Shells of twisted flag varieties and the Rost invariant

Abstract

We introduce two new general methods to compute the Chow motives of twisted flag varieties and settle a 20-year-old conjecture of Markus Rost about the Rost invariant for groups of type E7.Comment: 39 page

$E_8$, the most exceptional group

“…This combines the main results of [28] and [29]. (Those papers make some hypothesis on char F , but such an assumption is harmless in view of [75,Prop. 9.3] As the case p ≥ 7 was treated in §6, only p = 2 remains.…”

“…What is its image? Given a nonzero symbol s in H 3 (F, Z/3(2)), there is an Albert division algebra A constructed by the first Tits' construction such that r F 4 (A) = s. The inclusions (2)), [75] shows that there is a finite extension K of F of degree not divisible by 3 such that G × K has Tits index (8.7). (2)) nor does it become a symbol over any finite extension of F of degree not divisible by 3.…”

“…(In particular, such a group does not arise from Tits' construction.) To see this, one appeals to the theory of the J-invariant from [128] that there exists such a G with J 3 (G) = (1, 1); [75,Lemma 10.23] shows that such a group has the desired property.…”

Hasse Principle for Rost Motives

“…Chow motives were introduced by Grothendieck, and since then they became a fundamental tool for investigating the structure of algebraic varieties. Applications of the Chow motives include, among others, results on higher Witt indices of quadratic forms [Ka03], structure of the powers of the fundamental ideal in the Witt ring [Ka04], cohomological invariants of algebraic groups ( [GPS16] and [S16]), Kaplansky's problem on the u-invariants of fields [Vi07], and isotropy of involutions [KaZ13].…”

Integral motives, relative Krull–Schmidt principle, and Maranda-type theorems

“…A short overview of recent results proved using motivic methods is given e.g. in the introduction of [GaPS11].…”