The Brauer group of torsors and its arithmetic applications

Harari, David; Skorobogatov, Alexei N.

doi:10.5802/aif.1998

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)

Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 1987-2019.

Abstract
Résumé

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 $X$ .

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.

MR Zbl

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

Author's affiliations:

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)

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Harari, David; Skorobogatov, Alexei N. The Brauer group of torsors and its arithmetic applications. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 1987-2019. doi : 10.5802/aif.1998. http://archive.numdam.org/articles/10.5802/aif.1998/

References
Cited by

[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

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