Motives, cohomological invariants and Freudenthal magic square

TL;DR

This paper explores cohomological and motivic invariants of semisimple algebraic groups linked to the Freudenthal magic square, providing new proofs and interpretations of existing results.

Contribution

It offers new insights into invariants of algebraic groups, including a motivic interpretation and alternative proofs for known properties of groups of specific types.

Findings

Rost invariant of type E7 groups is isotropic over odd degree extensions if it is a sum of two symbols.

Provides a motivic interpretation of a degree 5 invariant for certain ^2E6 groups.

Shows that certain cohomological invariants detect isotropy in algebraic groups.

Abstract

We investigate cohomological invariants and motivic invariants of semisimple algebraic groups arising in the Freudenthal magic square. Besides, we show that if the Rost invariant of a strongly inner group of type $E_7$ is a sum of at most two symbols modulo $2$, then it is isotropic over an odd degree field extension, and use this fact to give a different proof of a result of Petrov and Rigby. Moreover, we give a motivic interpretation of a result of Garibaldi and Petersson about a cohomological invariant of degree $5$ for certain groups of type $^2E_6$ which detects their isotropy.