Abstract 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 Künneth 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 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 G is p -cohomologically smooth.
Hansen et al. (Mon,) studied this question.