The Picard group of vertex affinoids in the first Drinfeld covering

Abstract

AbstractLetFbe 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 thep-adic completion of$\mathcal{O}\!\left(\Sigma^1_v\right)^\times$.