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=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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.1064
Classification : 11G20,  11G45,  11R29
Mots clés : Class number, Hasse principle, Tamagawa number
@article{JTNB_2018__30_3_1037_0,
     author = {Bitan, Rony A.},
     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 = {http://www.numdam.org/articles/10.5802/jtnb.1064/}
}
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/

[1] Artin, Emil Quadratische Körper im Gebiete der höheren Kongruenzen, Math. Z., Volume 19 (1927), pp. 153-206 | Article | Zbl 50.0107.01

[2] Théorie des Topos et Cohomologie Étale des Schémas (SGA 4) (Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis, eds.), Lecture Notes in Mathematics, 269, 270, 305, Springer, 1972/1973

[3] Behrend, Kai; Dhillon, Ajneet Connected components of moduli stacks of torsors via Tamagawa numbers, Can. J. Math., Volume 61 (2009) no. 1, pp. 3-28 | Article | MR 2488447 | Zbl 1219.14030

[4] Bitan, Rony A. The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field, J. Number Theory, Volume 168 (2016), pp. 346-359 | Article | MR 3515823 | Zbl 1401.11078

[5] Bitan, Rony A. Between the genus and the Γ-genus of an integral quadratic Γ-form, Acta Arith., Volume 181 (2017) no. 2, pp. 173-183 | Article | MR 3726187 | Zbl 06814184

[6] Bitan, Rony A. On the classification of quadratic forms over an integral domain of a global function field, J. Number Theory, Volume 180 (2017), pp. 26-44 | Article | MR 3679786 | Zbl 1406.11030

[7] Bitan, Rony A.; Köhl, Ralf A building-theoretic approach to relative Tamagawa numbers of semisimple groups over global function fields, Funct. Approximatio, Comment. Math., Volume 53 (2015) no. 2, pp. 215-247 | MR 3435798 | Zbl 1396.20047

[8] Borel, Armand; Prasad, Gopal Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 69 (1989), pp. 119-171 | Article | Numdam | Zbl 0707.11032

[9] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Publ. Math., Inst. Hautes Étud. Sci., Volume 60 (1984), pp. 197-376 | Numdam | Zbl 0597.14041

[10] Calmès, Baptiste; Fasel, Jean Groupes Classiques, On group schemes (Panoramas et Synthèses), Volume 46, Société Mathématique de France, 2015, pp. 1-133 | MR 3525594 | Zbl 1360.20048

[11] Caprace, Pierre-Emmanuel; Monod, Nicolas Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol., Volume 2 (2009), pp. 701-746 | Article | MR 2574741 | Zbl 1187.53037

[12] Chernousov, Vladimir; Gille, Philippe; Pianzola, Arturo A classification of torsors over Laurent polynomial rings, Comment. Math. Helv., Volume 92 (2017) no. 1, pp. 37-55 | Article | MR 3615034 | Zbl 1362.14022

[13] Conrad, Brian Math 252. Properties of orthogonal groups http://math.stanford.edu/~conrad/252Page/handouts/O(q).pdf

[14] Conrad, Brian Math 252. Reductive group schemes (http://math.stanford.edu/~conrad/252Page/handouts/luminysga3.pdf) | Zbl 1349.14151

[15] Séminaire de Géométrie Algébrique du Bois Marie 1962-64 (SGA 3). Schémas en groupes, Tome II (Demazure, Michel; Grothendieck, Alexander, eds.), Documents Mathématiques, Société Mathématique de France, 2011

[16] Giraud, Jean Cohomologie non abélienne, Grundlehren der Mathematischen Wissenschaften, 179, Springer, 1971 | Zbl 0226.14011

[17] González-Avilés, Cristian D. Quasi-abelian crossed modules and nonabelian cohomology, J. Algebra, Volume 369 (2012), pp. 235-255 | Article | MR 2959794 | Zbl 1292.14016

[18] Grothendieck, Alexander Le groupe de Brauer III: Exemples et compléments, Dix Exposes Cohomologie Schemas (Advanced Studies Pure Math.), Volume 3, American Mathematical Society, 1968, pp. 88-188 | Zbl 0198.25901

[19] Harder, Günter Über die Galoiskohomologie halbeinfacher algebraischer Gruppen, III, J. Reine Angew. Math., Volume 274/275 (1975), pp. 125-138 | Zbl 0317.14025

[20] Jarden, Moshe The Čebotarev density theorem for function fields: An elementary approach, Math. Ann., Volume 261 (1982) no. 4, pp. 467-475 | Article | Zbl 0501.12018

[21] Knus, Max-Albert Quadratic and hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften, 294, Springer, 1991 | MR 1096299 | Zbl 0756.11008

[22] Lenstra, H. W. Galois theory for schemes (http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf)

[23] Lurie, Jacob Tamagawa Numbers of Algebraic Groups Over Function Fields

[24] Milne, James S. Étale Cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980 | MR 559531 | Zbl 0433.14012

[25] Milne, James S. Arithmetic Duality Theorems, BookSurge, 2006 | Zbl 1127.14001

[26] Nisnevich, Yevsey Étale Cohomology and Arithmetic of Semisimple Groups (1982) (Ph. D. Thesis)

[27] Ono, Takashi On the Relative Theory of Tamagawa Numbers, Ann. Math., Volume 82 (1965), pp. 88-111 | MR 177991 | Zbl 0135.08804

[28] Platonov, Vladimir; Rapinchuk, Andrei Algebraic Groups and Number Theory, Pure and Applied Mathematics, 139, Academic Press Inc., 1994 | MR 1278263 | Zbl 0841.20046

[29] Rosen, Michael Number Theory in Function Fields, Graduate Texts in Mathematics, 210, Springer, 2000 | Zbl 1043.11079

[30] Serre, Jean-Pierre Algebraic Groups and Class Fields, Graduate Texts in Mathematics, 117, Springer, 1988 | MR 918564 | Zbl 0703.14001

[31] Skorobogatov, Alexei N. Torsors and Rational Points, Cambridge Tracts in Mathematics, 144, Cambridge University Press, 2001 | MR 1845760 | Zbl 0972.14015

[32] Thǎńg, Nguyêñ Q. A Norm Principle for class groups of reductive group schemes over Dedekind rings, Vietnam J. Math., Volume 43 (2015) no. 2, pp. 257-281 | Article | Zbl 1370.11052

[33] Weil, André Adèles and Algebraic Groups, Progress in Mathematics, 23, Birkhäuser, 1982 | Zbl 0493.14028