Geodesic
Strong Approximation with Brauer–Manin Obstruction for Toric Varieties
[Approximation forte par rapport à l’obstruction de Brauer–Manin pour les variétés toriques]
Cao, Yang1 ; Xu, Fei1
1 School of Mathematical Sciences, Capital Normal University, 105 Xisanhuanbeilu, 100048 Beijing, China
Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 1879-1908.

Voir la notice de l'article provenant de la source Numdam

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3199
Classification : 11G35, 14G05, 20G30
Keywords: torus, toric variety, strong approximation, Brauer–Manin obstruction
Mots-clés : tore, variété torique, approximation forte, obstruction de Brauer–Manin

Cao, Yang 1 ; Xu, Fei 1

1 School of Mathematical Sciences, Capital Normal University, 105 Xisanhuanbeilu, 100048 Beijing, China
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2018__68_5_1879_0,
     author = {Cao, Yang and Xu, Fei},
     title = {Strong {Approximation} with {Brauer{\textendash}Manin} {Obstruction} for {Toric} {Varieties}},
     journal = {Annales de l'Institut Fourier},
     pages = {1879--1908},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     doi = {10.5802/aif.3199},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5802/aif.3199/}
}
TY  - JOUR
AU  - Cao, Yang
AU  - Xu, Fei
TI  - Strong Approximation with Brauer–Manin Obstruction for Toric Varieties
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 1879
EP  - 1908
VL  - 68
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://geodesic.mathdoc.fr/articles/10.5802/aif.3199/
DO  - 10.5802/aif.3199
LA  - en
ID  - AIF_2018__68_5_1879_0
ER  - 
%0 Journal Article
%A Cao, Yang
%A Xu, Fei
%T Strong Approximation with Brauer–Manin Obstruction for Toric Varieties
%J Annales de l'Institut Fourier
%D 2018
%P 1879-1908
%V 68
%N 5
%I Association des Annales de l’institut Fourier
%U http://geodesic.mathdoc.fr/articles/10.5802/aif.3199/
%R 10.5802/aif.3199
%G en
%F 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. http://geodesic.mathdoc.fr/articles/10.5802/aif.3199/

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

[2] Bosch, Siegfried; Lütkebohmert, Werner; Raynaud, Michel Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 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), Volume 58, American Mathematical Society, 1992, pp. 1-64 | Zbl

[5] Colliot-Thélène, Jean-Louis; Harari, David Approximation forte en famille, J. Reine Angew. Math., Volume 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, Volume 287 (1978), pp. 449-452 | Zbl

[7] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques La descente sur les variétés rationnelles II, Duke Math. J., Volume 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., Volume 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., Volume 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., Volume 157 (2013) no. 2, pp. 169-199 | Zbl

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

[12] Cox, David; Little, John; Schenck, Henry Toric Varieties, Graduate Studies in Mathematics, 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., Volume 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., 152, Springer, 1970, ix+654 pages | Zbl

[15] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 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), Volume 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, Volume 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., Volume 149 (2010), pp. 413-421 | Zbl

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

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

[21] Platonov, Vladimir; Rapinchuk, Andrei Algebraic groups and number theory, Pure and Applied Mathematics, 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., Volume 327 (1981), pp. 12-80 | Zbl

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

[24] Sumihiro, Hideyasu Equivariant completion, J. Math. Kyoto Univ., Volume 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., Volume 232 (2013) no. 1, pp. 36-56 | Zbl

Cité par Sources :