Line bundles on the first Drinfeld covering

Let Ωd be the d-dimensional Drinfeld symmetric space for a finite extension F of Qp. Let Σ1 be a geometrically connected component of the first Drinfeld covering of Ωd, and let F be the residue field of the unique degree d + 1 unramified extension of F. We show that the natural homomorphism ([F ,+) −→ Pic(Σ1)[p] determined by the second Drinfeld covering is injective. Here ([F ,+) is the group of characters of (F ,+). In particular, Pic(Σ1)[p], 0. We also show that all vector bundles on Ω1 are trivial, which extends the classical result that Pic(Ω1) = 0.