×

zbMATH — the first resource for mathematics

Insufficiency of the Brauer-Manin obstruction applied to étale covers. (English) Zbl 1284.11096
Summary: Let \(k\) be any global field of characteristic not 2. We construct a \(k\)-variety \(X\) such that \(X(k)\) is empty, but for which the emptiness cannot be explained by the Brauer-Manin obstruction or even by the Brauer-Manin obstruction applied to finite étale covers.

MSC:
11G35 Varieties over global fields
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
PDF BibTeX XML Cite
Full Text: DOI Link arXiv
References:
[1] J. Colliot-Thélène, ”Conjectures de type local-global sur l’image des groupes de Chow dans la cohomologie étale,” in Algebraic \(K\)-Theory, Providence, RI, 1999, pp. 1-12. · Zbl 0981.14003
[2] J. Colliot-Thélène, ”Zéro-cycles de degré 1 sur les solides de Poonen,” , preprint , 2009.
[3] J. Colliot-Thélène, J. Sansuc, and P. Swinnerton-Dyer, ”Intersections of two quadrics and Châtelet surfaces. I,” J. Reine Angew. Math., vol. 373, pp. 37-107, 1987. · Zbl 0622.14029 · crelle:GDZPPN00220410X · eudml:152886
[4] J. Colliot-Thélène, J. Sansuc, and P. Swinnerton-Dyer, ”Intersections of two quadrics and Châtelet surfaces. II,” J. Reine Angew. Math., vol. 374, pp. 72-168, 1987. · Zbl 0622.14030 · crelle:GDZPPN002204215 · eudml:152896
[5] C. Demarche, ”Obstruction de descente et obstruction de Brauer-Manin étale,” Algebra Number Theory, vol. 3, iss. 2, pp. 237-254, 2009. · Zbl 1247.11090 · doi:10.2140/ant.2009.3.237 · pjm.math.berkeley.edu
[6] A. Grothendieck, ”Le groupe de Brauer. III. Exemples et compléments,” in Dix Exposés sur la Cohomologie des Schémas, Amsterdam: North-Holland, 1968, pp. 88-188. · Zbl 0198.25901
[7] D. Harari, ”Groupes algébriques et points rationnels,” Math. Ann., vol. 322, iss. 4, pp. 811-826, 2002. · Zbl 1042.14004 · doi:10.1007/s002080100289
[8] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52. · Zbl 0367.14001
[9] V. A. Iskovskih, ”A counterexample to the Hasse principle for systems of two quadratic forms in five variables,” Mat. Zametki, vol. 10, pp. 253-257, 1971. · Zbl 0232.10015 · doi:10.1007/BF01464715
[10] C. Lind, ”Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins,” Thesis, University of Uppsala,, vol. 1940, p. 97, 1940. · Zbl 0025.24802
[11] Y. I. Manin, ”Le groupe de Brauer-Grothendieck en géométrie diophantienne,” in Actes du Congrès International des Mathématiciens , Tome 1, Paris: Gauthier-Villars, 1971, pp. 401-411. · Zbl 0239.14010
[12] B. Poonen, ”The Hasse principle for complete intersections in projective space,” in Rational Points on Algebraic Varieties, Basel: Birkhäuser, 2001, pp. 307-311. · Zbl 1079.14027
[13] B. Poonen, ”Existence of rational points on smooth projective varieties,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 11, iss. 3, pp. 529-543, 2009. · Zbl 1183.14032 · doi:10.4171/JEMS/159 · www.ems-ph.org · arxiv:0712.1782
[14] H. Reichardt, ”Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen,” J. Reine Angew. Math., vol. 184, pp. 12-18, 1942. · Zbl 0026.29701 · doi:10.1515/crll.1942.184.12 · crelle:GDZPPN002175266 · eudml:150100
[15] P. Sarnak and L. Wang, ”Some hypersurfaces in \({\mathbf P}^4\) and the Hasse-principle,” C. R. Acad. Sci. Paris Sér. I Math., vol. 321, iss. 3, pp. 319-322, 1995. · Zbl 0857.14013
[16] A. Grothendieck, Revêtements étales et groupe fondamental (SGA 1), Paris: Société Mathématique de France, 2003.
[17] A. N. Skorobogatov, ”Beyond the Manin obstruction,” Invent. Math., vol. 135, iss. 2, pp. 399-424, 1999. · Zbl 0951.14013 · doi:10.1007/s002220050291
[18] A. N. Skorobogatov, Torsors and Rational Points, Cambridge: Cambridge Univ. Press, 2001. · Zbl 0972.14015
[19] A. N. Skorobogatov, ”Descent obstruction is equivalent to étale Brauer-Manin obstruction,” Math. Ann., vol. 344, iss. 3, pp. 501-510, 2009. · Zbl 1180.14017 · doi:10.1007/s00208-008-0314-4
[20] M. Stoll, ”Finite descent obstructions and rational points on curves,” Algebra Number Theory, vol. 1, iss. 4, pp. 349-391, 2007. · Zbl 1167.11024 · doi:10.2140/ant.2007.1.349 · pjm.math.berkeley.edu · arxiv:math/0606465
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.