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Existence of \(F\)-crystals with supplementary structures. (Existence de \(F\)-cristaux avec structures supplémentaires.) (French. English summary) Zbl 1104.14013
Let \(k\) be an algebraically closed field of characteristic \(p\), let \(W(k)\) be the Witt vectors over \(k\), and let \(\sigma\) be the absolute Frobenius. A crystal is a \(W(k)\)-module \(M\) free of finite rank with a \(\sigma\)-semi-linear bijective map \(\phi : M[1/p] \to M[1/p]\). One can associate to such an object two polygons : the Newton polygon (coming from the Dieudonné-Manin decomposition of \(M[1/p]\)) and the Hodge polygon (coming from the elementary divisors of \(\phi(M) \subset M\)). It is a theorem of B. Mazur [Bull. Am. Math. Soc. 78, 653–667 (1972; Zbl 0258.14006)] that the Hodge polygon lies below the Newton polygon and that they have the same endpoints. Conversely, given two polygons satisfying Mazur’s condition, Kottwitz and Rapoport have shown that there exists a crystal having those two polygons as Newton and Hodge polygons.
The article under review is about the generalization of Kottwitz and Rapoport’s theorem to so-called \(G\)-isocrystals, where \(G\) is a connected reductive group over \(\mathbb{Q}_p\). The analogue of Mazur’s criterion in this situation is a theorem of M. Rapoport and M. Richartz [Compos. Math. 103, No. 2, 153–181 (1996; Zbl 0874.14008)], and the problem of the existence of isocrystals satisfying the criterion then becomes a problem about certain affine Deligne-Lusztig varieties being non-empty. The author states two conjectures of Kottwitz and Rapoport in the introduction, and proceeds to prove various cases of them. Some results had previously been obtained by R. Kottwitz and M. Rapoport [Comment. Math. Helv. 78, No. 1, 153–184 (2003; Zbl 1126.14023)] and J.-M. Fontaine and M. Rapoport [Bull. Soc. Math. Fr. 133, No. 1, 73–86 (2005; Zbl 1073.14025)] and C. Lucarelli (Leigh) [J. Inst. Math. Jussieu 3, No. 2, 165–183 (2004; Zbl 1054.14059); preprint

14F30 \(p\)-adic cohomology, crystalline cohomology
20G25 Linear algebraic groups over local fields and their integers
20G40 Linear algebraic groups over finite fields
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[1] A. Björner, Orderings of Coxeter groups, in: Combinatorics and algebra (Boulder, CO, 1983), Contemporary Mathematics, Vol. 34, American Mathematical Society, Providence, RI, 1984, pp. 172-195.
[2] A. Borel, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 126, 2nd Edition, Springer, New York, 1991. · Zbl 0726.20030
[3] Borel, A.; Tits, J., Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I, Invent. math, 12, 95-104, (1971) · Zbl 0238.20055
[4] Borovoi, M., Abelian Galois cohomology of reductive groups, Mem. amer. math. soc, 132, 626, viii+50, (1998) · Zbl 0918.20037
[5] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algébres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, Vol. 1337, Hermann, Paris, 1968. · Zbl 0186.33001
[6] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local, Inst. hautes études sci. publ. math, 41, 5-251, (1972) · Zbl 0254.14017
[7] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local. II. schémas en groupes, existence d’une donnée radicielle valuée, Inst. hautes études sci. publ. math, 60, 197-376, (1984)
[8] Colmez, P.; Fontaine, J.-M., Construction des représentations p-adiques semi-stables, Invent. math, 140, 1, 1-43, (2000) · Zbl 1010.14004
[9] P. Deligne, J. Milne, Tannakian Categories, Lectures Notes in Mathematics, Vol. 900, Springer, Berlin, 1982, Philosophical Studies Series in Philosophy, Vol. 20. · Zbl 0477.14004
[10] M. Demazure, A. Grothendieck (Eds.), Schémas en groupes. II: groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Mathematics, Vol. 152, Springer, Berlin, 1962/1964.
[11] G. Faltings, Mumford-Stabilität in der algebraischen Geometrie, in: Proceedings of the International Congress of Mathematicians, Vols. 1, 2, Zürich, 1994, Birkhäuser, Basel, 1995, pp. 648-655.
