×

zbMATH — the first resource for mathematics

Mazur’s inequality and Laffaille’s theorem. (English) Zbl 07087162
Summary: We look at various questions related to filtrations in \(p\)-adic Hodge theory, using a blend of building and Tannakian tools. Specifically, Fontaine and Rapoport used a theorem of Laffaille on filtered isocrystals to establish a converse of Mazur’s inequality for isocrystals. We generalize both results to the setting of (filtered) \(G\)-isocrystals and also establish an analog of Totaro’s \(\otimes \)-product theorem for the Harder-Narasimhan filtration of Fargues.

MSC:
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
[2] Chaoha, P.; Phon-on, A., A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl., 320, 983-987, (2006) · Zbl 1101.54040
[3] Chen, M.; Viehmann, E., Affine Deligne-Lusztig varieties and the action of \(J\), J. Algebraic Geom., 27, 273-304, (2018) · Zbl 1408.14141
[4] Colmez, P.; Fontaine, J-M, Construction des représentations \(p\)-adiques semi-stables, Invent. Math., 140, 1-43, (2000) · Zbl 1010.14004
[5] Cornut, C.: Filtrations and Buildings. To appear in Memoirs of the AMS
[6] Cornut, C., A fixed point theorem in Euclidean buildings, Adv. Geom., 16, 487-496, (2016) · Zbl 1392.51002
[7] Cornut, C.; Nicole, M-H, Cristaux et immeubles, Bull. Soc. Math. France, 144, 125-143, (2016) · Zbl 1387.20023
[8] Dat, J.-F., Orlik, S., Rapoport, M.: Period domains over finite and \(p\)-adic fields. Cambridge Tracts in Mathematics, vol. 183. Cambridge University Press, Cambridge (2010) · Zbl 1206.14001
[9] Faltings, G.: Mumford-Stabilität in der algebraischen Geometrie. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 648-655. Birkhäuser, Basel (1995)
[10] Fargues, L.: Théorie de la réduction pour les groupes p-divisibles (Preprint)
[11] Fontaine, J-M; Laffaille, G., Construction de représentations \(p\)-adiques, Ann. Sci. École Norm. Sup. (4), 15, 547-608, (1983) · Zbl 0579.14037
[12] Fontaine, J-M; Rapoport, M., Existence de filtrations admissibles sur des isocristaux, Bull. Soc. Math. France, 133, 73-86, (2005) · Zbl 1073.14025
[13] Gashi, QR, On a conjecture of Kottwitz and Rapoport, Ann. Sci. Éc. Norm. Supér. (4), 43, 1017-1038, (2010) · Zbl 1225.14037
[14] Kapovich, M.: Generalized triangle inequalities and their applications. In: International Congress of Mathematicians. Vol. II, pp. 719-741. Eur. Math. Soc., Zürich (2006) · Zbl 1130.53037
[15] Kottwitz, RE, Isocrystals with additional structure, Compos. Math., 56, 201-220, (1985) · Zbl 0597.20038
[16] Kottwitz, RE, On the Hodge-Newton decomposition for split groups, Int. Math. Res. Not., 26, 1433-1447, (2003) · Zbl 1074.14016
[17] Kumar, S., A survey of the additive eigenvalue problem, Transform. Groups, 19, 1051-1148, (2014) · Zbl 1354.14074
[18] Laffaille, G., Groupes \(p\)-divisibles et modules filtrés: le cas peu ramifié, Bull. Soc. Math. France, 108, 187-206, (1980) · Zbl 0453.14021
[19] Landvogt, E., Some functorial properties of the Bruhat-Tits building, J. Reine Angew. Math., 518, 213-241, (2000) · Zbl 0937.20026
[20] Rapoport, M.; Richartz, M., On the classification and specialization of \(F\)-isocrystals with additional structure, Compos. Math., 103, 153-181, (1996) · Zbl 0874.14008
[21] Rapoport, M., Zink, T.H.: Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996) · Zbl 0873.14039
[22] Rapoport, M.; Zink, T., A finiteness theorem in the Bruhat-Tits building: an application of Landvogt’s embedding theorem, Indag. Math. (N.S.), 10, 449-458, (1999) · Zbl 1029.20016
[23] Tits, J.: Reductive groups over local fields. In: Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1. Proc. Sympos. Pure Math., XXXIII, pp. 29-69. Amer. Math. Soc., Providence, R.I. (1979)
[24] Totaro, B., Tensor products in \(p\)-adic Hodge theory, Duke Math. J., 83, 79-104, (1996) · Zbl 0873.14019
[25] Vollaard, I.; Wedhorn, T., The supersingular locus of the Shimura variety of GU\((1, n-1)\) II, Invent. Math., 184, 591-627, (2011) · Zbl 1227.14027
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.