Hasse principle and weak approximation for multinorm equations

Abstract

In this note, we are interested in local-global principles for multinorm equations of the form ${\prod }_{i=1}^n{N}_{L_i/k}(z_i)=a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a\in k^{\ast }$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w)=a$ where $F:={\bigcap }_{i=1}^n{L}_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture concerning Hasse principle.

arXiv comment14 pages, Final version

References in corpus (1)

• The unramified Brauer group of norm one tori