@article{3cb427326a344a74969d3726b2fde43f,

title = "Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields",

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-Th{\'e}l{\`e}ne, 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.",

keywords = "Birational invariants, Linear algebraic group, R-equivalence, Tate-Shafarevick kernel, Two-dimensional geometric field, Weak approximation",

author = "Mikhail Borovoi and Boris Kunyavskiǐ and Philippe Gille",

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.",

year = "2004",

month = jun,

day = "1",

doi = "10.1016/j.jalgebra.2003.10.024",

language = "אנגלית",

volume = "276",

pages = "292--339",

journal = "Journal of Algebra",

issn = "0021-8693",

publisher = "Academic Press Inc.",

number = "1",

}