Campana points of bounded height on vector group compactifications

We initiate a systematic quantitative study on Fano orbifolds of subsets of rational points that are integral with respect to a weighted boundary divisor. 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.

Context. Manin's conjecture for rational points, extensively studied now for more than three decades, predicts an asymptotic formula for the counting function of rational points of bounded height on algebraic varieties (over number fields) that are rationally connected. The class of equivariant compactifications of homogeneous spaces has proved to be a particularly fertile testing ground for the conjecture, with two general methods available to study the conjecture in this case. The first one is the method of mixing: it comes from ergodic theory, and can be used to study homogeneous spaces acted upon by semisimple groups; see, for example, [GMO08,GO11,GTBT15]. The second one involves the spectral theory of automorphic forms applied to height zeta functions; in principle, this method can be used for arbitrary algebraic groups; see [FMT89,BT96a,BT98a,CLT02,STBT07,TT12,ST16]. The problem of counting integral points on homogeneous spaces has also received much attention, both classically (see, for example, [DRS93,EM93]), and more recently, as attested by [CLT10b,BO12,CLT12,TBT13,TT15,Cho19]. This body of work represents progress towards a "logarithmic version" of Manin's conjecture for integral points. It is important to note, however, that there are several difficulties in the formulation of such a Manin-type conjecture for integral points. For example, the exponents appearing in the asymptotics of the counting functions in these results depend heavily on the divisor chosen for the counting function, and not only on its numerical class (see, e.g., [CLT12] for the formulation of its exponent). More problematically, the set of integral points on a rational variety may well be thin, and one typically removes a thin set when counting points of bounded height, in order to get a count that is not dominated by accumulating subvarieties. Readers interested in this feature can consult recent refinements of Manin's conjecture for rational points using the notion of thin sets [Pey17,LST18].
There are several ways to "interpolate" between the classical notions of rational and integral points. Keeping Manin's conjecture in mind, the purpose of this article is to argue in favor of a compelling option that arises from Campana's theory of pairs, which he baptised orbifoldes géométriques 1 . Following ideas of Campana ([Cam04,§9], [Cam11, §12]) and Abramovich ([Abr09, Lecture 2]), we focus on Campana points in this paper. We refer to §2.3 for details, but morally one should think of a set of Campana points on a pair (X, D) over a number field as lying "halfway" between the set of rational points on X, and a suitable set of integral points on X \ supp(D), with integrality conditions being dictated by a set of weights on the components of D, a chosen set S of places of the ground field, and a model of (X, D) over the S-integers.
Surprisingly, analogues of Manin's conjecture for sets of Campana points have not been well-studied so far. The only results we are aware of are to be found in [BVV12], [VV12] and [BY19]; all of these papers address counting problems for "squareful" and "powerful" numbers on projective spaces. In this paper, we focus on Manin's conjecture for Campana points in a very different setting, with homogeneous structure.
1.2. Equivariant compactifications of vector groups. Seminal work of Chambert-Loir-Tschinkel establishes Manin's conjecture for rational points on equivariant compactifications of vector groups in [CLT02], using the height zeta function method. Analogues of these results for integral points were studied in [CLT12]. This important class of varieties also has been studied recently for the motivic version of Manin's conjecture in [CLL16], [Bil18].
The geometry of vector group compactifications is worked out in [HT99], where equivariant compactifications of a vector group on P n are classified. Surprisingly, there is more than one such compactification. There are classification results of equivariant compactifications that are del Pezzo surfaces and Fano 3-folds [DL10,DL15,HM18], but equivariant compactifications of vector groups need not be Mori dream spaces. Indeed, blow-ups of the standard equivariant compactification on P n along a smooth center on the boundary hyperplane inherit the group compactification structure, so examples with a Cox ring that is not finitely generated can be constructed by blowing up suitable centers (see [HTT15,Example 2.17]). This feature makes equivariant compactifications of vector groups difficult to study via universal torsors, showing once more the power of the height zeta function method. In addition, equivariant compactifications of vector groups admit deformations, whereas equivariant compactifications involving reductive groups typically do not; this feature also makes the former class of compactifications interesting objects from a geometric point of view.
1.3. Results. We prove a version of Manin's conjecture for Campana points on equivariant compactifications of vector groups, generalizing the results from [CLT02,CLT12].
