Campana points and powerful values of norm forms

TL;DR

This paper establishes an asymptotic count for weak Campana points on orbifolds linked to norm forms of Galois extensions, revealing significant differences between Campana points and elements with full norm, and relates findings to Manin-type conjectures.

Contribution

It provides the first asymptotic formulas for weak Campana points on orbifolds associated with norm forms, and compares these to classical norm form counts and conjectures.

Findings

Asymptotic formula for weak Campana points on orbifolds

Asymptotic count for elements with m-full norm

Comparison between Campana points and classical norm form counts

Abstract

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with $m$-full norm over a given Galois extension of $\mathbb{Q}$. We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and V\'arilly-Alvarado.