The étale Brauer-Manin obstruction to strong approximation on homogeneous spaces

Demeio, Julian

doi:10.1090/tran/8705

The étale Brauer-Manin obstruction to strong approximation on homogeneous spaces
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by Julian L. Demeio PDF

Trans. Amer. Math. Soc. 375 (2022), 8581-8634

Abstract:

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces $X$ under a linear group $G$ (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of $X$ are connected. In this work we prove, under similar assumptions, that the étale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilizers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients $G/H$, where $G$ and $H$ are connected algebraic groups and $H$ is linear, and certain abelianization morphisms associated with these spaces.

References

M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer-Verlag, Berlin, 1986. Reprint of the 1969 original. MR 883959
Francesca Balestrieri, Some remarks on strong approximation and applications to homogeneous spaces of linear algebraic groups, Proc. A.M.S., To appear, Preprint, https://fbalestrieri.github.io/files/SROSAAATHSOLAG.pdf
M. Borovoi, J.-L. Colliot-Thélène, and A. N. Skorobogatov, The elementary obstruction and homogeneous spaces, Duke Math. J. 141 (2008), no. 2, 321–364 (English, with English and French summaries). MR 2376817, DOI 10.1215/S0012-7094-08-14124-9
Mikhail Borovoi, The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer, J. Reine Angew. Math. 473 (1996), 181–194. MR 1390687, DOI 10.1515/crll.1995.473.181
Mikhail Borovoi, Abelianization of the first Galois cohomology of reductive groups, Internat. Math. Res. Notices 8 (1996), 401–407. MR 1393331, DOI 10.1155/S1073792896000268
Mikhail Borovoi and Cyril Demarche, Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv. 88 (2013), no. 1, 1–54. MR 3008912, DOI 10.4171/CMH/277
Mikhail Borovoi and Joost van Hamel, Extended Picard complexes for algebraic groups and homogeneous spaces, C. R. Math. Acad. Sci. Paris 342 (2006), no. 9, 671–674 (English, with English and French summaries). MR 2225874, DOI 10.1016/j.crma.2006.02.030
Mikhail V. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), no. 1, 217–239. MR 1242885, DOI 10.1215/S0012-7094-93-07209-2
Lawrence Breen, Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 401–476 (French). MR 1086889
Michel Brion, On the geometry of algebraic groups and homogeneous spaces, J. Algebra 329 (2011), no. 1, 52–71, Special issue celebrating the 60th birthday of Corrado De Concini.
Yang Cao, Sous-groupe de Brauer invariant et obstruction de descente itérée, Algebra Number Theory 14 (2020), no. 8, 2151–2183 (French, with English and French summaries). MR 4172704, DOI 10.2140/ant.2020.14.2151
Yang Cao, Cyril Demarche, and Fei Xu, Comparing descent obstruction and Brauer-Manin obstruction for open varieties, Trans. Amer. Math. Soc. 371 (2019), no. 12, 8625–8650. MR 3955558, DOI 10.1090/tran/7567
Yang Cao, Cyril Demarche, and Fei Xu, Comparing descent obstruction and Brauer-Manin obstruction for open varieties, Trans. Amer. Math. Soc. 371 (2019), no. 12, 8625–8650. MR 3955558, DOI 10.1090/tran/7567
Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492 (French). MR 899402, DOI 10.1215/S0012-7094-87-05420-2
Jean-Louis Colliot-Thélène and Fei Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, Compos. Math. 145 (2009), no. 2, 309–363. With an appendix by Dasheng Wei and Xu. MR 2501421, DOI 10.1112/S0010437X0800376X
Brian Conrad, A modern proof of Chevalley’s theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), no. 1, 1–18. MR 1906417
Cyril Demarche, Abélianisation des espaces homogenes et applications arithmétiques, Torsors, étale homotopy and applications to rational points, London Math. Soc. Lecture Note Ser., vol. 405, Cambridge Univ. Press, Cambridge, 2013, pp. 138–209 (French, with English and French summaries). MR 3077169
