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 $\text{ Pic}(\Sigma^1_v)[p] = 0$, which we establish by showing that $\text{ Pic}(\mathbf{ Y})[p] = 0$ for $\mathbf{ Y}$ the Deligne-Lusztig variety of $\text{ SL}_2(\mathbb{ F}_q)$. One formal consequence is a description of the representation $H^1_{ \text{ \'{ e}t}}(\Sigma^1_v, \mathbb{ Z}_p(1))$ of $\text{ GL}_2(\mathcal{ O}_F)$ as the $p$-adic completion of $\mathcal{ O}(\Sigma^1_v)^\times$.
Categories: math.RT math.AG math.NT