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Let $p$ be a prime number, let $\mathcal{O}_F$ be the ring of integers of afinite field extension $F$ of $\mathbb{Q}_p$ and let $\mathcal{O}_K$ be acomplete valuation ring of rank $1$ and mixed characterstic $(0,p)$. Weintroduce 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 theclassical Hodge polygon and only depends on the reduction of $(H,\iota)$ to theresidue field of $\mathcal{O}_K$. This is not the case in general, whence theattribute "integral". The new polygon lies between Fargues' Harder-Narasimhanpolygons of the $p$-power torsion parts of $H$ and another combinatorialinvariant of $(H,\iota)$ called the Pappas-Rapoport polygon. Furthermore, theintegral Hodge polygon behaves continuously in families over a $p$-adicanalytic space.

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