The Picard Group of Vertex Affinoids in the First Drinfeld Covering

Abstract

Let $F$ be a finite extension of $\mathbb{Q}_p$. Let $Ω$ be the Drinfeld upper half plane, and $Σ^1$ the first Drinfeld covering of $Ω$. We study the affinoid open subset $Σ^1_v$ of $Σ^1$ above a vertex of the Bruhat-Tits tree for $\text{GL}_2(F)$. Our main result is that $\text{Pic}(Σ^1_v)[p] = 0$, 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 $H^1_{\text{ét}}(Σ^1_v, \mathbb{Z}_p(1))$ of $\text{GL}_2(\mathcal{O}_F)$ as the $p$-adic completion of $\mathcal{O}(Σ^1_v)^\times$.