Motivic construction of cohomological invariants

Abstract

Let G be a group of type E8 over Q such that GR is a compact Lie group, let K be a field of characteristic 0, and q=⟨⟨−1,−1,−1,−1,−1⟩⟩ 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 GK is split if and only if qK is hyperbolic? In the present article we construct a cohomological invariant of degree 5 for groups of type E8 with trivial Rost invariant over any field k of characteristic 0, and putting k=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.