The twisted forms of a semisimple group over an $\mathbb{F}_q$-curve

Abstract

Let $C$ be a smooth, projective and geometrically connected curve defined over a finite field $\mathbb{F}_q(C)$. Given a semisimple $C-S$-group scheme $\underline{G}$ where $S$ is a finite set of closed points of $C$, we describe the set of ($\mathcal{O}_S$-classes of) twisted forms of $\underline{G}$ in terms of geometric invariants of its fundamental group $F(\underline{G})$.