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 wit
h tensor products. We finally verify our criterion in three cases, one of which is new.
We completely classify all subbundles of a given vector bundle on the Fargues-Fontaine curve. Our classification is given in terms of a simple and explicit condition on Harder-Narasimhan polygons. Our proof is inspired by the proof of the main theore
m in [Hon19], but also involves a number of nontrivial adjustments.
We show that the Kisin varieties associated to simple $phi$-modules of rank $2$ are connected in the case of an arbitrary cocharacter. This proves that the connected components of the generic fiber of the flat deformation ring of an irreducible $2$-d
imensional Galois representation of a local field are precisely the components where the multiplicities of the Hodge-Tate weights are fixed.
Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A in operatorname{M}_{2g}(mathbb{Z})$ such that each Tate module $T_ell X$ has a $mathbb{Z}_ell$-basis on which the action of $u$ is given by $A$.
We consider stacks of filtered phi-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack
containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrizing integral data and determine the image of this morphism.