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)[ 1 p ] of ramification degree e. Let R 0 be a relative base ring over W (k) t ±1 1 , . . . , t ±1 m satisfying some mild conditions, and let R = R 0 ⊗ W (k) O K . We show that if e < p − 1, then every crystalline representation of π ét 1 (SpecR[ 1 p ]) with Hodge-Tate weights in [0, 1] arises from a p-divisible group over R.