Cohomological invariants of odd degree Jordan algebras

TL;DR

This paper classifies cohomological invariants of automorphism torsors of split central simple Jordan algebras of odd degree, extending previous results to new algebra types and computing essential dimensions of related groups.

Contribution

It extends the classification of cohomological invariants to unitary and symplectic types of Jordan algebras, previously known only for orthogonal and exceptional types.

Findings

Classified all cohomological invariants for these Jordan algebras.

Computed the essential dimension of PSp(2n) as n+1 for odd n.

Abstract

In this paper we determine all possible cohomological invariants of 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>=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(PSp(2n))=n+1 for n odd.