Hasse principle and weak approximation for multinorm equations

Abstract

In this note, we are interested in local-global principles for multinorm equations $\prod\nolimits_{i = 1}^n {{N_{{L_i}/k}}({z_i}} ) = a$ where k is a global field, L _i /k are finite separable field extensions and a ∈ k*.

In particular, we prove a result relating the Hasse principle and weak approximation for this equation to the Hasse principle and weak approximation for some classical norm equation N _F/k (w) = a where $F: = \bigcap\nolimits_{i = 1}^n {{L_i}}$. It provides a proof of a “weak approximation” analogue of a recent conjecture by Pollio and Rapinchuk about the multinorm principle. We also provide a counterexample to the original conjecture concerning the Hasse principle.