Witt groups of Severi-Brauer varieties and of function fields of conics

Abstract

The Witt group of skew hermitian forms over a division algebra $D$ with symplectic involution is shown to be canonically isomorphic to the Witt group of symmetric bilinear forms over the Severi-Brauer variety of $D$ with values in a suitable line bundle. In the special case where $D$ is a quaternion algebra we extend previous work by Pfister and by Parimala on the Witt group of conics to set up two five-terms exact sequences relating the Witt groups of hermitian or skew-hermitian forms over $D$ with the Witt groups of the center, of the function field of the Severi-Brauer conic of $D$, and of the residue fields at each closed point of the conic.