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Matematicheskie Zametki, 2007, Volume 81, Issue 1, Pages 112–124
DOI: https://doi.org/10.4213/mzm3521 (Mi mzm3521) 		 
This article is cited in 7 scientific papers (total in 7 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
Full-text PDF (547 kB) Citations (7)
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DOI: https://doi.org/10.4213/mzm3521
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 ∪-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.
Received: 21.10.2005
Revised: 04.07.2006
English version:
Mathematical Notes, 2007, Volume 81, Issue 1, Pages 97–107
DOI: https://doi.org/10.1134/S0001434607010099
Bibliographic databases:
UDC: 512.74
Language: Russian
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
Citation in format AMSBIB
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    • This publication is cited in the following 7 articles:
    1. Ma Q., “Brauer Class Over the Picard Scheme of Totally Degenerate Stable Curves”, Math. Z., 298:3-4 (2021), 1641–1652  crossref  mathscinet  isi
    2. Harari D., Szamuely T., “Galois sections for abelianized fundamental groups”, Math. Ann., 344:4 (2009), 779–800  crossref  mathscinet  zmath  isi  elib  scopus
    3. Beauville A., “On the Brauer group of Enriques surfaces”, Math. Res. Lett., 16:5-6 (2009), 927–934  crossref  mathscinet  zmath  isi  elib  scopus
    4. Harari D., Szamuely T., “Local-global principles for 1-motives”, Duke Math. J., 143:3 (2008), 531–557  crossref  mathscinet  zmath  isi  elib  scopus
    5. Wittenberg O., “On Albanese torsors and the elementary obstruction”, Math. Ann., 340:4 (2008), 805–838  crossref  mathscinet  zmath  isi  elib  scopus
    6. 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
    7. 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
    Citing articles in Google Scholar: Russian citations, English citations
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