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\$

chernous@mathematik.uni-bielefeld.de

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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