Explicit non-abelian Lubin-Tate theory for GL(2)

Let $F$ be a non-Archimedean local field with residue field $k$ of odd characteristic, and let $B\mathrm{/}F$ be the division algebra of rank 4. We explicitly construct a stable curve $\mathfrak{X}$ over the algebraic closure of $k$ admitting an action of $G{L}_2(F)\times B^{\times }\times W_F$ which realizes the Jacquet-Langlands correspondence and the local Langlands correspondence in its cohomology.