Line Bundles on The First Drinfeld Covering

Abstract

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

v3: final published version