Dieudonné crystals and Wach modules for p‐divisible groups

Let k be a perfect field of characteristic p>2 and K an extension of F=FracW(k) contained in some F(μpr). Using crystalline Dieudonné theory, we provide a classification of p‐divisible groups over R=OK[[t1,…,td]] in terms of finite height (φ,Γ)‐modules over S:=W(k)[[u,t1,…,td]]. When d=0, such a...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2017-04, Vol.114 (4), p.733-763, Article 733
Main Authors: Cais, Bryden, Lau, Eike
Format: Article
Language:English
Subjects:
11F80 (secondary)
14F30 (primary)
14L05
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Summary:Let k be a perfect field of characteristic p>2 and K an extension of F=FracW(k) contained in some F(μpr). Using crystalline Dieudonné theory, we provide a classification of p‐divisible groups over R=OK[[t1,…,td]] in terms of finite height (φ,Γ)‐modules over S:=W(k)[[u,t1,…,td]]. When d=0, such a classification is a consequence of (a special case of) the theory of Kisin–Ren; in this setting, our construction gives an independent proof of this result, and moreover 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.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms.12021