Hasse principle and weak approximation for multinorm equations

TL;DR

This paper investigates local-global principles for multinorm equations over global fields, establishing connections to classical norm equations, proving a weak approximation analogue of a recent conjecture, and providing a counterexample to the Hasse principle conjecture.

Contribution

It relates weak approximation for multinorm equations to classical norm equations and offers a counterexample to the multinorm principle conjecture.

Findings

Weak approximation for multinorm equations reduces to classical norm equations.

Provided a proof of a weak approximation analogue of Pollio and Rapinchuk's conjecture.

Constructed a counterexample to the Hasse principle 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 $a \in k^*$. 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.