Stacks of p-adic shtukas and spatial kimberlites

TL;DR

This paper demonstrates that the Newton polygon map from p-adic shtukas to G-bundles is well-behaved in the category of diamonds and explores the properties of its fibers, facilitating advances in the local Langlands program.

Contribution

It introduces the concept of spatial kimberlites and proves the representability and henselianity properties of the Newton polygon map, crucial for the local Langlands correspondence.

Findings

The Newton polygon map is representable in diamonds.

The fibers of the map behave like formal schemes.

The properties enable comparison of schematic and analytic categories.

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, Louren\c{c}o 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.