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 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.

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

“…When a = 1, N(B) counts squareful values of the norm form x 2 + by 2 . This is a very special case of the result by Streeter in [18,Theorem 1.4]. The constant from [18,Theorem 1.4] and the constant c from Theorem 1.5 must therefore agree.…”

“…The constant from [18,Theorem 1.4] and the constant c from Theorem 1.5 must therefore agree. However, the proof of [18,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 [20], Xiao treats the case of biequivariant compactifications of the Heisenberg group over Q, using the height zeta function method. Finally, in [18], Streeter studies m-full values of norm forms by considering the orbifold…”

“…In [13], [12] and [20], 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 provides in [18,Section 7.3] an example where the leading constant in [18,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.…”

Sums of four squareful numbers

“…In [29], Xiao treats the case of biequivariant compactifications of the Heisenberg group over Q, using the height zeta function method. Finally, in [27], Streeter studies m-full values of norm forms by considering the orbifold (P d−1 K , (1− 1 m )V(N E/K )), where K is a number field, V(N E/K ) 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.…”