Duke Mathematical Journal

Local-global principles for $1$-motives

David Harari and Tamás Szamuely

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


Building upon our arithmetic duality theorems for $1$-motives, we prove that the Manin obstruction related to a finite subquotient ${\cyrille B} (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shafarevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate-type dual exact sequence for $1$-motives and give an application to weak approximation

Article information

Duke Math. J. Volume 143, Number 3 (2008), 531-557.

First available in Project Euclid: 3 June 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14G25: Global ground fields
Secondary: 14G05: Rational points


Harari, David; Szamuely, Tamás. Local-global principles for 1 -motives. Duke Math. J. 143 (2008), no. 3, 531--557. doi:10.1215/00127094-2008-028. http://projecteuclid.org/euclid.dmj/1212500466.

Export citation


  • M. Borovoi, J.-L. Colliot-ThéLèNe, and A. N. Skorobogatov, The elementary obstruction and homogeneous spaces, Duke Math. J. 141 (2008), 321--364.
  • M. Borovoi and J. Van Hamel, Extended Picard complexes and linear algebraic groups, preprint,\arxivmath/0612156v1[math.AG]
  • S. Bosch, W. LüTkebohmert, and M. Raynaud, Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.
  • J.-L. Colliot-ThéLèNe, Résolutions flasques des groupes linéaires connexes, or appear in J. Reine Angew. Math.
  • J.-L. Colliot-ThéLèNe and J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987), 375--492.
  • P. Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5--77.
  • D. Eriksson and V. Scharaschkin, On the Brauer-Manin obstruction for zero-cycles on curves, preprint,\arxivmath/0602355v1[math.NT]
  • D. Harari, The Manin obstruction for torsors under connected algebraic groups, Int. Math. Res. Not. 2006, no. 68632.
  • D. Harari and T. Szamuely, Arithmetic duality theorems for $1$-motives, J. Reine Angew. Math. 578 (2005), 93--128.
  • Y. I. Manin, ``Le groupe de Brauer-Grothendieck en géométrie diophantienne'' in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, 401--411.
  • J. S. Milne, Étale Cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • —, Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.
  • F. Oort, Commutative Group Schemes, Lecture Notes in Math. 15, Springer, Berlin, 1966.
  • N. Ramachandran, Duality of Albanese and Picard $1$-motives, $K$-Theory 21 (2001), 271--301.
  • 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.
  • J.-P. Serre, ``Morphismes universels et différentielles de troisième espèce'' in Séminaire C. Chevalley 3ième année: 1958/59: Variétès de Picard, École Norm. Sup., Paris, 1960, no. 11.
  • —, ``Morphismes universels et variété d'Albanese'' in Séminaire C. Chevalley 3ième année: 1958/59: Variétès de Picard, Ecole Norm. Sup., Paris, 1960, no. 10.
  • 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.
  • O. Wittenberg, On Albanese torsors and the elementary obstruction, Math. Ann. 340 (2008), 805--838.