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 |