Isometries of lattices and Hasse principles (Q6582326)

The famous Hasse-Minkowski theorem connects the existence of a solution to a homogeneous quadratic equation over the rational numbers to the existence of solutions over all local fields (\(p\)-adic numbers and the real numbers). Similar ``Hasse principles'' (local-global principles) are therefore anticipated in other contexts as well.\N\NThis paper deals with such principles for the following question: given a quadratic form \(q\) over a global field \(K\) and a monic polynomial \(F\) over \(K\) (with additional requirements such as no linear factors et cetera), does it follow that \(q\) has a (semisimple) isometry with characteristic polynomial \(F\) if and only if such an isometry exists everywhere locally. The author provides necessary and sufficient for this to hold in terms of an obstruction group. The author also studies an ``integral'' version of this question (the former being the ``rational'' version).