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

## TL;DR

This paper studies the classification of semisimple groups over rings of functions on curves over finite fields, providing criteria for local-global principles and formulas for Tamagawa numbers based on abelian group invariants.

## Contribution

It characterizes the genera of semisimple groups over Dedekind domains from function fields and relates Tamagawa numbers to Euler-Poincaré invariants, extending understanding of their arithmetic properties.

## Key findings

- Describes the set of genera of semisimple groups in terms of abelian groups.
- Provides a criterion for the Hasse local-global principle for these groups.
- Expresses Tamagawa numbers using Euler-Poincaré invariants.

## 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 $S \neq \emptyset$ 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 Poincar\'e invariant. This facilitates the computation of $\tau(G)$ for twisted $K$-groups.