Line Bundles on The First Drinfeld Covering

Let ${\mathrm{\Omega }}^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of ${\mathbb{Q}}_p$. Let ${\mathrm{\Sigma }}^1$ be a geometrically connected component of the first Drinfeld covering of ${\mathrm{\Omega }}^d$ and let $\mathbb{F}$ be the residue field of the unique degree $d+1$ unramified extension of $F$. We show that the natural homomorphism determined by the second Drinfeld covering from the group of characters of $(\mathbb{F},+)$ to $\text{Pic}({\mathrm{\Sigma }}^1)[p]$ is injective. In particular, $\text{Pic}({\mathrm{\Sigma }}^1)[p]\mathrm{\ne }0$. We also show that all vector bundles on ${\mathrm{\Omega }}^1$ are trivial, which extends the classical result that $\text{Pic}({\mathrm{\Omega }}^1)=0$.