Trialitarian groups and the Hasse principle (Q1279853)

The main object of the paper under review is a simply connected group \(G\) of type \({}^3D_4\) or \({}^6D_4\) defined over an arbitrary field \(F\) of characteristic \(\neq 2\). Note that these are the simplest unsettled cases for Serre's Conjecture II stating that \(H^1(F,G)=0\) for any semisimple simply connected group \(G\) defined over a perfect field \(F\) of cohomological dimension \(\leq 2\), as well as for its ``virtual'' generalization to fields with cohomological dimension of \(F(\sqrt{-1})\) at most two (in this case one expects the injectivity of the natural restriction map \[ H^1(F,G)\to \prod H^1(F_v,G), \tag{1} \] where the product is taken over all orderings of \(F\) and \(F_v\) stands for the real closure of \(F\) at \(v\)). These two conectures have been proved for all classical groups by \textit{E. Bayer-Fluckiger} and \textit{R. Parimala} [Invent. Math. 122, 195-229 (1995; Zbl 0851.11024) and Ann. Math. (2) 147, 651-693 (1998; Zbl 0909.20029)]. The author makes a step towards these conjectures for trialitarian groups. To be more precise, for \(G\) a simply connected group of type \({}^3D_4\) or \({}^6D_4\) denote by \(\overline G=G/Z(G)\) the adjoint group isogenous to \(G\), and let \(K\) denote the kernel of the natural map \(H^1(F,G)\to H^1(F,\overline G)\). Assume that the cohomological \(2\)-dimension of \(F(\sqrt{-1})\) is at most 2. Then the restriction of map (1) to \(K\) is injective. The proof relies upon an explicit description of triality in terms of central simple algebras; a key step consists in applying a norm principle for classes of \(R\)-equivalence established by \textit{Ph. Gille} [C. R. Acad. Sci., Paris Sér. I 316, 315-320 (1993; Zbl 0826.12003)] and \textit{A. S. Merkurjev} [St. Petersbg. Math. J. 7, 243-264~(1996; Zbl 0859.20039)]; see also \textit{V. I. Chernousov and A. S. Merkurjev} [J. Algebra 209, 175-198 (1998)].