Universal torsors and values of quadratic polynomials represented by norms

Abstract

Let \(K/k\) be an extension of number fields, and let \(P(t)\) be a quadratic polynomial over \(k\). Let \(X\) be the affine variety defined by \(P(t) = N_{K/k}(\mathbf {z})\). We study the Hasse principle and weak approximation for \(X\) in three cases. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\) and split in \(K\), we prove the Hasse principle and weak approximation. For \(k=\mathbb {Q}\) with arbitrary \(K\), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For \([K:k]=4\) and \(P(t)\) irreducible over \(k\), we determine the Brauer group of smooth proper models of \(X\). In a case where it is non-trivial, we exhibit a counterexample to weak approximation.