Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensions

Abstract

Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus $R_{K/k}^1{\mathbb{G}}_m$. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of $K/k$ has symmetric or alternating Galois group.

43 pages, final version. Major changes, including a restructure of the paper and the addition of a geometric interpretation of the first obstruction to the HNP, which improved several results and proofs. To appear in Math. Proc. Camb. Philos. Soc