Let be the global function field of rational functions over a smooth and projective curve defined over a finite field . The ring of regular functions on where is any finite set of closed points on is a Dedekind domain of . For a semisimple -group with a smooth fundamental group , we aim to describe both the set of genera of and its principal genus (the latter if is isotropic at ) in terms of abelian groups depending on and only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain . We also use it to express the Tamagawa number of a semisimple -group by the Euler–Poincaré invariant. This facilitates the computation of for twisted -groups.
Soit un corps de fonctions global, i.e. le corps des fonctions d’une courbe projective lisse définie sur un corps fini . L’anneau des fonctions régulières sur , où est un ensemble fini de points fermés sur , est un domaine de Dedekind de . Étant donné un -groupe semisimple dont le groupe fondamental est lisse, on aimerait décrire l’ensemble des genres de et encore (dans le cas où le groupe est isotrope à ) son genre principal en termes des groupes abéliens ne dépendant que de et de . Ceci conduit à une condition nécessaire et suffisante pour que le principe local-global de Hasse soit valable pour certains groupes . Nous l’utilisons aussi pour exprimer le nombre de Tamagawa d’un -groupe semisimple par l’invariant d’Euler–Poincaré et faciliter le calcul de pour les -groupes tordus.
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1064
Keywords: Class number, Hasse principle, Tamagawa number
Rony A. Bitan 1, 2
@article{JTNB_2018__30_3_1037_0, author = {Rony A. Bitan}, title = {On the genera of semisimple groups defined over an integral domain of a global function field}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {1037--1057}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {30}, number = {3}, year = {2018}, doi = {10.5802/jtnb.1064}, zbl = {1441.11289}, mrnumber = {3938641}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1064/} }
TY - JOUR AU - Rony A. Bitan TI - On the genera of semisimple groups defined over an integral domain of a global function field JO - Journal de théorie des nombres de Bordeaux PY - 2018 SP - 1037 EP - 1057 VL - 30 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1064/ DO - 10.5802/jtnb.1064 LA - en ID - JTNB_2018__30_3_1037_0 ER -
%0 Journal Article %A Rony A. Bitan %T On the genera of semisimple groups defined over an integral domain of a global function field %J Journal de théorie des nombres de Bordeaux %D 2018 %P 1037-1057 %V 30 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1064/ %R 10.5802/jtnb.1064 %G en %F JTNB_2018__30_3_1037_0
Rony A. Bitan. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1064/
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