On a Conjecture of Rapoport and Zink

In their book Rapoport and Zink constructed rigid analytic period spaces F for Fontaine’s filtered isocrystals and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism F → F of rigid analytic spaces and of interesting local systems of Qp-vector spaces on F. For those period spaces possessing period morphisms de Jong pointed out that one may take F as the image of the period morphism, viewed as a morphism of Berkovich spaces, and take the rational Tate module of the universal p-divisible group as the desired local system on F. In this article we construct for Hodge-Tate weights 0 and 1 an intrinsic Berkovich open subspace of F through which the period morphism factors and which we conjecture to be the image of the period morphism. We present indications supporting our conjecture and we show that only in exceptional cases our open subspace equals all of F. Mathematics Subject Classification (2000): 11S20, (14G22, 14L05, 14M15)