V. Chernousov: The kernel of the Rost invariant, Serre's Conjecture II and the Hasse principle for quasi-split groups $^{3,6}D_4,E_6,E_7$


Submission: 2002, Jan 13

We prove that for a simple simply connected quasi-split group of type $^{3,6}D_4, E_6, E_7$ defined over a perfect field $F$ of characteristic $\not= 2,3$ the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant. It follows immediately from the result above that if cohomological dimension (resp. virtual cohomological dimension) of $F$ is at most $2$ then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a $(C_2)$-field we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of type $^{3,6}D_4, E_6, E_7$.

2000 Mathematics Subject Classification: 20G10

Keywords and Phrases: Galois cohomology, Serre's Conjecture II, the Rost invariant, exceptional groups

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