Stacks of p-adic shtukas and spatial kimberlites
Ian Gleason
TL;DR
The paper advances the geometric Langlands program in characteristic $p$ by proving that the special Newton polygon map from the stack of $p$-adic shtukas to the Fargues–Fontaine curve is representable in locally spatial diamonds and satisfies fdcs, with fibers over $\bar{\mathbb{F}}_p$ exhibiting henselian behavior along their reduced loci. It introduces spatial kimberlites as a robust refinement of kimberlites, enabling control over cohomology and the analytic/schematic comparison in the local Langlands setting. The core results show that $\operatorname{Sht}_{\mathcal{G},\mu}(b)$ is a locally spatial kimberlite whose reduced fiber is the affine Deligne–Lusztig variety, and that the pushforward functor $\sigma_!$ is well-behaved for deriving Hecke operators and connecting categorical local Langlands descriptions. These developments lay the groundwork for the equivalence between schematic and analytic local Langlands categories, illuminating the interplay between $p$-adic Hodge theory, diamonds, and shtuka moduli.
Abstract
The main purpose of this article is to show that the special Newton polygon map from the stack of p-adic shtukas to the stack of G-bundles on the Fargues--Fontaine curve is representable in diamonds and sufficiently nice for cohomological considerations (i.e. fdcs). The second purpose is to show that the $\bar{\mathbb{F}}_p$-fibers of the special Newton polygon map behave like formal schemes, and in particular, satisfy henselianity properties with respect to their reduced locus. These two goals achieved in this article are two of the crucial ingredients used in our collaboration with Hamman, Ivanov, Lourenço and Zou to construct the equivalence that compares the schematic and analytic local Langlands categories of Zhu and of Fargues--Scholze. To achieve these goals, we introduce and study spatial kimberlites, which is a better behaved variant of the theory previously developed by the author.