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

Abstract

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.