The Brauer group of torsors and its arithmetic applications
[Le groupe de Brauer des torseurs et ses applications arithmétiques]
Annales de l'Institut Fourier, Tome 53 (2003) no. 7, pp. 1987-2019.

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.

DOI : 10.5802/aif.1998
Classification : 11G35, 14G05
Keywords: Brauer group, Hasse principle, universal torsor
Mot clés : groupe de Brauer, principe de Hasse, torseur universel
Harari, David 1 ; Skorobogatov, Alexei N. 2

1 École Normale Supérieure, DMA, 45 rue d'Ulm, 75230 Paris Cedex 05 (France)
2 Imperial College, Department of Mathematics, 180 Queen's Gate, London SW7 2BZ (Royaume-Uni)
@article{AIF_2003__53_7_1987_0,
     author = {Harari, David and Skorobogatov, Alexei N.},
     title = {The {Brauer} group of torsors and its arithmetic applications},
     journal = {Annales de l'Institut Fourier},
     pages = {1987--2019},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     doi = {10.5802/aif.1998},
     mrnumber = {2044165},
     zbl = {02093464},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1998/}
}
TY  - JOUR
AU  - Harari, David
AU  - Skorobogatov, Alexei N.
TI  - The Brauer group of torsors and its arithmetic applications
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 1987
EP  - 2019
VL  - 53
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1998/
DO  - 10.5802/aif.1998
LA  - en
ID  - AIF_2003__53_7_1987_0
ER  - 
%0 Journal Article
%A Harari, David
%A Skorobogatov, Alexei N.
%T The Brauer group of torsors and its arithmetic applications
%J Annales de l'Institut Fourier
%D 2003
%P 1987-2019
%V 53
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.1998/
%R 10.5802/aif.1998
%G en
%F 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) no. 7, pp. 1987-2019. doi : 10.5802/aif.1998. http://www.numdam.org/articles/10.5802/aif.1998/

[CF] J.W.S. Cassels and A. Fröhlich (ed.) Algebraic number theory (1967)

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

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

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

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

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

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

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

[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

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

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

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

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

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

[M] J.-S. Milne Étale Cohomology, 33, Princeton Univ. Press, Princeton, 1980 | MR | Zbl

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

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

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

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

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

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

Cité par Sources :