Dieudonne crystals and Wach modules for p-divisible fgroups

TL;DR

This paper classifies p-divisible groups over certain local fields using crystalline Dieudonné theory and Wach modules, providing an independent proof and linking these structures explicitly.

Contribution

It offers an independent classification of p-divisible groups via Wach modules, connecting Dieudonné crystals with Kisin--Ren and Berger--Breuil theories.

Findings

Provides a classification of p-divisible groups over al_K.

Recovers Dieudonne9 crystals from Wach modules.

Establishes an explicit link between different theoretical frameworks.

Abstract

Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(\mu_{p^r})$. Using crystalline Dieudonn\'e theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(\varphi,\Gamma)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows us to recover the Dieudonn\'e crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.