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Let   k   be a perfect field of characteristic    p  >  2    and   K   an extension of    F  =  Frac  W  (  k  )    contained in some    F  (   μ   p  r    )   . Using crystalline Dieudonné theory, we provide a classification of   p  ‐divisible groups over    R  =   O  K    [   [   t  1   ,  …  ,   t  d   ]   ]in terms of finite height    (  φ  ,  Γ  )   ‐modules over    S  :  =  W   (  k  )   [   [  u  ,   t  1   ,  …  ,   t  d   ]   ]   . 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.

Dieudonné crystals and Wach modules for p ‐divisible groups