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Mat. Zametki, 2007, Volume 81, Issue 1, Pages 112–124 (Mi mz3521)  

This article is cited in 6 scientific papers (total in 6 papers)

On the Elementary Obstruction to the Existence of Rational Points

A. N. Skorobogatovab

a Institute for Information Transmission Problems, Russian Academy of Sciences
b Imperial College, Department of Mathematics

Abstract: The differentials of a certain spectral sequence converging to the Brauer–Grothendieck group of an algebraic variety $X$ over an arbitrary field are interpreted as the $\cup$-product with the class of the so-called “elementary obstruction.” This class is closely related to the cohomology class of the first-degree Albanese variety of $X$. If $X$ is a homogeneous space of an algebraic group, then the elementary obstruction can be described explicitly in terms of natural cohomological invariants of $X$. This reduces the calculation of the Brauer–Grothendieck group to the computation of a certain pairing in the Galois cohomology.

Keywords: Brauer–Grothendieck group, algebraic variety over a field, elementary obstruction to the existence of rational points, Albanese variety, Picard variety, Galois cohomology

DOI: https://doi.org/10.4213/mzm3521

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English version:
Mathematical Notes, 2007, 81:1, 97–107

Bibliographic databases:

UDC: 512.74
Received: 21.10.2005
Revised: 04.07.2006

Citation: A. N. Skorobogatov, “On the Elementary Obstruction to the Existence of Rational Points”, Mat. Zametki, 81:1 (2007), 112–124; Math. Notes, 81:1 (2007), 97–107

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Harari D., Szamuely T., “Local-global principles for 1-motives”, Duke Math. J., 143:3 (2008), 531–557  crossref  mathscinet  zmath  isi  elib  scopus
    2. Wittenberg O., “On Albanese torsors and the elementary obstruction”, Math. Ann., 340:4 (2008), 805–838  crossref  mathscinet  zmath  isi  elib  scopus
    3. Borovoi M., Colliot-Thélène J.-L., Skorobogatov A. N., “The elementary obstruction and homogeneous spaces”, Duke Math. J., 141:2 (2008), 321–364  crossref  mathscinet  zmath  isi  elib  scopus
    4. Skorobogatov A. N., Zarhin Yu. G., “A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces”, J. Algebraic Geom., 17:3 (2008), 481–502  crossref  mathscinet  zmath  isi  elib  scopus
    5. Harari D., Szamuely T., “Galois sections for abelianized fundamental groups”, Math. Ann., 344:4 (2009), 779–800  crossref  mathscinet  zmath  isi  elib  scopus
    6. Beauville A., “On the Brauer group of Enriques surfaces”, Math. Res. Lett., 16:5-6 (2009), 927–934  crossref  mathscinet  zmath  isi  elib  scopus
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