Hasse principle for principal homogeneous spaces under classical groups over a field of virtual cohomological dimension at most one
The main objects of the 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 for principal homogeneous spaces under such groups. The author follows Scheiderer's approach 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, 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, 639-650 (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)]. In the latter paper it is conjectured that the Hasse principle should hold for principal 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 this conjecture for all classical groups by case-by-case consideration following the approach by \textit{E. Bayer-Fluckiger} and \textit{R. Parimala} [Invent. Math. 122, No. 2, 195-229 (1995)] who treated the case of a ground field of cohomological dimension at most 2. One should note that recently the results of the paper reviewed were generalized in two directions: the above conjecture was proved by C. Scheiderer for any \(G\) and by E. Bayer-Fluckiger and R. Parimala for all classical groups defined over \(k\) with cohomological dimension of \(k (\sqrt{-1})\) at most 2.
Hasse principle for principal homogeneous spaces
scientific article; zbMATH DE number 871039
scientific article