Norms as products of linear polynomials

TL;DR

This paper investigates the representation of products of linear polynomials as norms from a field extension, proving that the Brauer-Manin obstruction is the only barrier to the Hasse principle and weak approximation for related varieties.

Contribution

It combines the circle method with descent techniques to establish the sufficiency of the Brauer-Manin obstruction for these norm-related varieties.

Findings

Brauer-Manin obstruction is the only obstruction to the Hasse principle.

Results apply to smooth, projective models of the variety.

Method integrates circle method with descent for norm representations.

Abstract

Let F be a number field, and let F\subset K be a field extension of degree n. Suppose that we are given 2r sufficiently general linear polynomials in r variables over F. Let X be the variety over F such that the F-points of X bijectively correspond to the representations of the product of these polynomials by a norm from K to F. Combining the circle method with descent we prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.