On the genera of semisimple groups defined over an integral domain of a global function field

Bitan, Rony A.

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 1037-1057.

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

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

Classification : 11G20, 11G45, 11R29
Mots clés : Class number, Hasse principle, Tamagawa number

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | Numdam | Zbl

[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

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

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

[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 | DOI | MR | Zbl

[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

[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

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

[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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

[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 | DOI | Zbl

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