Excellence of function fields of conics

TL;DR

This paper proves that for certain algebraic forms over division algebras, the anisotropic kernel remains defined over the base field when extended to the function field of a smooth projective conic, regardless of characteristic.

Contribution

It establishes a characteristic-independent result about the behavior of anisotropic kernels of quadratic and hermitian forms over function fields of conics.

Findings

Anisotropic kernels are defined over the base field after scalar extension.

The result holds without any restriction on the characteristic.

Applicable to generalized quadratic and hermitian forms over division algebras.

Abstract

For every generalized quadratic form or hermitian form over a division algebra, the anisotropic kernel of the form obtained by scalar extension to the function field of a smooth projective conic is defined over the field of constants. The proof does not require any hypothesis on the characteristic.