Motivic construction of cohomological invariants

Abstract

Let $G$ be a group of type ${\mathrm{E}}_8$ over $\mathbb{Q}$ such that $G_{\mathbb{R}}$ is a compact Lie group, let $K$ be a field of characteristic 0, and

a 5-fold Pfister form. J.-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that $G_K$ is split if and only if $q_K$ is hyperbolic?

In the present article we construct a cohomological invariant of degree 5 for groups of type ${\mathrm{E}}_8$ with trivial Rost invariant over any field $k$ of characteristic 0, and putting $k=\mathbb{Q}$ answer positively this question of Serre. Aside from that, we show that a variety which possesses a special correspondence of Rost is a norm variety.