The Penrose Correspondence over a Non-Archimedean Field

Let $K$ be a non-Archimedean local field of characteristic different from $2$ and let $V$ be a four-dimensional symplectic $K$-vector space. We study the Penrose incidence correspondence $\mathbb{P}(V)\leftarrow \mathrm{F}\mathrm{l}\mathrm{a}\mathrm{g}(1,2;V)\to \mathrm{G}\mathrm{r}(2,V)$ after Berkovich analytification. The Klein quadric $\mathrm{G}\mathrm{r}(2,V)$ decomposes into the open orbit $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(5)\mathrm{/}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(4)$ and the Lagrangian quadric, which parametrize respectively the projective lines transverse and tangent to the standard contact distribution on $\mathbb{P}(V)$. We construct explicit free discrete subgroups $\mathrm{\Gamma }\subset \mathrm{P}\mathrm{S}\mathrm{p}(V)$ and analytic domains, cut out by Drinfeld upper half-spaces, in all three members of the incidence diagram. Both incidence projections are surjective on these domains, the three $\mathrm{\Gamma }$-actions are free and properly discontinuous, and the Penrose double correspondence descends to the analytic quotients. The quotient of the twistor domain is a contact analytic threefold, and the Grassmannian domain is contained in the analytification of the $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(5)\mathrm{/}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(4)$ open orbit.