The integral Hodge polygon for $p$-divisible groups with endomorphism structure

Abstract

Let $p$ be a prime number, let $\mathcal{O}_F$ be the ring of integers of a finite field extension $F$ of $\mathbb{Q}_p$ and let $\mathcal{O}_K$ be a complete valuation ring of rank $1$ and mixed characteristic $(0,p)$. We introduce and study the "integral Hodge polygon", a new invariant of $p$-divisible groups $H$ over $\mathcal{O}_K$ endowed with an action $\iota$ of $\mathcal{O}_F$. If $F|\mathbb{Q}_p$ is unramified, this invariant recovers the classical Hodge polygon and only depends on the reduction of $(H,\iota)$ to the residue field of $\mathcal{O}_K$. This is not the case in general, whence the attribute "integral". The new polygon lies between Fargues' Harder-Narasimhan polygons of the $p$-power torsion parts of $H$ and another combinatorial invariant of $(H,\iota)$ called the Pappas-Rapoport polygon. Furthermore, the integral Hodge polygon behaves continuously in families over a $p$-adic analytic space.