Let F be a number field and let G = G n a be the n-dimensional vector group. Let X be a smooth, projective, equivariant compactification of G defined over F , such that the boundary divisor D = X \ G is a strict normal crossings divisor on X, with irreducible components (D α ) α∈A . Let S be a finite set of places of F , containing all archimedean places, such that there is a smooth proper model (X , D) for (X, D) over the ring of S-integers O F,S of F in the sense of §2.3. We choose a weight vector ǫ = (ǫ α ) α∈A , where for all α, and we set Let L be a big line bundle on X, and let L denote L equipped with an adelic metrization such that for all v ∈ S, the metrization · v is induced by the integral model X . Our goal is to obtain precise results on the asymptotics of the counting function N(G(F ) ǫ , L, T ), (1.1) 1.4. A log Manin conjecture. Our investigation suggests that klt Campana points have quantitative properties strongly resembling those of rational points, while dlt Campana points with at least one ǫ α = 1 exhibit all the difficulties one observes when dealing with integral points. In §9, we postulate a conjecture on the asymptotic behavior of the counting function for klt Campana points of bounded height on Fano orbifolds over number fields. We view Theorem 1.1 as evidence towards this conjecture, and we discuss its consistency with recent work of Browning and Yamagishi [BY19].
1.5. Methods. To prove Theorems 1.1 and 1.3, we use the height zeta function method, as in the foundational papers [CLT02] and [CLT12]. Recall the height zeta function is given by where δ ǫ (x) is the indicator function detecting whether a given point in G(F ) belongs to G(F ) ǫ . Our goal is to obtain a meromorphic continuation of this analytic function, and to apply a Tauberian theorem. To this end, we consider the Fourier transform over the adèles: and we use the Poisson summation formula to obtain a meromorphic continuation of Z ǫ (s). To prove the absolute convergence of the right hand side, we estimate H ǫ (a, s) by combining work from [CLT02], [CLT10a], and [CLT12] on height integrals with oscillating phase.
1.6. Competing notions of Campana points. There are several competing notions of Campana points in the literature. Our study of local height integrals and Euler products shows that the notion considered in this paper (see §2.3) interacts well with the tools from harmonic analysis we are using: the regularization of the Euler product of local height integrals looks similar to the one used for the study of Manin's conjecture for rational points (see Proposition 5.2 and Corollary 5.3). The notion of Campana points appearing in [AVA18] is different from the one considered here. That notion enjoys good functoriality properties, but it seems ill-suited to the study points of bounded height: for example, if one were to use the height zeta function method to count points of bounded height, then the regularization of the Euler product of local height integrals for the main termĤ ǫ (0, s) would require a newfound set of ideas. We consider this clarification an important contribution of this paper.
1.7. Structure of the paper. After setting up some notation and recalling the notion of Campana orbifold, in §2 we discuss different notions of Campana points that appear in the literature -this is crucial, since only one of these works well for our purposes. We include an example in §2.3.1 that shows how different notions lead to different asymptotics for point counts on a single orbifold. In §2.4 we review a type of simplicial complex, called the Clemens complex, that helps keep track, in the presence of integrality conditions, of the contribution of local height integrals to the pole of the height zeta function. We then use these complexes to give birational invariance results (Lemmas 2.5 and 2.6) for the a and b-invariants that appear in the asymptotic formula of the counting function for Campana points.
