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 Dieudonn√© module functor for \(p\)-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).


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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