  We compute the p-torsion and p-adic etale cohomologies with compact support
of period domains over local fields in the case of basic isocrystals for
quasi-split reductive groups. For the p-torsion case, we follow the method used
by Orlik in his computations of the l-torsion etale cohomology using as a key
new ingredient the computation of Ext groups between mod p generalized
Steinberg representations of p-adic groups. For the p-adic case, we don't use
Huber's definition of etale cohomology with compact support as Orlik did since
it seems to give spaces that are much too big; instead we use continuous etale
cohomology with compact support.