In §3, we specialize to Campana orbifolds that are equivariant compactifications of vector groups. We recall basic facts about their geometry such as their Picard groups and effective cones of divisors, as well as results from harmonic analysis. After a discussion on local and global heights in §4, we define the height zeta function of an equivariant compactification of a vector group, and explain how to reduce the Poisson summation formula to the convergence of a sum of Fourier transforms of local height functions (local height integrals). Sections 5 and 6 contain the necessary estimates of local height integrals; before carrying on these technical estimates, we have included an interlude with a detailed explanation of the calculations in dimension 1, for the benefit of readers new to this type of analysis. we denote by F v the completion of F with respect to v. If v is non-archimedean, we denote by O v the corresponding ring of integers, with maximal ideal m v and residue field k v of size q v . We write A F for the ring of adeles of F . For each v ∈ Ω F , the additive group F v is locally compact, and carries a Haar measure µ v , also denoted by dx v . We denote a Haar measure on F n v by dx v . We define the absolute value | · | v by requiring that When v is real, |·| v is the usual absolute value. When v is complex, | · | v is the square of the usual norm on the complex numbers. For any prime number p, we have |p| p = 1/p. For any finite extension F v /Q p , we have We define the local zeta function by For non-archimedean places, the local zeta functions fit together to give the Dedekind zeta function 2.1.2. Varieties and divisors. Let F be a field with fixed algebraic closureF . An F -variety X is a geometrically integral separated F -scheme of finite type. We denote byX the base change of X toF . If F is a number field and v ∈ Ω F , we write X v for the base change of X to F v . Given a Weil R-divisor D = i a i D i on X, we denote by ⌊D⌋ = i ⌊a i ⌋D i its "integral part". We denote the reduced divisor a i =0 D i by D red .
2.1.3. Conventions for complex numbers. We denote the real part of a complex number s by ℜ(s), and the absolute value by |s|. Given s = (s 1 , . . . , s n ) ∈ C n and c ∈ R, by the expression ℜ(s) > c we mean that ℜ(s i ) > c for all i ∈ {1, . . . , n}. We also write |s| := max n i=1 |s i |.

2.2.
Orbifolds. We recall Campana's notion of orbifolds ("orbifoldes géométriques"), as introduced in his foundational papers [Cam04,Cam11]. In this article, we only consider those orbifolds which Campana calls "smooth"; in this section, we allow F to be any field.
Definition 2.1. A Campana orbifold over F is a pair (X, D) consisting of a smooth variety X and an effective Weil Q-divisor D on X, both defined over F , such that where the D α are prime divisors on X, and ǫ α belongs to the set of weights for all α ∈ A; (2) the support D red = α∈A D α is a divisor with strict normal crossings on X.
Condition (2) in this definition implies that the irreducible components D α of D red are smooth; it is important to note, however, that they may well be geometrically reducible. The definition also implies that any Campana orbifold (X, D) is a dlt (divisorial log terminal) pair, in the sense of birational geometry. We say that (X, D) is klt (Kawamata log terminal) if moreover ǫ α = 1 for all α ∈ A, i.e., if all weights are strictly smaller than 1.
Conversely, given a smooth F -variety X, a reduced divisor D = α∈A D α on X with strict normal crossings and a weight vector ǫ = (ǫ α ) α∈A , where ǫ α ∈ W for all α, we obtain a Campana orbifold (X, D ǫ ) over F by setting D ǫ = α∈A ǫ α D α .
2.3. Two types of Campana points. The notion of "orbifold rational point" is explored in Campana's papers [Cam04,Cam05,Cam11] and in Abramovich's survey [Abr09]. The adjective "rational" may create confusion, so we use the name Campana points here, to acknowledge that they are an intermediate notion between rational and integral points. In fact, [Abr09] defines two different notions of Campana points, one more restrictive than the other. It is essential for us to separate the two notions, since the orbifold analogue of Manin's conjecture seems to work well only for the more restrictive version; this is the one to which we will refer to simply as Campana points. The notion featuring in the recent paper [AVA18] is (a slight variant of) the less restrictive version, and we will refer to it as weak Campana points; it seems to be ill-behaved for the problem studied in this paper (see §2.3.1).
Remark 2.2. Surprisingly few results on the arithmetic of (weak) Campana points are available. Work on points of bounded height goes back to [VV12], followed immediately by [BVV12] and more recently by [BY19]. Forthcoming work of Damaris Schindler and the first author investigates the distribution of Campana points on toric varieties.
In dimension 1, where both notions of Campana points coincide, the analogue of Mordell's conjecture for Campana points has been proved over function fields, first in characteristic 0 by Campana himself [Cam05], and only very recently in arbitrary characteristic [KPS19]. Over number fields, the only known result says that the abc conjecture implies Mordell's conjecture for Campana points; see [Sme17,Appendix] for a detailed argument.
