Strong Approximation with Brauer–Manin Obstruction for Toric Varieties
Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 1879-1908.

For smooth open toric varieties, we establish strong approximation off infinity with Brauer–Manin obstruction.

Pour les variétés toriques lisses ouvertes, on établit l’approximation forte par rapport à l’obstruction de Brauer–Manin hors de infini.

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Published online:
DOI: 10.5802/aif.3199
Classification: 11G35,  14G05,  20G30
Keywords: torus, toric variety, strong approximation, Brauer–Manin obstruction
License: CC-BY-ND 4.0
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Cao, Yang; Xu, Fei. Strong Approximation with Brauer–Manin Obstruction for Toric Varieties. Annales de l'Institut Fourier, Volume 68 (2018) no. 5, pp. 1879-1908. doi : 10.5802/aif.3199. https://aif.centre-mersenne.org/articles/10.5802/aif.3199/

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

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

[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 (Proceedings of Symposia in Pure Mathematics) Tome 58, American Mathematical Society, 1992, pp. 1-64 | Zbl

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

[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

[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

[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

[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

[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

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

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

[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

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

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

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

[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

[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

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

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

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

[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

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

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

[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

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