In fact it is enough to prove this for one such X, since any two X have the same fundamental group. If p = 5 or 7 then X is known to be rational and therefore simply connected. For p = 5 this goes back to [HM]; for p = 7 it has recently been shown in [MS] that X is birationally equivalent to a Fano variety of type V22, which is known to be rational. It does… (More)