Uniformizing The Moduli Stacks of Global G-Shtukas

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.

48 pages. v2: existence of moduli space of global G-shtukas as DM-stack generalized to arbitrary flat affine group schemes G of finite type; uniformization theorem proved for smooth affine group schemes with connected fibers and reductive generic fiber; its compatibility with various group actions proved; minor errors corrected