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p-ADIC SHEAVES ON CLASSIFYING STACKS, AND THE p-ADIC JACQUET-LANGLANDS CORRESPONDENCE
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 25 / Issue 3 / May 2026
- Published online by Cambridge University Press:
- 30 March 2026, pp. 1441-1486
- Print publication:
- May 2026
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- Article
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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
$\mathcal {O}^+/p$-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.
