Local Models For The Moduli Stacks of Global $G$-Shtukas

Abstract

In this article we develop the theory of local models for the moduli stacks of global $G$-shtukas, the function field analogs for Shimura varieties. Here $G$ is a smooth affine group scheme over a smooth projective curve. As the first approach, we relate the local geometry of these moduli stacks to the geometry of Schubert varieties inside global affine Grassmannian, only by means of global methods. Alternatively, our second approach uses the relation between the deformation theory of global $G$-shtukas and associated local $P$-shtukas at certain characteristic places. Regarding the analogy between function fields and number fields, the first (resp. second) approach corresponds to the Beilinson-Drinfeld-Gaitsgory (resp. Rapoport-Zink) local model for (PEL-)Shimura varieties. As an application, we prove the flatness of these moduli stacks over their reflex rings, for tamely ramified group $G$. Furthermore, we introduce the Kottwitz-Rapoport stratification on these moduli stacks and discuss the intersection cohomology of the special fiber.

arXiv comment25 pages

References in corpus (7)

• The Newton stratification on deformations of local G-shtukas
• On parahoric subgroups
• Local P-shtukas and their relation to global G-shtukas
• Langlands-Rapoport Conjecture Over Function Fields
• Moduli spaces of principal F-bundles
• Twisted loop groups and their affine flag varieties
• Construction of central elements in the affine Hecke algebra via nearby cycles

Cited by in corpus (6)

• Category Of C-Motives Over Finite Fields
• Rapoport-Zink Spaces For Local P-Shtukas and Their Local Models
• Néron blowups and low-degree cohomological applications
• 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"
• Nearby cycles of parahoric shtukas, and a fundamental lemma for base change