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Degree $3$ relative invariant for unitary involutions

Demba Barry, Alexandre Masquelein and Anne Quéguiner-Mathieu

Vol. 7 (2022), No. 3, 549–597

Using the Rost invariant for nonsplit simply connected groups, we define a relative degree 3 cohomological invariant for pairs of orthogonal or unitary involutions having isomorphic Clifford or discriminant algebras. The main purpose of this paper is to study general properties of this invariant in the unitary case, that is, for torsors under groups of outer type A. If the underlying algebra is split, it can be reinterpreted in terms of the Arason invariant of quadratic forms, using the trace form of a hermitian form. When the algebra with unitary involution has a symplectic or orthogonal descent, or a symplectic or orthogonal quadratic extension, we provide comparison theorems between the corresponding invariants of unitary and orthogonal or symplectic types. We also prove the relative invariant is classifying in degree 4, at least up to conjugation by the nontrivial automorphism of the underlying quadratic extension. In general, choosing a particular base point, the relative invariant also produces absolute Arason invariants, under some additional condition on the underlying algebra. Notably, if the algebra has even co-index, so that it admits a hyperbolic involution which is unique up to isomorphism, we get a so-called hyperbolic Arason invariant. Assuming in addition the algebra has degree 8, we may also define a decomposable Arason invariant. It generally does not coincide with the hyperbolic Arason invariant, as the hyperbolic involution need not be totally decomposable.

central simple algebras, unitary involutions, cohomological invariants, Arason invariant, algebraic groups
Mathematical Subject Classification
Primary: 11E72, 16W10
Secondary: 11E81
Received: 17 November 2021
Revised: 24 June 2022
Accepted: 15 July 2022
Published: 19 December 2022
Demba Barry
Faculté des Sciences et Techniques
Université des Sciences, des Techniques et des Technologies de Bamako
Departement Wiskunde–Informatica
Universiteit Antwerpen
Alexandre Masquelein
ICTEAM Institute
Université Catholique de Louvain
Anne Quéguiner-Mathieu
LAGA - CNRS (UMR 7539)
Université Sorbonne Paris Nord