Dieudonne crystals and Wach modules for p-divisible fgroups

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 Dieudonné theory, we provide a classification of $p$-divisible groups over ${\mathcal{O}}_K$ in terms of finite height $(\phi ,\Gamma )$-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 Dieudonné crystal of a $p$-divisible group from the Wach module associated to its Tate module by Berger--Breuil or by Kisin--Ren.

References in corpus (2)

• Représentations galoisiennes p-adiques et (phi,tau)-modules
• Wach modules and critical slope p-adic L-functions

Cited by in corpus (2)

• Prismatic $G$-displays and descent theory
• The Geometry of Hida Families II: $\Lambda$-adic $(\phi ,\Gamma )$-modules and $\Lambda$-adic Hodge Theory