Local models for the moduli stacks of global $\mathfrak{G}$-shtukas

In this article we develop the theory of local models for the moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ 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 $\mathfrak{G}$-shtukas and associated local $\mathbb{P}$-shtukas at certain characteristic places. Regarding the analogy between function fields and number fields, the first (resp. second) approach corresponds to Beilinson–Drinfeld–Gaitsgory (resp. Rapoport–Zink) type local model for (PEL-) Shimura varieties. This discussion will establish a conceptual relation between the above approaches. Furthermore, as applications of this theory, we discuss the flatness of these moduli stacks over their reflex rings, we introduce the Kottwitz–Rapoport stratification on them, and we study the intersection cohomology of the special fiber.

Full Text (PDF format)

Received 2 November 2017

Accepted 2 April 2018

Published 12 August 2019