Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one
The main objects of the reviewed paper are connected linear groups defined over a perfect field \(k\) with cohomological dimension of \(k(\sqrt{-1})\) at most one. A typical example of such a field is the field of rational functions \(\mathbb{R}(C)\) of a smooth projective real curve \(C\). The goal is to establish the Hasse principle and approximation theorems for homogeneous spaces under such groups. The author's approach consists in taking as local objects the real closures of \(k\) rather than the completions at the closed points. Such a principle was established for quadrics by \textit{R. Elman}, \textit{T.-Y. Lam} and \textit{A. Prestel} [Math. Z. 134, 291-301 (1973; Zbl 0277.15013)] thus generalizing classical results of \textit{E. Witt} [J. Reine Angew. Math. 171, 4-11 (1934; Zbl 0009.29103), 176, 31-44 (1936; Zbl 0015.05701)], and for Hermitian forms by \textit{N. Q. Thang} [Manuscr. Math. 78, No. 1, 9-35 (1993; Zbl 0804.11034), 82, No. 3-4, 445-447 (1994; Zbl 0812.11026)]. In another direction, the Hasse principle is known to hold for norms of any finite extension of \(k=\mathbb{R}(C)\) (\textit{J. T. Knight} [Proc. Camb. Philos. Soc. 65, 635-650 (1969; Zbl 0176.50703)]) and more general for principal homogeneous spaces of \(k\)-tori (\textit{J.-L. Colliot-Thélène} [J. Reine Angew. Math. 474, 139-167 (1996; Zbl 0847.11017)]). In the latter paper it is conjectured that the Hasse principle and weak approximation should hold for homogeneous spaces under any connected linear group \(G\), and the problem is reduced to the case when \(G\) is semisimple simply connected. The author proves these conjectures. The Hasse principle is established for any homogeneous space, weak approximation is proved under the sole assumption that the (geometric) stabilizer is connected. He also proves component approximation (with real closures as local objects) without any restriction on stabilizers. These results have various applications (the author [Manuscr. Math. 89, No. 3, 373-394 (1996; Zbl 0846.20048)]). One should note that the Hasse principle for principal homogeneous spaces of classical groups has been independently established by \textit{A. Ducros} [Manuscr. Math. 89, No. 3, 335-354 (1996; Zbl 0846.20047)] by case-by-case consideration. Recently some of the above results have been generalized by \textit{E. Bayer-Fluckiger} and \textit{R. Parimala} [Invent. Math. 122, No. 2, 195-229 (1995; Zbl 0851.11024) and Hasse principle for the classical groups over fields of virtual cohomological dimension at most 2 (Preprint, 1995)].
scientific article; zbMATH DE number 922468