Campana points of bounded height on vector group compactifications

Abstract

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.

Hyperbola method on toric varieties

“…When we apply Theorem 1.1 to prove Theorem 1.2, we need to verify that both the exponent of B as well as the power of log B match the prediction in [PSTVA19]. The exponent a in Theorem 1.1 is the result of a linear optimization problem.…”

“…Campana points are a notion of points that interpolate between rational points and integral points on certain log smooth pairs, or orbifolds, introduced and first studied by Campana [Cam04,Cam11,Cam15]. The study of the distribution of Campana points over number fields was initiated only quite recently and the literature on this topic is still sparse [BVV12,VV12,BY19,PSTVA19]. In this paper we deal with toric varieties, which constitute a fundamental family of examples for the study of the distribution of rational points [BT95b,BT95a,BT96,BT98,Sal98,dlB01a], via a combination of the universal torsor method with a very general version of the hyperbola method, which we develop.…”

“…Theorem 1.2 could also be deduced from work of de la Bretèche [dlB01b], [dlB01a], who developed a multi-dimensional Dirichlet series approach to count rational points of bounded height on toric varieties. Another approach could be via harmonic analysis of the height zeta function, even though such a proof would probably be more involved than the case of compactifications of vector groups [PSTVA19]. Our proof proceeds via the universal torsor method introduced by Salberger in [Sal98] in combination with Theorem 1.1.…”

Campana points and powerful values of norm forms

“…They capture the idea of rational points which are integral with respect to a weighted boundary divisor. These two notions have been termed Campana points and weak Campana points in the recent paper [27] of Pieropan, Smeets, Tanimoto and Várilly-Alvarado, in which the authors initiate a systematic quantitative study of points of the former type on smooth Campana orbifolds and prove a logarithmic version of Manin's conjecture for Campana points on vector group compactifications. The only other quantitative results in the literature are found in [5], [31], [6], [26] and [32], and the former three of these indicate the close relationship between Campana points and m-full solutions of equations.…”

“…As observed in [27, §1.1], the study of weak Campana points of bounded height is challenging and requires new ideas for the regularisation of certain Fourier transforms, and these ideas for the orbifolds in consideration are the main innovation of this paper. We adopt a height zeta function approach, similar to the one employed in [27] and modelled on the work of Loughran in [20] and Batyrev and Tschinkel in [3] on toric varieties, in order to prove log Manin conjecture-type results for both types of Campana points on P d−1 K , 1 − 1 m Z(N ω ) , where N ω is a norm form associated to a K-basis ω of a Galois extension of number fields E/K of degree d ≥ 2 coprime to m ∈ Z ≥2 if d is not prime. When K = Q, we derive from the result for weak Campana points an asymptotic for the number of elements of E of bounded height with m-full norm over Q.…”