RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (547 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 512.74
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

Citation in format AMSBIB
\Bibitem{Sko07} \by A.~N.~Skorobogatov \paper On the Elementary Obstruction to the Existence of Rational Points \jour Mat. Zametki \yr 2007 \vol 81 \issue 1 \pages 112--124 \mathnet{http://mi.mathnet.ru/mz3521} \crossref{https://doi.org/10.4213/mzm3521} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2333868} \zmath{https://zbmath.org/?q=an:1134.14012} \elib{http://elibrary.ru/item.asp?id=9429669} \transl \jour Math. Notes \yr 2007 \vol 81 \issue 1 \pages 97--107 \crossref{https://doi.org/10.1134/S0001434607010099} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000244695200009} \elib{http://elibrary.ru/item.asp?id=13541175} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33947513603} 

• http://mi.mathnet.ru/eng/mz3521
• https://doi.org/10.4213/mzm3521
• http://mi.mathnet.ru/eng/mz/v81/i1/p112

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Wittenberg O., “On Albanese torsors and the elementary obstruction”, Math. Ann., 340:4 (2008), 805–838
3. Borovoi M., Colliot-Thélène J.-L., Skorobogatov A. N., “The elementary obstruction and homogeneous spaces”, Duke Math. J., 141:2 (2008), 321–364
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
5. Harari D., Szamuely T., “Galois sections for abelianized fundamental groups”, Math. Ann., 344:4 (2009), 779–800
6. Beauville A., “On the Brauer group of Enriques surfaces”, Math. Res. Lett., 16:5-6 (2009), 927–934
•  Number of views: This page: 308 Full text: 166 References: 53 First page: 1