Duke Mathematical Journal

The elementary obstruction and homogeneous spaces

M. Borovoi, J.-L. Colliot-Thélène, and A. N. Skorobogatov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let $k$ be a field of characteristic zero, and let ${\overline k}$ be an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\overline k}(X)$ for the function field of ${\overline X}=X\times_k{\overline k}$. If $X$ has a smooth $k$-point, the natural embedding of multiplicative groups ${\overline k}^*\hookrightarrow {\overline k}(X)^* $ admits a Galois-equivariant retraction.

In the first part of this article, equivalent conditions to the existence of such a retraction are given over local and then over global fields. Those conditions are expressed in terms of the Brauer group of $X$.

In the second part of the article, we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For $k$ local or global, and for such a variety $X$, in many situations but not all, the existence of a Galois-equivariant retraction to ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ ensures the existence of a $k$-rational point on $X$. For homogeneous spaces of linear algebraic groups, the technique also handles the case where $k$ is the function field of a complex surface.


Soient $k$ un corps de caractéristique nulle et ${\overline k}$ une clôture algébrique de $k$. Pour une $k$-variété $X$ géométriquement intègre, on note ${\overline k}(X)$ le corps des fonctions de ${\overline X}=X\times_k{\overline k}$. Si $X$ possède un $k$-point lisse, le plongement naturel de groupes multiplicatifs ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ admet une rétraction équivariante pour l'action du groupe de Galois de ${\overline k}$ sur $k$.

Dans la première partie de l'article, sur les corps locaux puis sur les corps globaux, on donne des conditions équivalentes à l'existence d'une telle rétraction équivariante. Ces conditions s'expriment en terme du groupe de Brauer de la variété $X$.

Dans la seconde partie de l'article, on considère le cas des espaces homogènes de groupes algébriques connexes, non nécessairement linéaires, avec groupes d'isotropie géométriques connexes. Pour $k$ local ou global, pour un tel espace homogène $X$, dans beaucoup de cas mais pas dans tous, l'existence d'une rétraction équivariante à ${\overline k}^*\hookrightarrow {\overline k}(X)^*$ implique l'existence d'un point $k$-rationnel sur $X$. Pour les espaces homogènes de groupes linéaires, la technique permet aussi de traiter le cas où $k$ est un corps de fonctions de deux variables sur les complexes

Article information

Duke Math. J. Volume 141, Number 2 (2008), 321-364.

First available in Project Euclid: 17 January 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G05: Rational points 11G99: None of the above, but in this section 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50]
Secondary: 11E72: Galois cohomology of linear algebraic groups [See also 20G10] 14F22: Brauer groups of schemes [See also 12G05, 16K50] 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 14M17: Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15] 20G99: None of the above, but in this section


Borovoi, M.; Colliot-Thélène, J.-L.; Skorobogatov, A. N. The elementary obstruction and homogeneous spaces. Duke Math. J. 141 (2008), no. 2, 321--364. doi:10.1215/S0012-7094-08-14124-9. http://projecteuclid.org/euclid.dmj/1200601794.

