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

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

Let K=𝔽q(C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field 𝔽q. The ring of regular functions on C-S where S is any finite set of closed points on C is a Dedekind domain 𝒪S of K. For a semisimple 𝒪S-group G̲ with a smooth fundamental group F̲, we aim to describe both the set of genera of G̲ and its principal genus (the latter if G̲𝒪SK is isotropic at S) in terms of abelian groups depending on 𝒪S and F̲ only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G̲. 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.

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DOI : 10.5802/jtnb.1064
Classification : 11G20, 11G45, 11R29
Mots-clés : 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, Tome 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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  • Bitan, Rony A.; Köhl, Ralf; Schoemann, Claudia The twisted forms of a semisimple group over an Fq-curve, Journal de Théorie des Nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 17-38 | DOI:10.5802/jtnb.1150 | Zbl:1507.11057

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