Line Bundles on The First Drinfeld Covering

Abstract

Let $Ω^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of $\mathbb{Q}_p$. Let $Σ^1$ be a geometrically connected component of the first Drinfeld covering of $Ω^d$ 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 $\text{Pic}(Σ^1)[p]$ is injective. In particular, $\text{Pic}(Σ^1)[p] \neq 0$. We also show that all vector bundles on $Ω^1$ are trivial, which extends the classical result that $\text{Pic}(Ω^1) = 0$.