Let (X, D ǫ ) be a Campana orbifold with X proper over F , where D ǫ = α∈A ǫ α D α and the ǫ α belong to the usual set W. Let S ⊆ Ω F be a finite set containing Ω ∞ F such that (X, D ǫ ) has good reduction outside S. By this we mean that there exists a smooth, proper model X over O F,S such that if D α denotes the Zariski closure of D α in X , then α∈A D α is a fibrewise strict normal crossings divisor on X , i.e., the reduction We say that (X , D ǫ ) is a smooth, proper model for (X, D ǫ ) over O F,S ; such a model always exists away from a sufficiently large set S. Campana points can only be defined once a suitable model has been fixed, so let us fix a smooth, proper model (X , D ǫ ) for (X, D ǫ ) over O F,S . Any rational point P ∈ X(F ) extends uniquely to an integral point P ∈ X (O F,S ) by the valuative criterion for properness.
Let X • = X \ α∈A D α . If P ∈ X • (F ) and if v ∈ S is a place of F , then we get an induced point P v ∈ X (O v ). For each α ∈ A, the pullback of D α via P v defines a non-zero ideal in O v . We denote its colength by n v (D α , P ); this is the intersection multiplicity of P and D α at v. The total intersection number of P with D is then The following definition goes back to [Abr09, §2.1.7] and features in [AVA18] as well.
Definition 2.3. With the notation introduced above, we say that P ∈ X We denote the set of weak Campana O F,S -points on (X , D ǫ ) by (X , D ǫ ) w (O F,S ). We obtain a more restrictive notion by imposing conditions for individual irreducible components of the support of D, in the spirit of [Abr09, Definition 2.4.17]: Definition 2.4. With the notation introduced above, we say that P ∈ X • (F ) is a Campana O F,S -point on (X , D ǫ ) if the following implications hold for all places v ∈ S: if α ∈ A satisfies ǫ α = 1, then n v (D α , P ) = 0. If α ∈ A satisfies ǫ α < 1 and if n v (D α , P ) > 0, then In other words, writing ǫ α = 1 − 1 mα , we require n v (D α , P ) ≥ m α whenever n v (D α , P ) > 0. We denote the set of Campana O F,S -points on (X , D ǫ ) by (X , D ǫ )(O F,S ). We have The leftmost two inclusions are equalities if ǫ α = 0 for all α ∈ A, and the rightmost inclusion is an equality if ǫ α = 1 for all α ∈ A.
2.3.1. An instructive example. The following example illustrates the difference between the two notions of Campana points introduced above. We show that these notions yield different asymptotics for counts of points of bounded height. Moreover, the difference is encoded not only in the leading constant, but also in the exponent of the logarithm. In §9.5 we use this example to discuss functoriality of Campana points under birational transformations. Let X = P 2 Q with coordinates (x 0 : x 1 : x 2 ), and let D i = {x i = 0} for i ∈ {0, 1, 2}. Taking X = P 2 Z and ǫ 0 , ǫ 1 , ǫ 2 ∈ W, the Campana orbifold (X, 2 i=0 ǫ i D i ) has the obvious smooth, proper model (X , 2 i=0 ǫ i D i ) over Z in the sense of §2.3. For 0 ≤ i ≤ 2, we write ǫ i = 1 − 1 m i with the convention that 1 m i = 0 if ǫ i = 1. A point in X (Z), represented by coprime integer coordinates (x 0 : for every prime p and every i ∈ {0, 1, 2}, or equivalently, if x i is m i -full for all i ∈ {0, 1, 2}, assuming ǫ 0 , ǫ 1 , ǫ 2 < 1. Let us specialize to the case where m 0 = m 1 = m 2 = 2. To count (weak) Campana points of bounded height we use the exponential Weil height H : P 2 (Q) → R (x 0 : x 1 : x 2 ) → max{|x 0 |, |x 1 |, |x 2 |} whenever x 0 , x 1 , x 2 are coprime integers.
In this case, the set of Campana Z-points is in bijection with the set of triples (x 0 , x 1 , x 2 ) ∈ Z 3 such that gcd(x 0 , x 1 , x 2 ) = 1 and x 0 , x 1 and x 2 are all squareful. The counting function of Campana Z-points of Weil height bounded by T has an upper bound given by the cardinality of the set obtained by removing the coprimality condition, which grows asymptotically like T 3 2 , up to multiplication by a positive constant, by [BG58]. This upper bound is in agreement with Conjecture 9.5 below (see Remark 9.6).
