Motivic equivalence and similarity of quadratic forms

Abstract

A result by Vishik states that given two anisotropic quadratic forms of the same dimension over a field of characteristic not 2, the Chow motives of the two associated projective quadrics are isomorphic iff both forms have the same Witt indices over all field extensions, in which case the two forms are called motivically equivalent. Izhboldin has shown that if the dimension is odd, then motivic equivalence implies similarity of the forms. This also holds for even dimension $\le 6$, but Izhboldin also showed that this generally fails in all even dimensions $\ge 8$ except possibly in dimension $12$. The aim of this paper is to show that motivic equivalence does imply similarity for fields over which quadratic forms can be classified by their classical invariants provided that in the case of formally real such fields the space of orderings has some nice properties. Examples show that some of the required properties for the field cannot be weakened.