## Moduli of $$p$$-divisible groups.(English)Zbl 1349.14149

Summary: We prove several results about moduli spaces of $$p$$-divisible groups such as Rapoport-Zink spaces. Our main goal is to prove that Rapoport-Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of $$p$$-adic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport-Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings. For this, we give a general classification of $$p$$-divisible groups over $$\mathcal{O}_C$$, where $$C$$ is an algebraically closed complete extension of $$\mathbb{Q}_p$$, in the spirit of Riemann’s classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonné module functor for $$p$$-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).

### MSC:

 14L05 Formal groups, $$p$$-divisible groups 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 11G25 Varieties over finite and local fields 14D20 Algebraic moduli problems, moduli of vector bundles
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