Mazur's inequality and laffaille's theorem

TL;DR

This paper generalizes key theorems in p-adic Hodge theory, extending Mazur's inequality and Laffaille's theorem to G-isocrystals and establishing an analog of Totaro's tensor product theorem for the Harder-Narasimhan filtration.

Contribution

It broadens the scope of Mazur's inequality and Laffaille's theorem to G-isocrystals and introduces a new tensor product theorem for the Harder-Narasimhan filtration.

Findings

Generalization of Mazur's inequality to G-isocrystals

Extension of Laffaille's theorem to filtered G-isocrystals

Establishment of an analog of Totaro's tensor product theorem

Abstract

We look at various questions related to filtrations in $p$-adic Hodgetheory, using a blend of building and Tannakian tools. Specifically,Fontaine and Rapoport used a theorem of Laffaille on filtered isocrystalsto establish a converse of Mazur's inequality for isocrystals. Wegeneralize both results to the setting of (filtered) $G$-isocrystalsand also establish an analog of Totaro's $\otimes$-product theoremfor the Harder-Narasimhan filtration of Fargues.