Strong Approximation with Brauer–Manin Obstruction for Toric Varieties  [ Approximation forte par rapport à l’obstruction de Brauer–Manin pour les variétés toriques ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, p. 1879-1908
Pour les variétés toriques lisses ouvertes, on établit l’approximation forte par rapport à l’obstruction de Brauer–Manin hors de infini.
For smooth open toric varieties, we establish strong approximation off infinity with Brauer–Manin obstruction.
Reçu le : 2015-12-09
Révisé le : 2016-12-20
Accepté le : 2017-11-07
Publié le : 2018-11-23
DOI : https://doi.org/10.5802/aif.3199
Classification:  11G35,  14G05,  20G30
Mots clés: tore, variété torique, approximation forte, obstruction de Brauer–Manin
@article{AIF_2018__68_5_1879_0,
     author = {Cao, Yang and Xu, Fei},
     title = {Strong Approximation with Brauer--Manin Obstruction for Toric Varieties},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     pages = {1879-1908},
     doi = {10.5802/aif.3199},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_5_1879_0}
}
Cao, Yang; Xu, Fei. Strong Approximation with Brauer–Manin Obstruction for Toric Varieties. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1879-1908. doi : 10.5802/aif.3199. https://aif.centre-mersenne.org/item/AIF_2018__68_5_1879_0/

[1] Borovoi, Mikhail; Demarche, Cyril Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv., Tome 88 (2013), pp. 1-54 | Zbl 1271.14073

[2] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 21 (1990), x+325 pages | Zbl 0705.14001

[3] Chambert-Loir, Antoine; Tschinkel, Yuri Integral points of bounded height on toric varieties (2012) (https://arxiv.org/abs/1006.3345v2 )

[4] Colliot-Thélène, Jean-Louis Birational invariants, purity and the Gersten conjecture, K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, Proceedings of Symposia in Pure Mathematics, Part I, American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 58 (1992), pp. 1-64 | Zbl 0834.14009

[5] Colliot-Thélène, Jean-Louis; Harari, David Approximation forte en famille, J. Reine Angew. Math., Tome 710 (2016), pp. 173-198 | Zbl 1334.14014

[6] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques Cohomologie des groupes de type multiplicatif sur les schémas réguliers, C. R. Math. Acad. Sci. Paris, Tome 287 (1978), pp. 449-452 | Zbl 0399.14011

[7] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques La descente sur les variétés rationnelles II, Duke Math. J., Tome 54 (1987), pp. 375-492 | Zbl 0659.14028

[8] Colliot-Thélène, Jean-Louis; Wittenberg, Olivier Groupe de Brauer et points entiers de deux familles de surfaces cubiques affines, Am. J. Math., Tome 134 (2012) no. 5, pp. 1303-1327 | Zbl 1295.14020

[9] Colliot-Thélène, Jean-Louis; Xu, Fei Brauer–Manin obstruction for integral points of homogeneous spaces and representations by integral quadratic forms, Compos. Math., Tome 145 (2009) no. 2, pp. 309-363 | Zbl 1190.11036

[10] Colliot-Thélène, Jean-Louis; Xu, Fei Strong approximation for the total space of certain quadric fibrations, Acta Arith., Tome 157 (2013) no. 2, pp. 169-199 | Zbl 1328.11060

[11] Conrad, Brian Weil and Grothendieck approaches to adelic points, Enseign. Math., Tome 58 (2012) no. 1-2, pp. 61-97 | Zbl 1316.14002

[12] Cox, David; Little, John; Schenck, Henry Toric Varieties, American Mathematical Society, Graduate Studies in Mathematics, Tome 124 (2011), xxiv+841 pages | Zbl 1223.14001

[13] Demarche, Cyril Le défaut d’approximation forte dans les groupes linéaires connexes, Proc. Lond. Math. Soc., Tome 102 (2011) no. 3, pp. 563-597 | Zbl 1216.11047

[14] Demazure, Michel; Grothendieck, Alexander Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux (SGA 3), Springer, Lecture Notes in Math., Tome 152 (1970), ix+654 pages | Zbl 0209.24201

[15] Fulton, William Introduction to toric varieties, Princeton University Press, Annals of Mathematics Studies, Tome 131 (1993), xi+157 pages | Zbl 0813.14039

[16] Grothendieck, Alexander Le groupe de Brauer (I, II, III), Dix exposés sur la cohomologie des schéma, North-Holland; Masson (Advanced Studies in Pure Mathematics) Tome 3 (1968), pp. 46-189 | Zbl 0193.21503;0198.25901

[17] Harari, David Le défaut d’approximation forte pour les groupes algébriques commutatifs, Algebra Number Theory, Tome 2 (2008) no. 5, pp. 595-611 | Zbl 1194.14067

[18] Harari, David; Voloch, José Felipe The Brauer-Manin obstruction for integral points on curves, Math. Proc. Camb. Philos. Soc., Tome 149 (2010), pp. 413-421 | Zbl 1280.11038

[19] Milne, James Stuart Étale cohomology, Princeton University Press, Princeton Mathematical Series, Tome 33 (1980) | Zbl 0433.14012

[20] Oda, Tadao Convex bodies and algebraic geometry, Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Tome 15 (1987), viii+212 pages | Zbl 0628.52002

[21] Platonov, Vladimir; Rapinchuk, Andrei Algebraic groups and number theory, Academic Press, Pure and Applied Mathematics, Tome 139 (1994), xi+614 pages | Zbl 0841.20046

[22] Sansuc, Jean-Jacques Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math., Tome 327 (1981), pp. 12-80 | Zbl 0468.14007

[23] Skorobogatov, Alexei N. Torsors and rational points, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 144 (2001), viii+187 pages | Zbl 0972.14015

[24] Sumihiro, Hideyasu Equivariant completion, J. Math. Kyoto Univ., Tome 14 (1974), pp. 1-28 | Zbl 0277.14008

[25] Wei, Dasheng Strong approximation for the variety containing a torus (2014) (https://arxiv.org/abs/1403.1035 )

[26] Wei, Dasheng; Xu, Fei Integral points for groups of multiplicative type, Adv. Math., Tome 232 (2013) no. 1, pp. 36-56 | Zbl 1282.11022