Vishik equivalence and similarity of quadratic forms over fields of characteristic 2

Abstract

An important aspect in the algebraic theory of quadratic forms is the study of equivalence relations based on algebraic-geometric properties of the associated quadrics. A well-known criterion originally proved by Vishik in characteristic zero states that two nonsingular quadratic forms of the same dimension have identical Witt indices over all field extensions if and only if their motives are isomorphic in the category of (integral or mod 2) Chow motives. In characteristic 2, it is meaningful to include singular forms. We therefore define two quadratic forms (including singular ones) of the same dimension to be Vishik-equivalent if they share the same isotropy behavior (in a suitably defined way) over all field extensions. Similar quadratic forms are always Vishik-equivalent, but the converse need not hold. We determine various classes of quadratic forms in characteristic 2 where Vishik equivalence implies similarity and give nonsingular counterexamples in all dimensions $2^n\ge 8$, and also singular counterexamples in dimension 8. To construct the counterexamples, we use a generalized notion of so-called half-neighbors in characteristic 2.