Export citation


  • M. Borovoi, ``On weak approximation in homogeneous spaces of simply connected algebraic groups'' in Automorphic Functions and Their Applications (Khabarovsk, U.S.S.R., 1988), Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, U.S.S.R., 1990, 64--81., available at http://www.math.tau.ac.il/$^\sim$borovoi/publ.html
  • —, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), 217--239.
  • —, The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer, J. Reine Angew. Math. 473 (1996), 181--194.
  • —, A cohomological obstruction to the Hasse principle for homogeneous spaces, Math. Ann. 314 (1999), 491--504.
  • J.-L. Colliot-Thélène, ``Birational invariants, purity and the Gersten conjecture'' in K-theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Santa Barbara, 1992), Proc. Sympos. Pure Math. 58 (1995), Amer. Math. Soc., Providence, 1995, 1--64.
  • J.-L. Colliot-Thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over $2$-dimensional geometric fields, Duke Math. J. 121 (2004), 285--341.
  • J.-L. Colliot-Thélène and B. è. Kunyavskiĭ, Groupe de Picard et groupe de Brauer des compactifications lisses d'espaces homogènes, J. Algebraic Geom. 15 (2006), 733--752.
  • J.-L. Colliot-Thélène, M. Ojanguren, and R. Parimala, ``Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes'' in Algebra, Arithmetic and Geometry, I, II (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math. 16, Tata Inst. Fund. Res., Bombay, 2002, 185--217.
  • J.-L. Colliot-Thélène and R. Parimala, Real components of algebraic varieties and étale cohomology, Invent. Math. 101 (1990), 81--99.
  • J.-L. Colliot-Thélène and S. Saito, Zéro-cycles sur les variétés $p$-adiques et groupe de Brauer, Internat. Math. Res. Notices 1996, no. 4, 151--160.
  • J.-L. Colliot-Thélène and J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375--492.
  • J.-L. Colliot-Thélène and F. Xu, Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms, preprint, \arxiv0712.1957v1 [math.NT]
  • B. Conrad, A modern proof of Chevalley's theorem on algebraic groups, J. Ramanujan Math. Soc. 17 (2002), 1--18.
  • D. F. Coray, Algebraic points on cubic hypersurfaces, Acta Arith. 30 (1976), 267--296.
  • A. J. De Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), 71--94.
  • A. J. De Jong, Xuhua He, and J. Starr, Points of rationally simply connected varieties over the function field of a surface and torsors for semisimple groups, in preparation.
  • M. Florence, Zéro-cycles de degré un sur les espaces homogènes, Int. Math. Res. Not. 2004, no. 54, 2897--2914.
  • K. Fujiwara, ``A proof of the absolute purity conjecture (after Gabber)'' in Algebraic Geometry 2000, Azumino (Hotaka, Japan, 2000), Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002, 153--183.
  • P. Gabriel, ``Généralités sur les groupes algébriques'' in Schémas en groupes, I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3), Lecture Notes in Math. 151, Springer, Berlin, 1970, fasc. 2a, no. 6a.
  • W.-D. Geyer, Dualität bei abelschen Varietäten über reell abgeschlossenen Körpern, J. Reine Angew. Math. 293/294 (1977), 62--66.
  • M. J. Greenberg, Rational points in Henselian discrete valuation rings, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 59--64.
  • A. Grothendieck, ``Le groupe de Brauer, I: Algèbres d'Azumaya et interprétations diverses,'' ``Le groupe de Brauer, II: Théorie cohomologique,'' ``Le groupe de Brauer, III: Exemples et compléments'' in Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. 3, North-Holland, Amsterdam, 1968, 46--188.
  • —, ``Techniques de construction et théorèmes d'existence en géométrie algébrique, III: Préschémas quotients'' in Séminaire Bourbaki, Vol. 6 (1960/1961), no. 212, Soc. Math. France, Montrouge, 1995, 99--118.
  • D. Harari, The Manin obstruction for torsors under connected algebraic groups, Int. Math. Res. Not. 2006, no. 68632.
  • D. Harari and T. Szamuely, Local-global principles for $1$-motives, to appear in Duke Math. J., preprint,\arxivmath/0703845v2 [math.NT]
  • S. Lichtenbaum, Duality theorems for curves over $p$-adic fields, Invent. Math. 7 (1969), 120--136.
  • S. Mac Lane, Homology, Grundlehren Math. Wiss. 114, Springer, Berlin, 1963.
  • Y. I. Manin, ``Le groupe de Brauer-Grothendieck en géométrie diophantienne'' in Actes du Congrès International des Mathématiciens, I (Nice, 1970), Gauthier-Villars, Paris, 1971, 401--411.
  • B. Mazur, On the passage from local to global in number theory, Bull. Amer. Math. Soc. (N.S.) 29 (1993), 14--50.
  • A. S. Merkur'Ev [Merkurjev], Simple algebras and quadratic forms (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 55, no. 1 (1991), 218--224.; English translation in Math. USSR-Izv. 38 (1992), 215--221.
  • J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • —, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.
  • J. S. Milne and K.-Y. Shih, ``Conjugates of Shimura varieties'' in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982, 280--356.
  • D. Mumford, Abelian Varieties, Tata Inst. Fund. Res. Stud. Math. 5, Oxford Univ. Press, London, 1970.
  • M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401--443.
  • J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12--80.
  • C. Scheiderer, Real and Étale Cohomology, Lecture Notes in Math. 1588, Springer, Berlin, 1994.
  • J-P. Serre, Groupes algébriques et corps de classes, Publ. Inst. Math. Univ. Nancago 7, Hermann, Paris, 1959.
  • —, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin, 1994.
  • J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, New York, 1986.
  • A. N. Skorobogatov, Torsors and Rational Points, Cambridge Tracts in Math. 144, Cambridge Univ. Press, Cambridge, 2001.
  • —, On the elementary obstruction to the existence of rational points (in Russian), Mat. Zametki 81, no. 1 (2007), 112--124.; English translation in Math. Notes 81 (2007), 97--107.
  • T. A. Springer, ``Nonabelian $\H^2$ in Galois cohomology'' in Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence, 1966, 164--182.
  • J. Tate, ``$WC$-groups over ${\mathfrak{p}}$-adic fields'' in Séminaire Bourbaki, Vol. 4 (1956/ 57--1957./58), reprint of the 1966 ed., no. 156, Soc. Math. France, Montrouge, 1995, 265--277.
  • J. Van Hamel, Divisors on real algebraic varieties without real points, Manuscripta Math. 98 (1999), 409--424.
  • —, Lichtenbaum-Tate duality for varieties over $p$-adic fields, J. Reine Angew. Math. 575 (2004), 101--134.
  • V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, Transl. Math. Monogr. 179, Amer. Math. Soc., Providence, 1998.
  • L. Wang, Brauer-Manin obstruction to weak approximation on abelian varieties, Israel J. Math. 94 (1966), 189--200.
  • O. Wittenberg, On Albanese torsors and the elementary obstruction to the existence of $0$-cycles of degree $1$, to appear in Math. Ann., preprint.
  • X. L. Wu, ``On the extensions of abelian varieties by affine group schemes'' in Group Theory (Beijing, 1984), Lecture Notes in Math. 1185, Springer, Berlin, 1986, 361--387.