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

Abstract

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 : (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 R G has trivial kernel if G is quasi-split of type E 6 or E 7 . A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank.

Serre’s Conjecture II: A Survey

“…We need to know that a trialitarian algebra whose underlying algebra is split arises as the endomorphism of a twisted composition [49, 44.16] and to use results on degree 3 invariants of twisted compositions (ibid, 40.16). For type E 6 and E 7 , this is due independently to Chernousov [17] and Garibadi [27].…”

Totaro's Question on Zero-Cycles on $G_2$, $F_4$ and $E_6$ Torsors

“…The stabilizer of the identity element in J is of type F 4 in all characteristics by [Sp,4.6], hence the stabilizer of the identity element in P(J) is F 4 × µ 3 . The conclusion now follows by [Ga,Lemma 3.1], where the last paragraph of the proof is replaced with an appeal to [DG,p. 373 …”

“…This has been shown in [Ga,3.4] under the assumption that k has characteristic different from 2, 3, but this restriction is unnecessary by the following reasoning. In all characteristics G sp is the group of isometries of a cubic form-a norm for an Albert algebra J-and the elements of norm 1 in J form an open orbit in P(J) [As,3.16(3)].…”

“…Over a separable closure of k, the subgroup H from Lemma 5.1 is similar to the standard inclusion of the split F 4 into the split E 6 , and the integer multiple for this inclusion is 1 by [Ga,2.4]. Since this multiplying factor does not change under scalar extensions [M 03,Prop.…”

$E_8$, the most exceptional group

“…The conclusion of Theorem 9.1 is false in that case, as can be seen already from Table B, where we see that the compact E 8 has Rost invariant zero. For context, this phenomenon is different from that of other exceptional groups; if one considers not E 8 but any other split simply connected group G of exceptional type, then the Rost invariant (2)) has zero kernel [65]. Recently in [137], Semenov produced a newer, finer invariant that can be used to probe the kernel of the Rost invariant.…”