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

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.

37 pages, v3: generalization to flat affine group schemes of finite type, v3: final version which appears in Muenster J. of Mathematics

References in corpus (2)

• Revêtements étales et groupe fondamental (SGA 1)
• On parahoric subgroups

Cited by in corpus (16)

• Local Shtukas and Divisible Local Anderson Modules
• Local Shtukas, Hodge-Pink Structures and Galois Representations
• Local Models For The Moduli Stacks of Global $G$-Shtukas
• Foliations and the cohomology of moduli spaces of bounded global $G$-shtukas
• Langlands-Rapoport Conjecture Over Function Fields
• The generic fiber of moduli spaces of bounded local $G$-shtukas
• Functoriality of Moduli Spaces of Global $\mathbb{G}$-Shtukas
• Category Of C-Motives Over Finite Fields
• Rapoport-Zink Spaces For Local P-Shtukas and Their Local Models
• Compactification of Level Maps of Moduli Spaces of Drinfeld Shtukas
• Some Motivic Remarks On The Moduli Stacks Of global G-Shtukas And Their Local Models
• A Note On Two Fiber Bundles and The Manifestations Of "Shtuka"
• On Langlands program, global fields and shtukas
• The local Langlands correspondence for $\DeclareMathOperator{\GL}{GL}\GL_n$ over function fields
• Point Counting on Igusa Varieties for function fields
• On Langlands program, related representation and $G$-shtukas