Hasse principle for $G$-quadratic forms

Eva Bayer-Fluckiger ; Nivedita Bhaskhar ; Raman Parimala

Documenta mathematica, Tome 18 (2013), pp. 383-392

Résumé

Let k be a global field of characteristic =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]).

Détail

DOI : 10.4171/dm/402