Cyril Demarche, Méthodes cohomologiques pour l’étude des points rationnels sur les espaces homogènes, Ph.D. Thesis, Université de Paris-Sud, Faculté des Sciences d’Orsay (Essonne), 2009.
Cyril Demarche, Une formule pour le groupe de Brauer algébrique d’un torseur, J. Algebra 347 (2011), 96–132 (French, with English summary). MR 2846400, DOI 10.1016/j.jalgebra.2011.09.003
Cyril Demarche, Obstructions de Brauer-Manin entières sur les espaces homogènes à stabilisateurs finis nilpotents, Bull. Soc. Math. France 145 (2017), no. 2, 225–236 (French, with English and French summaries). MR 3749784, DOI 10.24033/bsmf.2735
Santiago Encinas and Herwig Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–845. MR 1949115, DOI 10.1007/PL00012443
Stefan Gille, On the Brauer group of a semisimple algebraic group, Adv. Math. 220 (2009), no. 3, 913–925. MR 2483231, DOI 10.1016/j.aim.2008.10.004
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5, Institut des Hautes Études Scientifiques, Paris, 1963. Troisième édition, corrigée; Séminaire de Géométrie Algébrique, 1960/61. MR 0217087
David Harari, The Manin obstruction for torsors under connected algebraic groups, Int. Math. Res. Not. , posted on (2006), Art. ID 68632, 13. MR 2211158, DOI 10.1155/IMRN/2006/68632
David Harari, Cohomologie galoisienne et théorie du corps de classes, Savoirs Actuels (Les Ulis). [Current Scholarship (Les Ulis)], EDP Sciences, Les Ulis; CNRS Éditions, Paris, 2017 (French). MR 3561631
David Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75 (1994), no. 1, 221–260 (French). MR 1284820, DOI 10.1215/S0012-7094-94-07507-8
David Harari, Le défaut d’approximation forte pour les groupes algébriques commutatifs, Algebra Number Theory 2 (2008), no. 5, 595–611 (French, with English and French summaries). MR 2429455, DOI 10.2140/ant.2008.2.595
David Harari and Tamás Szamuely, Arithmetic duality theorems for 1-motives, J. Reine Angew. Math. 578 (2005), 93–128. MR 2113891, DOI 10.1515/crll.2005.2005.578.93
David Harari and Tamás Szamuely, Local-global principles for 1-motives, Duke Math. J. 143 (2008), no. 3, 531–557. MR 2423762, DOI 10.1215/00127094-2008-028
Yonatan Harpaz and Olivier Wittenberg, Zéro-cycles sur les espaces homogènes et problème de Galois inverse, J. Amer. Math. Soc. 33 (2020), no. 3, 775–805 (French). MR 4127903, DOI 10.1090/jams/943
Peter Jossen, On the arithmetic of 1-motives, Ph.D. Thesis, 2009.
Friedrich Knop, Hanspeter Kraft, and Thierry Vust, The Picard group of a $G$-variety, Algebraische Transformationsgruppen und Invariantentheorie, DMV Sem., vol. 13, Birkhäuser, Basel, 1989, pp. 77–87. MR 1044586
V. A. Kolyvagin, Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523–541. MR 954295, DOI 10.1070/IM1989v032n03ABEH000779
J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field. MR 3729270, DOI 10.1017/9781316711736
Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg, Cohomology of number fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, Springer-Verlag, Berlin, 2000. MR 1737196
V. Platonov, A. Rapinchuk, and R. Rowen, Algebraic groups and number theory, Pure and Applied Mathematics, Elsevier Science, 1993.
Bjorn Poonen and Michael Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 1740984, DOI 10.2307/121064
V. L. Popov, Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322 (Russian). MR 0357399
Claus Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics, vol. 1588, Springer-Verlag, Berlin, 1994. MR 1321819, DOI 10.1007/BFb0074269
Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. MR 1845760, DOI 10.1017/CBO9780511549588
Alexei Skorobogatov, Descent obstruction is equivalent to étale Brauer-Manin obstruction, Math. Ann. 344 (2009), no. 3, 501–510. MR 2501300, DOI 10.1007/s00208-008-0314-4
Michael Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), no. 4, 349–391. MR 2368954, DOI 10.2140/ant.2007.1.349
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Olivier Wittenberg, On Albanese torsors and the elementary obstruction, Math. Ann. 340 (2008), no. 4, 805–838. MR 2372739, DOI 10.1007/s00208-007-0170-7