Algebra & Number Theory

Le défaut d'approximation forte pour les groupes algébriques commutatifs

David Harari

Full-text: Open access

Abstract

On établit une suite exacte décrivant l’adhérence des points rationnels d’un 1-motif dans ses points adéliques. On en déduit ensuite que le défaut d’approximation forte pour un groupe algébrique commutatif G est essentiellement mesuré par son groupe de Brauer algébrique via l’obstruction de Brauer-Manin entière.

We give an exact sequence describing the closure of the set of rational points of a 1-motive in its adelic points. From this we deduce that for a commutative algebraic group, the defect of strong approximation is essentially controlled by its algebraic Brauer group, by means of the integral Brauer-Manin obstruction.

Article information

Source
Algebra Number Theory, Volume 2, Number 5 (2008), 595-611.

Dates
Received: 22 April 2008
Revised: 26 May 2008
Accepted: 26 June 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513797290

Digital Object Identifier
doi:10.2140/ant.2008.2.595

Mathematical Reviews number (MathSciNet)
MR2429455

Zentralblatt MATH identifier
1194.14067

Subjects
Primary: 14L15: Group schemes
Secondary: 12G05: Galois cohomology [See also 14F22, 16Hxx, 16K50] 11G09: Drinfelʹd modules; higher-dimensional motives, etc. [See also 14L05]

Keywords
approximation forte groupe de Brauer $1$-motif strong approximation Brauer group $1$-motive

Citation

Harari, David. Le défaut d'approximation forte pour les groupes algébriques commutatifs. Algebra Number Theory 2 (2008), no. 5, 595--611. doi:10.2140/ant.2008.2.595. https://projecteuclid.org/euclid.ant/1513797290


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References

  • J.-L. Colliot-Thélène and X. Fei, “Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms”, prépublication, 2007, http://www.math.u-psud.fr/~colliot/CTXuFei4feb08.pdf.
  • D. Harari, “The Manin obstruction for torsors under connected algebraic groups”, Int. Math. Res. Not. 2006 (2006), Art. ID 68632.
  • D. Harari and T. Szamuely, “Arithmetic duality theorems for 1-motives”, J. Reine Angew. Math. 578 (2005), 93–128.
  • D. Harari and T. Szamuely, “Local-global principles for $1$-motives”, Duke Math. J. 143:3 (2008), 531–557.
  • J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, NJ, 1980.
  • J. S. Milne, Arithmetic duality theorems, Second ed., BookSurge, LLC, Charleston, SC, 2006.
  • \leavevmode\hskip-2pt N. Naumann, “Arithmetically defined dense subgroups of Morava stabilizer groups”, Compos. Math. 144:1 (2008), 247–270.
  • J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Math. Wiss. 323, Springer, Berlin, 2008.
  • V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Math. 139, Academic Press, Boston, 1994.
  • 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, Cohomologie galoisienne, 5ème ed., Lecture Notes in Mathematics 5, Springer, Berlin, 1994.
  • A. Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics 144, Cambridge University Press, Cambridge, 2001.