijenga, E.; Rapoport, M. Weights in the local cohomology of a Baily–Borel compactification, Complex geometry and Lie theory (Sundance, UT, 1989) (Proc. Sympos. Pure Math.), Volume 53, American Mathematical Society, Providence, RI, 1991, pp. 223-260 | DOI | MR | Zbl

[27] Looijenga, Eduard L 2 -cohomology of locally symmetric varieties, Compos. Math., Volume 67 (1988) no. 1, pp. 3-20 | MR | Zbl

[28] Mantovan, Elena On the cohomology of certain PEL-type Shimura varieties, Duke Math. J., Volume 129 (2005) no. 3, pp. 573-610 | DOI | MR | Zbl

[29] Mantovan, Elena On non-basic Rapoport–Zink spaces, Ann. Sci. Éc. Norm. Supér. (4), Volume 41 (2008) no. 5, pp. 671-716 | DOI | MR | Zbl

[30] Mantovan, Elena l-adic étale cohomology of PEL type Shimura varieties with non-trivial coefficients, WIN – women in numbers (Fields Inst. Commun.), Volume 60, American Mathematical Society, Providence, RI, 2011, pp. 61-83 | MR | Zbl

[31] Mœglin, Colette Classification et changement de base pour les séries discrètes des groupes unitaires p-adiques, Pacific J. Math., Volume 233 (2007) no. 1, pp. 159-204 | DOI | MR | Zbl

[32] Mok, Chung Pang Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc., 235, American Mathematical Society, Providence, RI, 2015 no. 1108, vi+248 pages | DOI | MR | Zbl

[33] Mokrane, Abdellah; Tilouine, Jacques Cohomology of Siegel varieties with p-adic integral coefficients and applications, Cohomology of Siegel varieties (Astérisque), Société Mathématique de France, 2002 no. 280, pp. 1-95 | MR | Zbl

[34] Morel, Sophie On the cohomology of certain noncompact Shimura varieties, Annals of Mathematics Studies, 173, Princeton University Press, Princeton, NJ, 2010, xii+217 pages (With an appendix by Robert Kottwitz) | DOI | MR | Zbl

[35] Rapoport, M.; Zink, Th. Period spaces for p-divisible groups, Annals of Mathematics Studies, 141, Princeton University Press, Princeton, NJ, 1996, xxii+324 pages | DOI | MR | Zbl

[36] Rapoport, Michael Non-Archimedean period domains, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel (1995), pp. 423-434 | MR | Zbl

[37] Saper, Leslie; Stern, Mark L 2 -cohomology of arithmetic varieties, Ann. of Math. (2), Volume 132 (1990) no. 1, pp. 1-69 | DOI | MR | Zbl

[38] Scholze, Peter; Weinstein, Jared Berkeley lectures on p-adic geometry, 2020

[39] Shen, Xu On the Hodge–Newton filtration for p-divisible groups with additional structures, Int. Math. Res. Not. IMRN (2014) no. 13, pp. 3582-3631 | DOI | MR | Zbl

[40] Shin, Sug Woo Automorphic Plancherel density theorem, Israel J. Math., Volume 192 (2012) no. 1, pp. 83-120 | DOI | MR | Zbl

[41] Shin, Sug Woo On the cohomology of Rapoport–Zink spaces of EL-type, Amer. J. Math., Volume 134 (2012) no. 2, pp. 407-452 | DOI | MR | Zbl

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