On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic
Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 177-190.

Voir la notice de l'article provenant de la source Math-Net.Ru

We extend to global function fields some Hasse principles for homogeneous spaces of connected linear algebraic groups proved earlier by several authors in the case of number fields. We also give some applications.
@article{TRSPY_2016_292_a10,
     author = {Ng\^o Thị Ngoan and Nguyẽn Qu\H{o}c Th\r{a}ng},
     title = {On some {Hasse} principles for homogeneous spaces of algebraic groups over global fields of positive characteristic},
     journal = {Informatics and Automation},
     pages = {177--190},
     publisher = {mathdoc},
     volume = {292},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a10/}
}
TY  - JOUR
AU  - Ngô Thị Ngoan
AU  - Nguyẽn Quőc Thång
TI  - On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic
JO  - Informatics and Automation
PY  - 2016
SP  - 177
EP  - 190
VL  - 292
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a10/
LA  - en
ID  - TRSPY_2016_292_a10
ER  - 
%0 Journal Article
%A Ngô Thị Ngoan
%A Nguyẽn Quőc Thång
%T On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic
%J Informatics and Automation
%D 2016
%P 177-190
%V 292
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a10/
%G en
%F TRSPY_2016_292_a10
Ngô Thị Ngoan; Nguyẽn Quőc Thång. On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic. Informatics and Automation, Algebra, geometry, and number theory, Tome 292 (2016), pp. 177-190. http://geodesic.mathdoc.fr/item/TRSPY_2016_292_a10/

[1] Borovoi M.V., “The Hasse principle for homogeneous spaces”, J. reine angew. Math., 426 (1992), 179–192 | MR

[2] Borovoi M.V., “Abelianization of the second nonabelian Galois cohomology”, Duke Math. J., 72:1 (1993), 217–239 | DOI | MR | Zbl

[3] Borovoi M.V., “The Brauer–Manin obstructions for homogeneous spaces with connected or abelian stabilizer”, J. reine angew. Math., 473 (1996), 181–194 | MR | Zbl

[4] Borovoi M., Abelian Galois cohomology of reductive groups, Mem. AMS, 132, Amer. Math. Soc., Providence, RI, 1998 | MR

[5] Borovoi M., “A cohomological obstruction to the Hasse principle for homogeneous spaces”, Math. Ann., 314 (1999), 491–504 | DOI | MR | Zbl

[6] Borovoi M., “Homogeneous spaces of Hilbert type”, Int. J. Number Theory, 11:2 (2015), 397–405, arXiv: 1304.1872v1 [math.NT] | DOI | MR | Zbl

[7] Breen L., On the classification of 2-stacks and 2-gerbes, Astérisque, 225, Soc. math. France, Paris, 1994 | MR | Zbl

[8] Bruhat F., Tits J., “Groupes algébriques simples sur un corps local: cohomologie galoisienne, décomposition d'Iwasawa et de Cartan”, C. r. Acad. sci. Paris A, 263 (1966), 867–869 | MR | Zbl

[9] Colliot-Thélène J.-L., “Résolutions flasques des groupes linéaires connexes”, J. reine angew. Math., 618 (2008), 77–133 | MR | Zbl

[10] Colliot-Thélène J.-L., Gille P., Parimala R., “Arithmetic of linear algebraic groups over 2-dimensional geometric fields”, Duke Math. J., 121 (2004), 285–341 | DOI | MR | Zbl

[11] Douai J.-C., 2-cohomologie galoisienne des groupes semi-simples: Applications de la cohomologie non abelienne, Thèse d'Etat. Univ. Lille 1, 1976; Éd. Univ. Eur., Südwestdtsch.-Verl., Saabrücken, 2010

[12] Douai J.-C., “Espaces homogènes et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind”, J. Théor. Nombres Bord., 7 (1995), 21–26 | DOI | MR | Zbl

[13] Giraud J., Cohomologie non abélienne, Grundl. Math. Wiss., 169, Springer, Berlin, 1971 | MR

[14] González-Avilés C.D., “Quasi-abelian crossed modules and nonabelian cohomology”, J. Algebra, 369 (2012), 235–255 | DOI | MR | Zbl

[15] González-Avilés C.D., “Flasque resolutions of reductive group schemes”, Cent. Eur. J. Math., 11:7 (2013), 1159–1176 | MR | Zbl

[16] Harder G., “Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen”, Jahresber. Dtsch. Math.-Ver., 70 (1967/1968), 182–216 | MR

[17] Harder G., “Über die Galoiskohomologie der halbeinfacher Matrizengruppen. III”, J. reine angew. Math., 274–275 (1975), 125–138 | MR | Zbl

[18] Humphreys J.E., Linear algebraic groups, Grad. Texts Math., 52, 2nd ed., Springer, New York, 1981 | MR

[19] Kneser M., “Galois-Kohomologie halbeinfacher algebraischer Gruppen über $p$-adischen Körpern. I, II”, Math. Z., 88 (1965), 40–47 ; 89, 250–272 | DOI | MR | Zbl | DOI | MR

[20] Kneser M., Lectures on Galois cohomology of classical groups, Tata Inst. Fundam. Res., Bombay, 1969 | MR | Zbl

[21] Lee T.-Y., “Embedding functors and their arithmetic properties”, Comment. math. Helv., 89 (2014), 671–717, arXiv: 1211.3564v1 [math.GR] | DOI | MR | Zbl

[22] Ngoan N.T., Tháng N.Q., “On some Hasse principles for algebraic groups over global fields”, Proc. Jpn. Acad. A, 90:5 (2014), 73–78 | DOI | MR | Zbl

[23] Prasad G., Rapinchuk A.S., “Local–global principles for embedding of fields with involution into simple algebras with involution”, Comment. math. Helv., 85 (2010), 583–645 | DOI | MR | Zbl

[24] Rapinchuk A.S., “Printsip Khasse dlya simmetricheskikh prostranstv”, DAN BSSR, 31:9 (1987), 773–776 | MR | Zbl

[25] Saavedra Rivano N., Catégories tannakiennes, Lect. Notes Math., 265, Springer, Berlin, 1972 | MR

[26] Sansuc J.-J., “Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres”, J. reine angew. Math., 327 (1981), 12–80 | MR | Zbl

[27] Schémas en groupes (SGA 3), T. 3, Lect. Notes Math., 153, Ed. by M. Demazure, A. Grothendieck, Springer, Berlin, 1970

[28] Shatz S.S., “The cohomological dimension of certain Grothendieck topologies”, Ann. Math. Ser. 2, 83 (1966), 572–595 | DOI | MR | Zbl

[29] Shih S.-C., Yang T.-C., Yu C.-F., “Embeddings of fields into simple algebras over global fields”, Asian J. Math., 18 (2014), 365–386, arXiv: 1108.0830v3 [math.NT] | DOI | MR | Zbl

[30] Springer T.A., “Nonabelian $H^2$ in Galois cohomology”, Algebraic groups and discontinuous subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc., Providence, RI, 1966, 164–182 | DOI | MR

[31] Steinberg R., Endomorphisms of linear algebraic groups, Mem. AMS, 90, Amer. Math. Soc., Providence, RI, 1968 | MR

[32] Tháng N.Q., “On Galois cohomology of semisimple groups over local and global fields of positive characteristic”, Math. Z., 259 (2008), 457–470 | DOI | MR | Zbl