$p$-adic sheaves on classifying stacks, and the $p$-adic Jacquet-Langlands correspondence

We establish several new properties of the $p$-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze's basic finiteness theorems, prove a duality theorem, and show a kind of partial K\"unneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results. Our key new tool is the six functor formalism with solid almost $\mathcal{O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain "stacky" maps. In the language of this extended formalism, we show that if $G$ is a $p$-adic Lie group, the structure map of the classifying small v-stack $B\underline{G}$ is $p$-cohomologically smooth.