Campana points and powerful values of norm forms

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 Várilly-Alvarado.

arXiv comment37 pages; minor revisions; to appear in Mathematische Zeitschrift

References in corpus (5)

• Birational geometry for number theorists
• On the square-free representation function of a norm form and nilsequences
• Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
• Campana points on biequivariant compactifications of the Heisenberg group
• Weak approximation and the Hilbert property for Campana points

Cited by in corpus (4)

• Weak approximation and the Hilbert property for Campana points
• Sums of four squareful numbers
• Campana points, Height zeta functions, and log Manin's conjecture
• Semi-integral Brauer-Manin obstruction and quadric orbifold pairs