Hasse principles for multinorm equations

TL;DR

This paper investigates the Hasse principle for multinorm equations over global fields, providing explicit descriptions of obstructions and criteria for when the principle holds, especially under cyclic and meta-cyclic extension assumptions.

Contribution

It offers explicit descriptions of the Brauer-Manin obstruction and criteria for the Hasse principle in multinorm equations with cyclic and meta-cyclic extensions.

Findings

Explicit description of the Brauer-Manin obstruction for cyclic extensions.

Complete criterion for the Hasse principle with meta-cyclic extensions.

Conditions under which the Hasse principle holds for multinorm equations.

Abstract

Let $k$ be a global field and let $L_0$,...,$L_m$ be finite separable field extensions of $k$. In this paper, we are interested in the Hasse principle for the multinorm equation $\underset{i=0}{\overset{m}{\prod}}N_{L_i/k}(t_i)=c$. Under the assumption that $L_0$ is a cyclic extension, we give an explicit description of the Brauer-Manin obstruction to the Hasse principle. We also give a complete criterion for the Hasse principle for multinorm equations to hold when $L_0$ is a meta-cyclic extension.