On the other hand, the set of weak Campana Z-points is in bijection with the set of triples (x 0 , x 1 , x 2 ) ∈ Z 3 such that gcd(x 0 , x 1 , x 2 ) = 1 and x 0 x 1 x 2 is squareful. We claim that the counting function of weak Campana Z-points of Weil height at most T satisfies the lower bound ≫ T 3/2 log T as T → +∞. To see this, it suffices to count points of bounded height in the subset A of coprime triples (x 0 , x 1 , x 2 ) ∈ Z 3 >0 such that x 0 is a square and x 1 x 2 is a square. The size of this subset is estimated by where µ denotes the Möbius function. The number of squares up to T that are divisible by a given squarefree integer d is T 1/2 /d + O(1). To estimate the cardinality of the set B of pairs (x 1 , x 2 ) ∈ (dZ >0 ) 2 such that x 1 , x 2 ≤ T and x 1 x 2 is a square, we write u = gcd(x 1 /d, x 2 /d) and y i = x i /(du) for i ∈ {1, 2}. Then x 1 x 2 is a square if and only if both y 1 and y 2 are squares. Writing 2.4. Analytic Clemens complexes. Clemens complexes are simplicial sets that keep track of containment relations between the intersections of components of a divisor in a variety. As in [CLT12], Clemens complexes will be used in §8 to keep track of the contribution of the local height integrals to the pole of the height zeta function when some integrality conditions appear, that is, when some component of the boundary has weight 1. For a more detailed treatment, we refer the reader to [CLT10a, §3.1].
In this section X will be a smooth, proper variety over a number field F , and D = α∈A D α will be a reduced divisor on X with strict normal crossings. Let v ∈ Ω F , and fix an embeddinḡ

acts onX andD. WriteĀ for the indexing set ofD, and
A v for the set of orbits ofĀ under the action of Given a divisor D ′ on X such thatD ′ = α∈AD α for some A ⊆Ā, we denote by A v the set of orbits of A under the action of Γ v . As a set, the F v -analytic Clemens complex associated to D ′ consists of irreducible components Z of intersections β∈B D v,β for B ⊆ A v such that Z(F v ) = ∅. The complex enjoys additional structure, e.g., as a poset; see [CLT10a, §3.1] for details. The dimension of the Clemens complex of D ′ is We may now define the a-and b-invariants of the pair (X, D) at v with respect to a linear combination of boundary components with positive coefficients. These invariants will come up in the calculation of the position and order of the rightmost pole of a local height integral of X at v, in the case where X is an equivariant compactification of G = G n a . Keeping the notation introduced above, we assume further that −K Xv ∼ β∈Av ρ β D v,β , where ρ β ∈ Z, and we set L = β∈Av λ β D v,β for some λ β > 0. We define the a-invariant of the pair (X, D) at v with respect to L by Let us denote the the sum of the boundary components which do not appear in the support of a((X, D), L)L + K X + D by D ′ ; in other words, we set Writing C an Fv (D, L) for the F v -analytic Clemens complex associated to D ′ , we define the binvariant of (X, D) at v with respect to L as follows: Fv (D, L). We will now prove that the a-and b-invariants are birational invariants in a suitable sense. While this result is certainly of independent interest, we will use it to prove the meromorphic continuation of certain local height integrals in §5.
Lemma 2.5. Let X, D and L be as above. Let ( X, D) be another pair satisfying the same hypotheses as (X, D), namely: (i) D is a reduced divisor on X with strict normal crossings, Proof. First, we observe that birational invariance of the a-invariant follows from the fact that the pair (X, D) is log canonical: indeed, we can write where E ≥ 0 is an effective divisor supported on the exceptional locus of ϕ.
From now on, we denote a((X, D), L) simply by a and we work over F v , for a fixed place v. To prove birational invariance of the b-invariant, we first use [AKMW02, Theorem 0.3.1] to reduce to the case where the morphism ϕ is a blow-up of a smooth center having normal crossings with D. Let E be an exceptional divisor of ϕ. If the image of E is not a component of the intersection of some of the boundary components, then [Kol97, (3.11.1)] shows that the log discrepancy of the exceptional divisor E is greater than −1, hence that E appears in the support of aϕ * L + K X + D. Let Z be a maximal element in C an Fv (D, L) and assume that Z is a component of ∩ r i=1 D v,β i . If the image T of E does not contain Z, then the result is clear. If T contains Z, then by rearranging indices, we may assume that On the other hand, the strict transforms of D v,β i for i > k all contain ϕ −1 (Z). Thus our assertion follows.
