On the genera of semisimple groups defined over an integral domain of a global function field
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1037-1057.

Let K=𝔽 q (C) 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 is any finite set of closed points on C is a Dedekind domain 𝒪 S of K. For a semisimple 𝒪 S -group G ̲ with a smooth fundamental group F ̲, we aim to describe both the set of genera of G ̲ and its principal genus (the latter if G ̲ 𝒪 S K is isotropic at S) in terms of abelian groups depending on 𝒪 S and F ̲ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G ̲. We also use it to express the Tamagawa number τ(G) of a semisimple K-group G by the Euler–Poincaré invariant. This facilitates the computation of τ(G) for twisted K-groups.

Soit K=𝔽 q (C) 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 est un ensemble fini de points fermés sur C, est un domaine de Dedekind 𝒪 S de K. Étant donné un 𝒪 S -groupe G ̲ semisimple dont le groupe fondamental F ̲ est lisse, on aimerait décrire l’ensemble des genres de G ̲ et encore (dans le cas où le groupe G ̲ 𝒪 S K est isotrope à S) son genre principal en termes des groupes abéliens ne dépendant que de 𝒪 S et de F ̲. Ceci conduit à une condition nécessaire et suffisante pour que le principe local-global de Hasse soit valable pour certains groupes G ̲. Nous l’utilisons aussi pour exprimer le nombre de Tamagawa τ(G) d’un K-groupe semisimple G ̲ par l’invariant d’Euler–Poincaré et faciliter le calcul de τ(G) pour les K-groupes tordus.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1064
Classification: 11G20, 11G45, 11R29
Keywords: Class number, Hasse principle, Tamagawa number
Bitan, Rony A. 1, 2

1 Afeka, Tel-Aviv Academic College of Engineering Tel-Aviv, Israel
2 Bar-Ilan University Ramat-Gan, Israel
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Bitan, Rony A. On the genera of semisimple groups defined over an integral domain of a global function field. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 1037-1057. doi : 10.5802/jtnb.1064. http://www.numdam.org/articles/10.5802/jtnb.1064/

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