Abstract
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.
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References
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)
Chaoha, P., Phon-on, A.: A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 320(2), 983–987 (2006)
Chen, M., Viehmann, E.: Affine Deligne–Lusztig varieties and the action of \(J\). J. Algebraic Geom. 27(2), 273–304 (2018)
Colmez, P., Fontaine, J.-M.: Construction des représentations \(p\)-adiques semi-stables. Invent. Math. 140(1), 1–43 (2000)
Cornut, C.: Filtrations and Buildings. To appear in Memoirs of the AMS
Cornut, C.: A fixed point theorem in Euclidean buildings. Adv. Geom. 16(4), 487–496 (2016)
Cornut, C., Nicole, M.-H.: Cristaux et immeubles. Bull. Soc. Math. France 144(1), 125–143 (2016)
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)
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)
Fargues, L.: Théorie de la réduction pour les groupes p-divisibles (Preprint)
Fontaine, J.-M., Laffaille, G.: Construction de représentations \(p\)-adiques. Ann. Sci. École Norm. Sup. (4) 15(4), 547–608 (1983). 1982
Fontaine, J.-M., Rapoport, M.: Existence de filtrations admissibles sur des isocristaux. Bull. Soc. Math. France 133(1), 73–86 (2005)
Gashi, Q.R.: On a conjecture of Kottwitz and Rapoport. Ann. Sci. Éc. Norm. Supér. (4) 43(6), 1017–1038 (2010)
Kapovich, M.: Generalized triangle inequalities and their applications. In: International Congress of Mathematicians. Vol. II, pp. 719–741. Eur. Math. Soc., Zürich (2006)
Kottwitz, R.E.: Isocrystals with additional structure. Compos. Math. 56(2), 201–220 (1985)
Kottwitz, R.E.: On the Hodge-Newton decomposition for split groups. Int. Math. Res. Not. 26, 1433–1447 (2003)
Kumar, S.: A survey of the additive eigenvalue problem. Transform. Groups 19(4), 1051–1148 (2014). (With an appendix by M. Kapovich)
Laffaille, G.: Groupes \(p\)-divisibles et modules filtrés: le cas peu ramifié. Bull. Soc. Math. France 108(2), 187–206 (1980)
Landvogt, E.: Some functorial properties of the Bruhat–Tits building. J. Reine Angew. Math. 518, 213–241 (2000)
Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103(2), 153–181 (1996)
Rapoport, M., Zink, T.H.: Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)
Rapoport, M., Zink, T.: A finiteness theorem in the Bruhat–Tits building: an application of Landvogt’s embedding theorem. Indag. Math. (N.S.) 10(3), 449–458 (1999)
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)
Totaro, B.: Tensor products in \(p\)-adic Hodge theory. Duke Math. J. 83(1), 79–104 (1996)
Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of GU\((1, n-1)\) II. Invent. Math. 184, 591–627 (2011)
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Communicated by Toby Gee.
