Motivic construction of cohomological invariants

TL;DR

This paper constructs a degree 5 cohomological invariant for E8 groups over fields of characteristic 0, providing a positive answer to Serre's question about splitting over the rationals and hyperbolicity of a specific form.

Contribution

It introduces a new cohomological invariant for E8 groups with trivial Rost invariant, resolving a longstanding question posed by Serre.

Findings

Constructed a degree 5 cohomological invariant for E8 groups.

Proved that G is split over K if and only if q is hyperbolic over K.

Showed that varieties with a Rost correspondence are norm varieties.

Abstract

Let G be a group of type E8 of compact type over the field of rational numbers, let K be a field of characteristic 0, and q the 5-fold Pfister form which is the sum of 32 squares. J-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that G is split over K if and only if q is hyperbolic over K? 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 the field of rational numbers for k 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.