The subject matter of the talk are the non-archimedean analogues of
the classical period domains studied by Griffiths, Schmid and others.
Whereas the classical period domains classify Hodge structures of a
fixed type (which are objects arising from the study of the singular
cohomology groups of algebraic varieties over the field of complex
numbers), their $p$-adic analogoues classify weakly admissible
filtered isocrystals of a fixed type (wich are objects arising from
the $p$-adic cohomology groups of algebraic varieties over a $p$-adic
field). The $p$-adic period domains are rigid-analytic open subsets of
homogeneous projective algebraic varieties under linear algebraic
groups, the simplest example of which was found by Drinfeld,
$$\Omega=\{x\in{\bf P}^n;\ x\text{ not on any ${\bf Q}_p$-rational
hyperplane}\}.$$
I will discuss the geometry of these domains and a conjecture on the
existence of non-trivial étale coverings of them. I will survey
results in this direction, obtained jointly with Th. Zink, in the
cases related to $p$-divisible groups and discuss examples due to
Swork, Drinfeld, Lubin and Tate, Gross and Hopkins, and van der Put
and Voskuil.