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 $$\mathbb {Q}$$ 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.

Campana points, Height zeta functions, and log Manin's conjecture

“…One can also consider a similar counting problem for weak Campana points, however this problem is much harder than Conjecture 4.5. At the moment of writing this paper, we do not know how one should formulate a log Manin's conjecture for weak Campana points, but [Str20] takes the first step towards to this problem.…”

“…The counting problem of Campana points has been originally featured in [VV11], [BVV12], and [VV12]. Recently many mathematicians started to look at this problem and develop a series of results, attested by [BY20], [PSTVA20], [PS20], [Xia20], and [Str20]. In [PSTVA20], Pieropan, Smeets, Várilly-Alvarado, and the author initiated a systematic study of the counting problem for Campana points, and formulated a log Manin's conjecture for klt Campana points.…”

On the leading constant in the Manin-type conjecture for Campana points

“…Xiao [23] treats the case of biequivariant compactifications of the Heisenberg group over Q, using the height zeta function method. Finally, Streeter [21] studies m-full values of norm forms by counting Campana points on the orbifold (P d−1 K , (1−1/m)V (N E/K )), where K is a number field, V (N E/K ) is the divisor cut out by a norm form associated to a degree-d Galois extension E/K, and m ≥ 2 is an integer which is coprime to d if d is not prime.…”

“…The constant from [21,Theorem 1.4] and the constant c from Theorem 1.5 must therefore agree. However, the proof of [21,Theorem 1.4] proceeds via very different methods, using height zeta functions and Fourier analysis, leading to a constant that involves a sum of limits of global Fourier transforms of 2-torsion toric characters.…”

“…In [16], [15] and [23], the leading constants for the counting problems considered were reconciled with the prediction from the PSTV-A conjecture. In the case of Campana points for norm forms, Streeter [21,Section 7.3] provides an example where the leading constant in [21,Theorem 1.4] differs from the constant defined in the PSTV-A conjecture. It remains unclear whether this could be explained by the removal of a thin set.…”