Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Good reduction of the Brauer-Manin obstruction

Authors: Jean-Louis Colliot-Thélène and Alexei N. Skorobogatov
Journal: Trans. Amer. Math. Soc. 365 (2013), 579-590
MSC (2010): Primary 14F22, 14G05, 11G35, 11G25
Published electronically: September 19, 2012
MathSciNet review: 2995366
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a smooth and projective variety over a number field with torsion-free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer-Manin set.

References [Enhancements On Off] (What's this?)

  • 1. M. Bright, Efficient evaluation of the Brauer-Manin obstruction. Math. Proc. Cambridge Philos. Soc. 142 (2007) 13-23. MR 2296387 (2007k:14026)
  • 2. M. Bright, Evaluating Azumaya algebras on cubic surfaces. Manuscripta Math. 134 (2011) 405-421. MR 2765718 (2012c:11136)
  • 3. J.-L. Colliot-Thélène et J.-J. Sansuc, La descente sur les variétés rationnelles, II, Duke Math. J. 54 (1987) 375-492. MR 899402 (89f:11082)
  • 4. O. Debarre, Higher-Dimensional Algebraic Geometry, Universitext, Springer-Verlag, 2001. MR 1841091 (2002g:14001)
  • 5. B. Fantechi, L. Göttsche, L. Illusie, S.L. Kleiman, N. Nitsure, and A. Vistoli, Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs 123, AMS, Providence, RI, 2005. MR 2222646 (2007f:14001)
  • 6. K. Fujiwara, A proof of the absolute purity conjecture (after Gabber). In: Algebraic geometry 2000, Azumino, Adv. Stud. Pure Math. 36 (2002), pp. 153-183. MR 1971516 (2004d:14015)
  • 7. O. Gabber, Some theorems on Azumaya algebras. In: The Brauer group (Séminaire, Les Plans-sur-Bex, 1980) Lecture Notes in Math. 844, Springer-Verlag, Berlin-New York, 1981, pp. 129-209. MR 611868 (83d:13004)
  • 8. A. Grothendieck, Le groupe de Brauer, I, II, III. In: Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 46-188. MR 0244269 (39:5586a)
  • 9. E. Ieronymou, A.N. Skorobogatov and Yu.G. Zarhin, On the Brauer group of diagonal quartic surfaces. J. London Math. Soc. (2) 83 (2011), 659-672. arXiv:0912.2865 MR 2802504
  • 10. K. Kato, A Hasse principle for two-dimensional global fields. J. reine angew. Math. 366 (1986) 142-181. MR 833016 (88b:11036)
  • 11. N. Katz, Applications of the weak Lefschetz theorem. Appendix to B. Poonen and J.F. Voloch, Random Diophantine equations. Progr. Math. 226 Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), Birkhäuser, Boston, MA, 2004, pp. 175-184. MR 2029869 (2005g:11055)
  • 12. J. S. Milne, Arithmetic duality theorems, Persp. Math. 1, Academic Press, 1986. MR 881804 (88e:14028)
  • 13. J.S. Milne, Étale cohomology. Princeton University Press, 1980. MR 559531 (81j:14002)
  • 14. J-P. Serre, Corps locaux. Hermann, Paris, 1968. MR 0354618 (50:7096)
  • 15. A.N. Skorobogatov, Descent on fibrations over the projective line. Amer. J. Math. 118 (1996) 905-923. MR 1408492 (97k:11099)
  • 16. A. Skorobogatov, Torsors and rational points. Cambridge University Press, 2001. MR 1845760 (2002d:14032)
  • 17. A.N. Skorobogatov and Yu.G. Zarhin, A finiteness theorem for the Brauer group of abelian varieties and $ K3$ surfaces. J. Alg. Geom. 17 (2008) 481-502. MR 2395136 (2009d:14016)
  • 18. A.N. Skorobogatov and Yu.G. Zarhin, The Brauer group of Kummer surfaces and torsion of elliptic curves. J. reine angew. Math. 666 (2012) 115-140.
  • 19. Sir Peter Swinnerton-Dyer, Density of rational points on certain surfaces. Preprint, 2010.
  • 20. J. Tate, Relations between $ K_{2}$ and Galois cohomology, Invent. Math. 36 (1976) 257-274. MR 0429837 (55:2847)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14F22, 14G05, 11G35, 11G25

Retrieve articles in all journals with MSC (2010): 14F22, 14G05, 11G35, 11G25

Additional Information

Jean-Louis Colliot-Thélène
Affiliation: CNRS, UMR 8628, Mathématiques, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France

Alexei N. Skorobogatov
Affiliation: Department of Mathematics, South Kensington Campus, Imperial College London, SW7 2BZ England, United Kingdom – and – Institute for the Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow, 127994 Russia

Received by editor(s): September 1, 2010
Published electronically: September 19, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society