Uniformizing The Moduli Stacks of Global G-Shtukas

TL;DR

This paper establishes that moduli stacks of global G-shtukas are algebraic Deligne-Mumford stacks, providing a foundation for their uniformization and applications to the Langlands program over function fields.

Contribution

It proves the algebraic nature and uniformization of moduli stacks of global G-shtukas, extending their structure and applications in the Langlands correspondence.

Findings

Moduli stacks of global G-shtukas are algebraic Deligne-Mumford stacks.

Established uniformization of these stacks using Rapoport-Zink spaces.

Applications to the Langlands-Rapoport conjecture for function fields.

Abstract

This is the second in a sequence of articles, in which we explore moduli stacks of global G-shtukas, the function field analogs for Shimura varieties. Here G is a flat affine group scheme of finite type over a smooth projective curve C over a finite field. Global G-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global G-shtukas are algebraic Deligne-Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the first article we explained the relation between global G-shtukas and local P-shtukas, which are the function field analogs of p-divisible groups. Here P is the base change of G to the complete local ring at a point of C. When P is smooth with connected reductive generic fiber we proved the existence of Rapoport-Zink spaces for local P-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global G-shtukas for smooth G with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands-Rapoport conjecture for our moduli stacks.