[12] Fontaine, J.-M., Représentations p-adiques semi-stables, Astérisque, 223, 113-184, (1994), With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988) · Zbl 0865.14009
[13] Fontaine, J.-M.; Laffaille, G., Construction de représentations p-adiques, Ann. sci. école norm. sup. (4), 15, 4, 547-608, (1983), (1982) · Zbl 0579.14037
[14] J.-M. Fontaine, R. Rapoport, Existence de filtrations admissibles sur des isocristaux, Preprint Mathematisches Institut des Universität zu Köln, à paraı̂tre au Bull. Soc. Math. France (2002) 1-10.
[15] Gel’fand, I.M.; Ponomarev, V.A., Indecomposable representations of the Lorentz group, Russ. math. surv, 23, 1-58, (1968) · Zbl 0236.22012
[16] Gross, B.H., Modular forms (mod p) and Galois representations, Internat. math. res. notices, 16, 865-875, (1998) · Zbl 0978.11018
[17] T. Haines, R. Rapoport, Papier en préparation.
[18] Iwahori, N.; Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Inst. hautes études sci. publ. math, 25, 5-48, (1965) · Zbl 0228.20015
[19] Kottwitz, R.E., Isocrystals with additional structure, Compositio math, 56, 2, 201-220, (1985) · Zbl 0597.20038
[20] Kottwitz, R.E., Isocrystals with additional structure. II, Compositio math, 109, 3, 255-339, (1997) · Zbl 0966.20022
[21] Kottwitz, R.E., On the Hodge-Newton decomposition for split groups, Int. math. res. not, 26, 1433-1447, (2003) · Zbl 1074.14016
[22] Kottwitz, R.; Rapoport, M., On the existence of F-crystals, Comment. math. helv, 78, 1, 153-184, (2003) · Zbl 1126.14023
[23] H. Kraft, Kommutative algebraische p-Gruppen (mit Anwendungen auf p-divisible Gruppen und abelsche Varietäten), Manuscrit non publié, 1975.
[24] Laffaille, G., Groupes p-divisibles et modules filtrésle cas peu ramifié, Bull. soc. math. France, 108, 2, 187-206, (1980) · Zbl 0453.14021
[25] C. Leigh, A converse to Mazur’s inequality for split clasical groups, arXiv:math.NT/0211327, 2002, pp. 1-16.
[26] Mazur, B., Frobenius and the Hodge filtration, Bull. amer. math. soc, 78, 653-667, (1972) · Zbl 0258.14006
[27] B. Moonen, Group schemes with additional structures and Weyl group cosets, in: Moduli of Abelian Varieties (Texel Island, 1999), Progress in Mathematics, Vol. 195, Birkhäuser, Basel, 2001, pp. 255-298. · Zbl 1084.14523
[28] M. Rapoport, A guide to the reduction modulo p of Shimura varieties, preprint, 02. · Zbl 1084.11029
[29] Rapoport, M., A positivity property of the Satake isomorphism, Manuscripta math, 101, 2, 153-166, (2000) · Zbl 0941.22006
[30] Rapoport, M.; Richartz, M., On the classification and specialization of F-isocrystals with additional structure, Compositio math, 103, 2, 153-181, (1996) · Zbl 0874.14008
[31] M. Rapoport, Th. Zink, Period spaces for p-divisble groups, Annals of Mathematics Studies, Vol. 141, Princeton University Press, Princeton, NJ, 1996. · Zbl 0873.14039
[32] D.C. Reuman, Determining whether certain affine Deligne-Lusztig sets are empty, arXiv:math.RT/0211434, 2002, pp. 1-135.
[33] N.S. Rivano, Catégories Tannakiennes, Lecture Notes in Mathematics, Vol. 265, Springer, Berlin, 1972. · Zbl 0241.14008
[34] J.-P. Serre, Groupes algébriques associés aux modules de Hodge-Tate, in: Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. III, Astérisque, Vol. 65, Society of Mathematics, France, Paris, 1979, pp. 155-158.
[35] J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics, 5th Edition, Vol. 5, Springer, Berlin, 1994.
[36] J. Tits, Reductive groups over local fields, in: Automorphic forms, representations and L-functions, Proceedings of the Symposium on Pure Mathematics, Oregon State University, Corvallis, OR, 1977, Part 1, American Mathematical Society, Providence, RI, 1979, pp. 29-69.
[37] J.-P. Wintenberger, Un scindage de la filtration de Hodge pour certaines variétés algébriques sur les corps locaux, Ann. of Math. (2) (1984) 119(3) 511-548. · Zbl 0599.14018
[38] Wintenberger, J.-P., Propriétés du groupe tannakien des structures de Hodge p-adiques et torseur entre cohomologies cristalline et étale, Ann. inst. Fourier (Grenoble), 47, 1289-1334, (1997) · Zbl 0888.14008
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