On a Conjecture of Rapoport and Zink

Abstract

In their book Rapoport and Zink constructed rigid analytic period spaces $F^{wa}$ for Fontaine's filtered isocrystals, and period morphisms from PEL moduli spaces of $p$-divisible groups to some of these period spaces. They conjectured the existence of an étale bijective morphism $F^a\to F^{wa}$ of rigid analytic spaces and of a universal local system of $Q_p$-vector spaces on $F^a$. For Hodge-Tate weights $n-1$ and $n$ we construct in this article an intrinsic Berkovich open subspace $F^0$ of $F^{wa}$ and the universal local system on $F^0$. We conjecture that the rigid-analytic space associated with $F^0$ is the maximal possible $F^a$, and that $F^0$ is connected. We give evidence for these conjectures and we show that for those period spaces possessing PEL period morphisms, $F^0$ equals the image of the period morphism. Then our local system is the rational Tate module of the universal $p$-divisible group and enjoys additional functoriality properties. We show that only in exceptional cases $F^0$ equals all of $F^{wa}$ and when the Shimura group is $G{L}_n$ we determine all these cases.

v2: 48 pages; many new results added, v3: final version that will appear in Inventiones Mathematicae

References in corpus (4)

• Repr{é}sentations p-adiques et {é}quations diff{é}rentielles
• Limites de représentations cristallines
• A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic
• On period spaces for p-divisible groups

Cited by in corpus (2)

• On some generalized Rapoport-Zink spaces
• Relative crystalline representations and $p$-divisible groups in the small ramification case