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 $R_G:H^1(F,G)\to H^3(F,\mathbb{Q}/\mathbb{Z}(2))$ . 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.