Metadata
Abstract
Let be a homogeneous space of a reductive group with reductive stabilizers, defined over a global field of positive characteristic. Using duality theorems for complexes of tori, we study cohomological obstructions to various arithmetic properties.
References
[BD13] Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv., Volume 88 (2013) no. 1, pp. 1-54 | DOI | MR | Zbl
[BDH13] Complexes de groupes de type multiplicatif et groupe de Brauer non ramifié des espaces homogènes, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 4, pp. 651-692 | DOI | Numdam | Zbl
[Bor91] Linear Algebraic Groups, Graduate Texts in Mathematics, 126, Springer, 1991 | Zbl
[Bor93] Abelianization of the second nonabelian Galois cohomology, Duke Math. J., Volume 72 (1993) no. 1, pp. 217-239 | MR | Zbl
[Bor96] The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer, J. Reine Angew. Math., Volume 473 (1996), pp. 181-194 | MR | Zbl
[Bor98] Abelian Galois cohomology of reductive groups, Mem. Am. Math. Soc., Volume 132 (1998) no. 626 | DOI | MR | Zbl
[Bor99] A cohomological obstruction to the Hasse principle for homogeneous spaces, Math. Ann., Volume 314 (1999) no. 3, pp. 491-504 | DOI | MR | Zbl
[BT87] Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 34 (1987) no. 3, pp. 671-698 | Zbl
[BvH09] Extended Picard complexes and linear algebraic groups, J. Reine Angew. Math., Volume 627 (2009), pp. 53-82 | MR | Zbl
[BvH12] Extended equivariant Picard complexes and homogeneous spaces, Transform. Groups, Volume 17 (2012) no. 1, pp. 51-86 | DOI | MR | Zbl
[Con12] Finiteness theorems for algebraic groups over function fields, Compos. Math., Volume 148 (2012) no. 2, pp. 555-639 | DOI | MR | Zbl
[CT08] Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math., Volume 618 (2008), pp. 77-133 | Zbl
[DeB06] Parameterizing conjugacy classes of maximal unramified tori via Bruhat–Tits theory, Mich. Math. J., Volume 54 (2006) no. 1, pp. 157-178 | MR | Zbl
[Dem11a] Le défaut d’approximation forte dans les groupes linéaires connexes, Proc. Lond. Math. Soc., Volume 102 (2011) no. 3, pp. 563-597 | DOI | MR | Zbl
[Dem11b] Suites de Poitou–Tate pour les complexes de tores à deux termes, Int. Math. Res. Not., Volume 2011 (2011) no. 1, pp. 135-174 | DOI | MR | Zbl
[Dem11c] Une formule pour le groupe de Brauer algébrique d’un torseur, J. Algebra, Volume 347 (2011) no. 1, pp. 96-132 | DOI | MR | Zbl
[Dem13] Abélianisation des espaces homogènes et applications arithmétiques, Torsors, étale homotopy and applications to rational points (London Mathematical Society Lecture Note Series), Volume 405, Cambridge University Press, 2013, pp. 138-209 | DOI | MR | Zbl
[Dem20] The étale Brauer–Manin obstruction to strong approximation on homogeneous spaces (2020) (https://arxiv.org/abs/2008.00570)
[DH19] Artin–Mazur–Milne duality for fppf cohomology, Algebra Number Theory, Volume 13 (2019) no. 10, pp. 2323-2357 | DOI | MR | Zbl
[DH20] Duality for complexes of tori over a global field of positive characteristic, J. Éc. Polytech., Math., Volume 7 (2020), pp. 831-870 | DOI | MR | Zbl
[FSS98] Grothendieck’s theorem on non-abelian and local-global principles, J. Am. Math. Soc., Volume 11 (1998) no. 3, pp. 731-750 | DOI | MR | Zbl
[GA12] Quasi-abelian crossed modules and nonabelian cohomology, J. Algebra, Volume 369 (2012), pp. 235-255 | DOI | MR | Zbl
[Gil02] Torseurs sur la droite affine, Transform. Groups, Volume 7 (2002) no. 3, pp. 231-245 | DOI | MR | Zbl
[Har75] Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III, J. Reine Angew. Math., Volume 274–275 (1975), pp. 125-138 | Zbl
[HS05] Arithmetic duality theorems for 1-motives, J. Reine Angew. Math., Volume 578 (2005), pp. 93-128 | MR | Zbl
[HS13] Descent theory for open varieties, Torsors, étale homotopy and applications to rational points (London Mathematical Society Lecture Note Series), Volume 405, Cambridge University Press, 2013, pp. 250-279 | DOI | MR | Zbl
[Mil06] Arithmetic Duality Theorems, second, BookSurge, LLC; Charleston, SC, 2006 | Zbl
[Pra77] Strong approximation for semi-simple groups over function fields, Ann. Math., Volume 105 (1977) no. 3, pp. 553-572 | DOI | MR | Zbl
[Ray70] Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, 119, Springer, 1970 | DOI | Zbl
[Ros21a] Tamagawa numbers and other invariants of pseudo-reductive groups over global function fields, Algebra Number Theory, Volume 15 (2021) no. 8, pp. 1865-1920 | DOI | MR | Zbl
[Ros21b] Tate duality in positive dimension over function fields (2021) (https://arxiv.org/abs/1805.00522, to appear in Memoirs of the American Mathematical Society)
[San81] Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math., Volume 327 (1981), pp. 12-80 | Zbl
[Tha13] On Galois cohomology of semisimple groups over local and global fields of positive characteristic, III, Math. Z., Volume 275 (2013) no. 3-4, pp. 1287-1315 | DOI | MR | Zbl
[Čes15] Poitou–Tate without restrictions on the order, Math. Res. Lett., Volume 22 (2015) no. 6, pp. 1621-1666 | DOI | MR | Zbl