Cohomological invariants of hermitian forms that detect hyperbolicity

TL;DR

This paper develops a comprehensive set of cohomological invariants for hermitian forms, capable of detecting hyperbolicity across various algebraic settings and extending previous results to arbitrary characteristic.

Contribution

It introduces a full sequence of invariants using unramified cohomology for hermitian forms of any type, generalizing prior work and applicable to fields of any characteristic.

Findings

Invariants can detect hyperbolicity of hermitian forms.

Over fields of separable dimension 3, certain hermitian pairs are hyperbolic.

Extension of Berhuy's result to arbitrary characteristic.

Abstract

By using unramified cohomology groups, we construct a full sequence of cohomological invariants for hermitian forms of any type (orthogonal, symplectic or unitary) that can be used to detect hyperbolicity. The base central simple algebras can have arbitrary degree and the base field can have arbitrary characteristic. In the orthogonal case, we work with hermitian pairs, and we apply our construction to show that over fields of separable dimension 3, hermitian pairs over quaternion algebras with trivial classical invariants are hyperbolic. This last result extends a result of Berhuy to arbitrary characteristic.