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.

21 pages, major revision containing additional results, to appear in Mathematische Annalen

Cited by in corpus (4)

• The local-global exact sequence for Chow groups of zero-cycles
• Cubic polynomials represented by norm forms
• Local-global principle for 0-cycles on fibrations over rationally connected bases
• Towards the Brauer-Manin obstruction on varieties fibred over the projective line