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

Bitan, Rony A.

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 1037-1057.

Soit $K={𝔽}_q\left(C\right)$ un corps de fonctions global, i.e. le corps des fonctions d’une courbe projective lisse $C$ définie sur un corps fini ${𝔽}_q$. L’anneau des fonctions régulières sur $C-S$, où $S\ne \varnothing$ est un ensemble fini de points fermés sur $C$, est un domaine de Dedekind ${𝒪}_S$ de $K$. Étant donné un ${𝒪}_S$-groupe $\underline{G}$ semisimple dont le groupe fondamental $\underline{F}$ est lisse, on aimerait décrire l’ensemble des genres de $\underline{G}$ et encore (dans le cas où le groupe $\underline{G}{\otimes }_{{𝒪}_S}K$ est isotrope à $S$) son genre principal en termes des groupes abéliens ne dépendant que de ${𝒪}_S$ et de $\underline{F}$. Ceci conduit à une condition nécessaire et suffisante pour que le principe local-global de Hasse soit valable pour certains groupes $\underline{G}$. Nous l’utilisons aussi pour exprimer le nombre de Tamagawa $\tau \left(G\right)$ d’un $K$-groupe semisimple $\underline{G}$ par l’invariant d’Euler–Poincaré et faciliter le calcul de $\tau \left(G\right)$ pour les $K$-groupes tordus.

Let $K={𝔽}_q\left(C\right)$ be the global function field of rational functions over a smooth and projective curve $C$ defined over a finite field ${𝔽}_q$. 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 ${𝒪}_S$ of $K$. For a semisimple ${𝒪}_S$-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 $\underline{G}{\otimes }_{{𝒪}_S}K$ is isotropic at $S$) in terms of abelian groups depending on ${𝒪}_S$ 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 $\tau \left(G\right)$ of a semisimple $K$-group $G$ by the Euler–Poincaré invariant. This facilitates the computation of $\tau \left(G\right)$ for twisted $K$-groups.

Classification : 11G20, 11G45, 11R29
Mots-clés : Class number, Hasse principle, Tamagawa number

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