The twisted forms of a semisimple group over an 𝔽 q -curve
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 17-38.

Let C be a smooth, projective and geometrically connected curve defined over a finite field 𝔽 q . Given a semisimple C-S-group scheme G ̲ where S is a finite set of closed points of C, we describe the set of (𝒪 S -classes of) twisted forms of G ̲ in terms of geometric invariants of its fundamental group F(G ̲).

Soit C une courbe projective, lisse et connexe définie sur un corps fini 𝔽 q . Étant donné un C-S-schéma en groupes semisimples où S est un ensemble fini de points fermés de C, nous décrivons l’ensemble de (𝒪 S -classes de) formes tordues de G ̲ en termes d’invariants géométriques de son groupe fondamental F(G ̲).

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1150
Classification: 11G20, 11G45, 11R29
Keywords: Class number, Hasse principle, Tamagawa number, étale cohomology
Bitan, Rony A. 1; Köhl, Ralf 2; Schoemann, Claudia 3

1 Afeka, Tel-Aviv Academic College of Engineering Tel-Aviv, Israel Bar-Ilan University Ramat-Gan, Israel
2 JLU Giessen Mathematisches Institut Arndtstr. 2 35392 Giessen, Germany
3 Leibniz University Hannover Institute for Algebraic Geometry Welfengarten 1 30167 Hannover, Germany
@article{JTNB_2021__33_1_17_0,
     author = {Bitan, Rony A. and K\"ohl, Ralf and Schoemann, Claudia},
     title = {The twisted forms of a semisimple group over an $\protect \mathbb{F}_q$-curve},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {17--38},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {1},
     year = {2021},
     doi = {10.5802/jtnb.1150},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.1150/}
}
TY  - JOUR
AU  - Bitan, Rony A.
AU  - Köhl, Ralf
AU  - Schoemann, Claudia
TI  - The twisted forms of a semisimple group over an $\protect \mathbb{F}_q$-curve
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 17
EP  - 38
VL  - 33
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - http://archive.numdam.org/articles/10.5802/jtnb.1150/
DO  - 10.5802/jtnb.1150
LA  - en
ID  - JTNB_2021__33_1_17_0
ER  - 
%0 Journal Article
%A Bitan, Rony A.
%A Köhl, Ralf
%A Schoemann, Claudia
%T The twisted forms of a semisimple group over an $\protect \mathbb{F}_q$-curve
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 17-38
%V 33
%N 1
%I Société Arithmétique de Bordeaux
%U http://archive.numdam.org/articles/10.5802/jtnb.1150/
%R 10.5802/jtnb.1150
%G en
%F JTNB_2021__33_1_17_0
Bitan, Rony A.; Köhl, Ralf; Schoemann, Claudia. The twisted forms of a semisimple group over an $\protect \mathbb{F}_q$-curve. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 17-38. doi : 10.5802/jtnb.1150. http://archive.numdam.org/articles/10.5802/jtnb.1150/

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

[2] Artin, Michael; Grothendieck, Alexander; Verdier, Jean-Louis Théorie des Topos et Cohomologie Étale des Schémas (SGA 4), Lecture Notes in Mathematics, 269, Springer, 1972 | Zbl

[3] 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 | DOI | MR | Zbl

[4] Bitan, Rony A. On the genera of semisimple groups defined over an integral domain of a global function field, J. Théor. Nombres Bordeaux, Volume 30 (2018) no. 3, pp. 1037-1057 | DOI | Numdam | MR | Zbl

[5] 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 | DOI | Numdam | Zbl

[6] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. III: Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo, Volume 34 (1987), pp. 671-688 | Zbl

[7] Calmès, Baptiste; Fasel, Jean Groupes Classiques, On group schemes. A celebration of SGA3 (Panoramas et Synthèses), Volume 46, Société Mathématique de France, 2015, pp. 1-133 | MR | Zbl

[8] Colliot-Thélène, Jean-Louis; Skorobogatov, Alexei N. The Brauer-Grothendieck group (https://wwwf.imperial.ac.uk/~anskor/brauer.pdf)

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

[10] Conrad, Brian Reductive group schemes (http://math.stanford.edu/~conrad/papers/luminysga3.pdf) | Zbl

[11] Conrad, Brian Non-split reductive groups over , On group schemes. A celebration of SGA3 (Panoramas et Synthèses), Volume 46, Société Mathématique de France, 2015, pp. 193-253 | MR | Zbl

[12] Séminaire de Géométrie Algébrique du Bois Marie - 1962–64 - Schémas en groupes Tome II (Demazure, Michel; Grothendieck, Alexander, eds.), Documents Mathématiques, Société Mathématique de France, 2011 (réédition de SGA3) | Zbl

[13] Douai, Jean-Claude Cohomologie des schémas en groupes semi-simples sur les anneaux de Dedekind et sur les courbes lisses, complètes, irréductibles, C. R. Math. Acad. Sci. Paris, Volume 285 (1977), pp. 325-328 | MR | Zbl

[14] Gille, Philippe Sur la classification des schémas en groupes semi-simples, On group schemes. A celebration of SGA3 (Panoramas et Synthèses), Volume 47, Société Mathématique de France, 2015, pp. 39-110 | Zbl

[15] Gillibert, Jean; Gillibert, Pierre On the splitting of the Kummer sequence, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 2019 (2019) no. 2, pp. 19-27 | Numdam | Zbl

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

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

[18] Harder, Günter Halbeinfache Gruppenschemata über Dedekindringen, Invent. Math., Volume 4 (1967), pp. 165-191 | DOI | Zbl

[19] Ivanyos, Gábor; Kutas, Péter; Rónyai, Lajos Explicit equivalence of quadratic forms over 𝔽 q (t), Finite Fields Appl., Volume 55 (2019), pp. 33-63 | DOI | MR | Zbl

[20] McCrimmon, Kevin The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Am. Math. Soc., Volume 139 (1969), pp. 495-510 | DOI | MR | Zbl

[21] Nisnevich, Yevsey Étale Cohomology and Arithmetic of Semisimple Groups, Ph. D. Thesis, Harvard University (USA) (1982)

[22] Petrov, Victor; Stavrova, Anastasia Tits indices over semilocal rings, Transform. Groups, Volume 16 (2011) no. 1, pp. 193-217 | DOI | MR | Zbl

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

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

[25] Tits, Jacques Classification of algebraic semisimple groups (Proceedings of Symposia in Pure Mathematics), Volume 9, American Mathematical Society, 1966, pp. 33-62 | Zbl

[26] Tits, Jacques Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., Volume 247 (1971), pp. 196-220 | Zbl

[27] Tits, Jacques Reductive groups over local fields, Automorphic Forms, Representations and L-Functions (Proceedings of Symposia in Pure Mathematics), Volume 33, American Mathematical Society, 1979, pp. 29-69 | DOI | Zbl

Cited by Sources: