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.

26 pages, v3: Added a short exposition of related works in the introduction; elaborated on the localisation map of $\sigma$-conjugacy classes and fixed a minor mistake in the formula of Newton point of the localisation of a $\sigma$-conjugacy class