The Rost invariant has trivial kernel for quasi-split groups of low rank

Abstract

For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map R_G : H^1(F, G) \rightarrow H^3(F, \mathbb{Q} / \mathbb{Z}(2)) . This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that RG has trivial kernel if G is quasi-split of type E6 or E7. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.