Local P-shtukas and their relation to global G-shtukas

TL;DR

This paper explores the relationship between global G-shtukas and local P-shtukas, establishing deformation equivalences, and introduces Rapoport--Zink spaces for local P-shtukas, advancing the understanding of moduli stacks in the function field Langlands program.

Contribution

It demonstrates the deformation equivalence between global G-shtukas and local P-shtukas and constructs Rapoport--Zink spaces for bounded local P-shtukas.

Findings

Proved the deformation equivalence between global G-shtukas and local P-shtukas.

Established the existence of Rapoport--Zink spaces for local P-shtukas.

Connected local P-shtukas with Galois representations via Tate modules.

Abstract

This is the first in a sequence of two articles investigating moduli stacks of global G-shtukas, which are function field analogs for Shimura varieties. Here G is a flat affine group scheme of finite type over a smooth projective curve, and global G-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. Our moduli stacks generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the present article we explain the relation between global G-shtukas and local P-shtukas, which are the function field analogs of p-divisible groups with additional structure. We prove the analog of a theorem of Serre and Tate stating the equivalence between the deformations of a global G-shtuka and its associated local P-shtukas. We also investigate local P-shtukas alone and explain their relation with Galois representations through their Tate modules. And if P is a smooth affine group scheme with connected reductive generic fiber we prove the existence of Rapoport--Zink spaces for bounded local P-shtukas as formal schemes locally formally of finite type. In the sequel to this article we use these Rapoport--Zink spaces to uniformize the moduli stacks of global G-shtukas.