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

Liu, Tong; Moon, Yong Suk

doi:10.2140/ant.2020.14.2773

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

Tong Liu and Yong Suk Moon

Vol. 14 (2020), No. 10, 2773–2789

DOI: 10.2140/ant.2020.14.2773

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 $R = R_{0} \otimes_{W (k)} 𝒪_{K}$ . We show that if $e < p - 1$ , then every crystalline representation of $π_{1}^{\overset{́}{e} t} (Spec R [\frac{1}{p}])$ with Hodge–Tate weights in $[0, 1]$ arises from a $p$ -divisible group over $R$ .

Keywords

crystalline representation, $p$-divisible group, relative $p$-adic Hodge theory

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11S20, 14L05

Milestones

Received: 2 October 2019

Revised: 14 May 2020

Accepted: 24 June 2020

Published: 19 November 2020

Authors

Tong Liu
Purdue University West Lafayette, IN United States

Yong Suk Moon
University of Arizona Tucson, AZ United States

Vol. 14, No. 10, 2020