On $\mathsf{G}$-isoshtukas over function fields

Abstract

In this paper we classify isogeny classes of global $\mathsf{G}$-shtukas over a smooth projective curve $C/\mathbb{F}_q$ (or equivalently $\sigma$-conjugacy classes in $\mathsf{G}(\mathsf{F} \otimes_{\mathbb{F}_q} \overline{\mathbb{F}_q})$ where $\mathsf{F}$ is the field of rational functions of $C$) by two invariants $\bar\kappa,\bar\nu$ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $\mathsf{G}$-shtukas and thus is helpful to calculate their cohomology.