Hasse principle for $G$-quadratic forms

Abstract

Let $k$ be a global field of characteristic $\ne 2$. The classical Hasse–Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of $k$, then they are isomorphic over $k$ as well. It is natural to ask whether this is true for $G$-quadratic forms, where $G$ is a finite group. In the case of number fields the Hasse principle for $G$-quadratic forms does not hold in general, as shown by J. Morales [M 86]. The aim of the present paper is to study this question when $k$ is a global field of positive characteristic. We give a sufficient criterion for the Hasse principle to hold (see th. 2.1.), and also give counter-examples. These counter-examples are of a different nature than those for number fields: indeed, if $k$ is a global field of positive characteristic, then the Hasse principle does hold for $G$-quadratic forms on projective $k[G]$- modules (see cor. 2.3), and in particular if $k[G]$ is semi-simple, then the Hasse principle is true for $G$-quadratic forms, contrarily to what happens in the case of number fields. On the other hand, there are counter-examples in the non semi-simple case, as shown in §3. Note that the Hasse principle holds in all generality for $G$-trace forms (cf. [BPS 13]).