Hasse Principle for G-quadratic forms

TL;DR

This paper investigates the validity of the Hasse principle for G-quadratic forms over global fields of positive characteristic, providing conditions for when it holds and presenting new counterexamples distinct from those in number fields.

Contribution

It offers a sufficient criterion for the Hasse principle to hold for G-quadratic forms over positive characteristic global fields and constructs novel counterexamples.

Findings

Hasse principle can hold under certain conditions

Counterexamples are different from those in number fields

Provides criteria for the principle's validity

Abstract

Let k be a global field of characteristic not 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 also 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 Jorge Morales. The aim of this 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, and also counter examples : note that these are of different nature than those for number fields.