  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 \'etale 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 $GL_n$ we determine all these cases.

Number Theory;Algebraic Geometry