The Brauer group of torsors and its arithmetic applications
[Le groupe de Brauer des torseurs et ses applications arithmétiques]
Harari, David ; Skorobogatov, Alexei N.
Annales de l'Institut Fourier, Tome 53 (2003), p. 1987-2019 / Harvested from Numdam

Soit X une variété algébrique définie sur un corps de caractéristique zéro k. Soit Y un X-torseur sous un tore. Nous calculons le groupe de Brauer de Y et nous en déduisons des conséquences arithmétiques pour X quand k est un corps de nombres.

Let X be an algebraic variety defined over a field k of characteristic 0, and let Y be an X-torsor under a torus. We compute the Brauer group of Y. In the case of a number field k we deduce results concerning the arithmetic of X.

Publié le : 2003-01-01
DOI : https://doi.org/10.5802/aif.1998
Classification:  11G35,  14G05
Mots clés: groupe de Brauer, principe de Hasse, torseur universel
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     author = {Harari, David and Skorobogatov, Alexei N.},
     title = {The Brauer group of torsors and its arithmetic applications},
     journal = {Annales de l'Institut Fourier},
     volume = {53},
     year = {2003},
     pages = {1987-2019},
     doi = {10.5802/aif.1998},
     mrnumber = {2044165},
     zbl = {02093464},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2003__53_7_1987_0}
}
Harari, David; Skorobogatov, Alexei N. The Brauer group of torsors and its arithmetic applications. Annales de l'Institut Fourier, Tome 53 (2003) pp. 1987-2019. doi : 10.5802/aif.1998. http://gdmltest.u-ga.fr/item/AIF_2003__53_7_1987_0/

[CF] J.W.S. Cassels And A. Fröhlich (Ed.) Algebraic number theory, Academic Press, London and New-York (1967)

[CS00] J.-L. Colliot-Thélène; A. N. Skorobogatov Descent on fibrations over P k 1 revisited, Math. Proc. Camb. Phil. Soc, Tome 128 (2000), pp. 383-393 | Article | MR 1744112 | Zbl 1024.14003

[CS77] J.-L. Colliot-Thélène; J.-J. Sansuc La R-équivalence sur les tores., Ann. Sci. École Norm. Sup., Tome 10 (1977), pp. 175-230 | Numdam | MR 450280 | Zbl 0356.14007

[CS87a] J.-L. Colliot-Thélène; J.-J. Sansuc La descente sur les variétés rationnelles, II, Duke Math. J., Tome 54 (1987), pp. 375-492 | Article | MR 899402 | Zbl 0659.14028

[CS87b] J.-L. Colliot-Thélène; J.-J. Sansuc Principal homogeneous spaces under flasque tori; applications, J. Algebra, Tome 106 (1987), pp. 148-205 | Article | MR 878473 | Zbl 0597.14014

[CSS87] J.-L. Colliot-Thélène; J.-J. Sansuc; Sir Peter Swinnerton-Dyer Intersections of two quadrics and Châtelet surfaces. I., J. reine angew. Math., Tome 373 (1987), pp. 37-107 | MR 870307 | Zbl 0622.14029

[CSS87] J.-L. Colliot-Thélène; J.-J. Sansuc; Sir Peter Swinnerton-Dyer Intersections of two quadrics and Châtelet surfaces. II., J. Reine Angew. Math., Tome 374 (1987), pp. 72-168 | MR 876222 | Zbl 0622.14030

[CT] J.-L. Colliot-Thélène; 4 Surfaces rationnelles fibrées en coniques de degré 4, Séminaire de Théorie des Nombres, Paris 1988-1989, Birkhäuser, Boston (Progr. Math.) Tome 91 (1990), pp. 43-55 | Zbl 0731.14033

[EGA4] A. Grothendieck Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas (EGA), Publ. Math. IHES (1964-1967) | Numdam | Zbl 0135.39701

[G] A. Grothendieck Le groupe de Brauer. III. Exemples et compléments., Dix exposés sur la cohomologie des schémas, Masson-North-Holland, Amsterdam (1968) | Zbl 0198.25901

[GHS] T. Graber; J. Harris; J. Starr Families of rationally connected varieties, J. Amer. Math. Soc, Tome 16 (2003), pp. 57-67 | Article | MR 1937199 | Zbl 01832408

[H94] D. Harari Méthode des fibrations et obstruction de Manin, Duke Math. J, Tome 75 (1994), pp. 221-260 | Article | MR 1284820 | Zbl 0847.14001

[H97] D. Harari Flèches de spécialisation en cohomologie étale et applications arithmétiques, Bull. Soc. Math. France, Tome 125 (1997), pp. 143-166 | Numdam | MR 1478028 | Zbl 0906.14014

[KST] B. Kunyavskiǐ; A.N. Skorobogatov; M.A. Tsfasman Del Pezzo surfaces of degree four, Mém. Soc. Math. France, Tome 37 (1989) | Numdam | MR 1016354 | Zbl 0705.14039

[M] J.-S. Milne Étale Cohomology, Princeton Univ. Press, Princeton Tome 33 (1980) | MR 559531 | Zbl 0433.14012

[S01] A.N. Skorobogatov Torsors and rational points, Cambridge Univ. Press, Cambridge (2001) | MR 1845760 | Zbl 0972.14015

[S90] A.N. Skorobogatov Arithmetic on certain quadric bundles of relative dimension. I., J. reine angew. Math, Tome 407 (1990), pp. 57-74 | Article | MR 1048528 | Zbl 0692.14001

[S96] A.N. Skorobogatov Descent on fibrations over the projective line, Amer. J. Math, Tome 118 (1996), pp. 905-923 | Article | MR 1408492 | Zbl 0880.14008

[SD] Sir Peter Swinnerton; - Dyer Rational points on some pencils of conics with 6 singular fibres, Ann. Fac. Sci. Toulouse, Tome 8 (1999), pp. 331-341 | Article | Numdam | MR 1751446 | Zbl 0976.14014

[SS] P. Salberger; A.N. Skorobogatov Weak approximation for surfaces defined by two quadratic forms, Duke Math. J, Tome 63 (1991), pp. 517-536 | MR 1115119 | Zbl 0770.14019

[V] V. E. Voskresenskiǐ Algebraic groups and their birational invariants, Amer. Math. Soc., Providence, Translations of Mathematical Monographs, Tome 179 (1998) | MR 1634406 | Zbl 0974.14034