Mark MacDonald: Cohomological invariants of odd degree Jordan algebras

Submission: 2007, Aug 20, revised: 2007, Oct. 22

In this paper we determine all possible cohomological invariants of $\mathbf{Aut}(J)$-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for $J$ a split central simple Jordan algebra of odd degree $n\geq 3$. This has already been done for $J$ of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed$(\mathbf{PSp}_{2n})=n+1$ for $n$ odd.

2000 Mathematics Subject Classification: 11E72, 16W10, 17C10, 20G15

Keywords and Phrases: Cohomological invariants, essential dimension, Jordan algebras, Stiefel-Whitney classes, algebras with involution

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