Mazur’s inequality and Laffaille’s theorem

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.