On the equation N_{K/k}(\Xi)=P(t)

TL;DR

This paper investigates the arithmetic of varieties defined by norm equations, focusing on the unramified Brauer group, rational points, and zero-cycles over number fields, with new results under abelian extension assumptions.

Contribution

It extends previous work by analyzing varieties with norm equations over arbitrary fields, providing new insights under the assumption that the extension K/k is abelian.

Findings

Computed the unramified Brauer group for these varieties.

Analyzed rational points and Brauer-Manin obstruction over number fields.

Studied zero-cycles and their obstructions in the context of abelian extensions.

Abstract

For varieties given by an equation N_{K/k}(\Xi)=P(t), where N_{K/k} is the norm form attached to a field extension K/k and P(t) in k[t] is a polynomial, three topics have been investigated: (1) computation of the unramified Brauer group of such varieties over arbitrary fields; (2) rational points and Brauer-Manin obstruction over number fields (under Schinzel's hypothesis); (3) zero-cycles and Brauer-Manin obstruction over number fields. In this paper, we produce new results in each of three directions. We obtain quite general results under the assumption that K/k is abelian (as opposed to cyclic in earlier investigation).