Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields

Mikhail Borovoi, Boris Kunyavskiǐ, Philippe Gille

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24 Scopus citations

Abstract

Let G be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by Colliot-Thélène, Gille and Parimala. Basing on their results, we compute the group of classes of R-equivalence G(k /R, the defect of weak approximation A Σ(G), the first Galois cohomology H1 (k, G), and the Tate-Shafarevich kernel III1 (k, G) (for suitable k) in terms of the algebraic fundamental group π1 (G). We prove that the groups G(k)/R and A Σ(G) and the set III1 (k, G) are stably k-birational invariants of G.

Original languageEnglish
Pages (from-to)292-339
Number of pages48
JournalJournal of Algebra
Volume276
Issue number1
DOIs
StatePublished - 1 Jun 2004

Bibliographical note

Funding Information:
Keywords: Two-dimensional geometric field; Linear algebraic group; Birational invariants; R-equivalence; Weak approximation; Tate–Shafarevich kernel ✩ This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities—Center of Excellence Program and by EU RTN HPRN-CT-2002-00287. * Corresponding author. E-mail addresses: borovoi@post.tau.ac.il (M. Borovoi), kunyav@macs.biu.ac.il (B. Kunyavski˘ı), philippe.gille@math.u-psud.fr (P. Gille). 1 The author was partially supported by the Hermann Minkowski Center for Geometry. 2 The author was partially supported by the Ministry of Absorption (Israel), the Minerva Foundation through the Emmy Noether Research Institute of Mathematics, and INTAS 00-566. 3 The author of the Appendix.

Keywords

  • Birational invariants
  • Linear algebraic group
  • R-equivalence
  • Tate-Shafarevick kernel
  • Two-dimensional geometric field
  • Weak approximation

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