# Dieudonné modules over rings of charateristic zero

Dear Colleagues,

would appreciate if you could recommend references, if such a theory exits, for the following question.

Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of an algebraic closure of $\mathbb{Q}_{p}$ ($p$-adic field). Is there any reference which describe group scheme $A[p]$ as it does the theory of the Dieudonné modules for schemes over fields or over a ring of characteristic zero with $p$ being a nilpotent.

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Dear Alexey, Can you say more about $R$. For example, if the ring $R$ is the ring of integers in a finite extension of $\mathbb Q_p$, then the answer is "yes". More generally, if the ring $R$ is $p$-adically complete, you can hope to say something. Regards, –  Emerton Oct 21 '12 at 13:27
Dear Emerton, thank you for your comment! The most interesting case for me now, when R is the ring of integers of a finite extension of $\mathbb{Q}_p$. And I would be glad to any information for a $p-$addically complete case. –  Alexey Zaytsev Oct 21 '12 at 15:14
The literature on this subject is rather vast. Some names worth checking are Fontaine, Raynaud, Zink, Breuil, Kisin, Vasiu... If $R$ is an unramified ring of integers, then the description is due to Fontaine (in terms of 'finite Honda systems'; see B. Conrad's article here: math.stanford.edu/~conrad/papers/gpscheme.pdf. For other rings of integers, there is a complete solution conjectured by Breuil and proved by Kisin in 'Moduli of finite flat group schemes and modularity'. –  Keerthi Madapusi Pera Oct 21 '12 at 21:35
The case when $R$ is an unramified extension of $\mathbb{Z}_p$ is well understood; for example, there is a book of Fontain written in 1977; a certain generalization could be found in the Brian's Conrad Compositio paper. The situation becomes more complicated when the ramification index is $\ge p-1$; then you should consider either Breuil's modules or Cartier modules. –  Mikhail Bondarko Oct 21 '12 at 21:36
Dear Keerthi and Mikhail, thank you very much! –  Alexey Zaytsev Oct 22 '12 at 5:48