Galois cohomology of quasi-split groups over fields of cohomological dimension \(\leqslant 2\) (Q2711351)

The main object of the paper under review is an absolutely almost simple simply connected quasi-split group \(G\) defined over a perfect field \(k\) of cohomological dimension \(\leq 2\). The main result of the paper says that the first Galois cohomology of such a \(G\) is trivial (assuming that \(G\) is not of type \(E_8\)) thus establishing Serre's Conjecture II for such groups. (For groups of type \(E_8\) it is shown that the result remains true provided the absolute Galois group of \(k\) is a \(p\)-group.) NEWLINENEWLINENEWLINESince Serre's Conjecture II was previously proved for the groups of type \({}^1 A_n\) [\textit{A. S. Merkurev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011-1046 (1982; Zbl 0525.18008)] and all the other classical groups [\textit{E. Bayer-Fluckiger} and \textit{R. Parimala}, Invent. Math. 122, 195-229 (1995; Zbl 0851.11024)], it only remains to treat the exceptional groups. The author's method is based on considering the map \(H^1(k,T)\to H^1(k,G)\) where \(T\) is a maximal \(k\)-torus of \(G\) split over a prime degree cyclic extension; in particular, using the Bruhat-Tits theory and the Rost invariant, he proves that under the hypotheses of Serre's Conjecture II this map is zero. To finish the proof, the author uses Steinberg's theorem and Harder's results on the Hasse principle. NEWLINENEWLINENEWLINEAnother interesting application of the author's methods is a new ``quasi-uniform'' proof of the Hasse principle for quasi-split groups defined over number fields (the \(E_8\)-case fits into its framework!). NEWLINENEWLINENEWLINENote that the main result of the paper was independently obtained by V. Chernousov who used different methods.