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

TL;DR

This paper advances the understanding of the $p$-adic Jacquet-Langlands correspondence by establishing new properties, duality, and bounds using the six functor formalism on classifying stacks, extending previous formalism to stacky maps.

Contribution

It introduces an extension of the six functor formalism to stacky maps and applies it to the $p$-adic Jacquet-Langlands functor, proving new properties and bounds.

Findings

Reproves Scholze's finiteness theorems.

Establishes a duality theorem for the functor.

Provides bounds on Gelfand-Kirillov dimension.

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 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.