On Harder-Narasimhan filtrations and their compatibility with tensor products

We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new

Mazur's inequality and laffaille's theorem

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

Liftings of Reduction Maps for Quaternion Algebras

We construct liftings of reduction maps from CM points to supersingular points for general quaternion algebras and use these liftings to establish a precise correspondence between CM points on indefinite quaternion algebras with a given conductor and CM points on certain corresponding totally definite quaternion algebras.Comment: 17 page