Hodge-Newton filtration for $p$-divisible groups with ramified endomorphism structure.

Let $\mathcal{O}_K$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ with perfect residue field. We prove the existence of the Hodge-Newton filtration for $p$-divisible groups over $\mathcal{O}_K$ with additional endomorphism structure for the ring of integers of a finite, possibly ramified field extension of $\mathbb{Q}_p$. The argument is based on the Harder-Narasimhan theory for finite flat group schemes over $\mathcal{O}_K$. In particular, we describe a sufficient condition for the existence of a filtration of $p$-divisible groups over $\mathcal{O}_K$ associated to a break point of the Harder-Narasimhan polygon.