MODULI OF p-DIVISIBLE GROUPS

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, [Fal10]. For this, we give a general classification of p-divisible groups over OC, where C is an algebraically closed complete extension of Qp, in the spirit of Riemann’s classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonne ́ module functor for p-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective)