On the genera of semisimple groups defined over an integral domain of a global function field

Abstract

Let $K={\mathbb{F}}_q(C)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field ${\mathbb{F}}_q$. The ring of regular functions on $C-S$ where $S\ne \varnothing$ is any finite set of closed points on $C$ is a Dedekind domain ${\mathcal{O}}_S$ of $K$. For a semisimple ${\mathcal{O}}_S$-group $\underline{G}$ with a smooth fundamental group $\underline{F}$, we aim to describe both the set of genera of $\underline{G}$ and its principal genus (the latter if $\underline{G}{\otimes }_{{\mathcal{O}}_S}K$ is isotropic at $S$) in terms of abelian groups depending on ${\mathcal{O}}_S$ and $\underline{F}$ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain $\underline{G}$. We also use it to express the Tamagawa number $\tau (G)$ of a semisimple $K$-group $G$ by the Euler Poincaré invariant. This facilitates the computation of $\tau (G)$ for twisted $K$-groups.

18 pages

References in corpus (4)

• Isometry groups of non-positively curved spaces: discrete subgroups
• The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field
• Connected components of moduli stacks of torsors via Tamagawa numbers
• On the classification of quadratic forms over an integral domain of a global function field

Cited by in corpus (1)

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