Relative crystalline representations and $p$-divisible groups in the small ramification case

Abstract

Let $k$ be a perfect field of characteristic $p>2$, and let $K$ be a finite totally ramified extension over $W(k)[\frac{1}{p}]$ of ramification degree $e$. Let $R_0$ be a relative base ring over $W(k)⟨t_1^{\pm 1},\dots ,t_m^{\pm 1}⟩$ satisfying some mild conditions, and let $R=R_0{\otimes }_{W(k)}{\mathcal{O}}_K$. We show that if $e<p-1$, then every crystalline representation of ${\pi }_1^{\text{eˊt}}(\mathrm{Spec}R[\frac{1}{p}])$ with Hodge-Tate weights in $[0,1]$ arises from a $p$-divisible group over $R$.

19 pages; changed the title; added section 6 and more details