The Picard group of vertex affinoids in the first Drinfeld covering

Abstract

Abstract Let F be a finite extension of ${\mathbb Q}_p$ . Let $\Omega$ be the Drinfeld upper half plane, and $\Sigma^1$ the first Drinfeld covering of $\Omega$ . We study the affinoid open subset $\Sigma^1_v$ of $\Sigma^1$ above a vertex of the Bruhat–Tits tree for $\text{GL}_2(F)$ . Our main result is that $\text{Pic}\!\left(\Sigma^1_v\right)[p] = 0$ , which we establish by showing that $\text{Pic}({\mathbf Y})[p] = 0$ for ${\mathbf Y}$ the Deligne–Lusztig variety of $\text{SL}_2\!\left({\mathbb F}_q\right)$ . One formal consequence is a description of the representation $H^1_{{\acute{\text{e}}\text{t}}}\!\left(\Sigma^1_v, {\mathbb Z}_p(1)\right)$ of $\text{GL}_2(\mathcal{O}_F)$ as the p-adic completion of $\mathcal{O}\!\left(\Sigma^1_v\right)^\times$ .