A Projective Twist on the Hasse Norm Theorem

TL;DR

This paper introduces the projective Hasse norm principle, extending the classical Hasse norm principle to multiple fields and analyzing its implications and independence in various field extension scenarios.

Contribution

It generalizes the Hasse norm principle to a projective setting involving multiple fields and establishes its relationship with the classical principle, including cases where it holds or is independent.

Findings

The projective Hasse norm principle is independent from the conjunction of all classical principles in general.

When all fields are Galois and independent, the classical and projective principles imply each other.

The principle holds in all cyclic extensions, extending Hasse's original theorem.

Abstract

A finite extension of global fields $L/K$ satisfies the Hasse norm principle if any nonzero element of $K$ has the property that it is a norm locally if and only if it is a norm globally. In 1931, Hasse proved that any cyclic extension satisfies the Hasse norm principle, providing a novel approach to extending the local-global principle to equations with degree greater than $2$. In this paper, we introduce the projective Hasse norm principle, generalizing the Hasse norm principle to multiple fields and asking whether a projective line that contains a norm locally in every field must also contain a norm globally in every field. We show that the projective Hasse norm principle is independent from the conjunction of Hasse norm principles in all of the constituent fields in the general case, but that the latter implies the former when the fields are all Galois and independent. We also prove an analogue of the Hasse norm theorem for the projective Hasse norm theorem, namely that the projective Hasse norm principle holds in all cyclic extensions.