If T is a component of the intersection of some of the boundary components, then E does not appear in the support of the difference of aϕ * L + K X + D and ϕ * (aL + K X + D). We distinguish two cases. First, if E does not appear in the support of ϕ * (aL + K X + D), we denote by Z a maximal element of C an Fv (D, L) and we assume that Z is a component of ∩ r i=1 D v,β i . Either T and Z do not meet, or T contains Z; in the latter case, the strict transforms of the D v,β i 's do not meet in ϕ −1 (Z), but E and D v,β 1 , · · · , D v,β r−1 intersect. Second, if E does appear in the support of ϕ * (aL + K X + D), then T does not contain Z, and therefore T and Z do not meet. Thus our assertion follows.
We will now introduce a version of the b-invariant for rational functions. If f is an arbitrary rational function on X, then for every α ∈ A, we denote by d α (f ) the coefficient of D α in the principal divisor div(f ). Let D ′′ be the sum of boundary components D α such that D α does not appear in the support of aL + K X + D and d α (f ) ≤ 0. We denote by C an Fv (D, L, f ) the F v -analytic Clemens complex associated to D ′′ , and we define the b-invariant by Fv (D, L, f ). Using the same methods, we obtain the following analogue of Lemma 2.5: Lemma 2.6. Let X, D, L and f be as above. Let ( X, D) be another pair satisfying the same hypotheses as (X, D), namely (i) D is a reduced divisor on X with strict normal crossings,

Geometry of equivariant compactifications of vector groups
We now recall some basic facts on the geometry of equivariant compactifications of vector groups from [HT99] and [CLT02]. Let X be a smooth equivariant compactification of G = G n a defined over a field F of characteristic 0. By definition, X contains G as a dense Zariski open, and its complement D = X \ G is divisorial, i.e., it is the union of prime divisors: The irreducible divisors D α need not be geometrically irreducible, so we also consider the decomposition ofD into irreducible components: There is a natural action of the Galois group Γ = Gal(F /F ) on the index setĀ, and Galois orbits are in one-to-one correspondence with elements of A.

Picard groups and the anticanonical class.
Proposition 3.1. [CLT02, Proposition 1.1] With the above notation, the following hold.
(1) There are natural isomorphisms of Galois modules where Eff 1 (X) is the cone of effective divisors onX.
(2) By taking Γ-invariant parts, we have where Eff 1 (X) is the cone of Γ-invariant effective divisors on X.
Let f be a non-zero linear form on G = G n a , defined over F . Considering f as an element of the function field F (X), we can write div(f ) uniquely as where E(f ) is the hyperplane along which f vanishes in G, and the d α (f ) are integers.
Finally, the anticanonical divisor turns out to be linearly equivalent to an integral linear combination of boundary components: we have −K X ∼ α∈A ρ α D α for certain integers ρ α , and by [CLT02, Lemma 2.4], we know that ρ α ≥ 2 for all α.
Remark 3.3. With the above notation, if (ǫ α ) α∈A is any vector of weights chosen from the allowed set is automatically big. This follows from the fact that the cone of big divisors is the interior of the pseudo-effective cone, together with Proposition 3.1.

3.2.
Harmonic analysis on vector groups. In this section, we recall some of the basic elements of harmonic analysis on adelic vector groups as developed in [Tat67].
These Haar measures satisfy µ v (O v ) = 1 for all but finitely many non-archimedean places v; they induce a self-dual measure dx = µ on A F and the product measure on G(A F ). For any non-archimedean place v such that the completion F v is a finite extension of Q p , we define the local additive unitary character by When v is an archimedean place, we define the local additive character by The Euler product ψ := v ψ v is an automorphic character of A F . otherwise.
If j = 0 the integral above vanishes whenever id − j ≥ c + 2.
To each adelic point a ∈ G(A F ), we associate the linear functional f a : x → a · x, where a · x denotes the inner product. The composition ψ a = ψ • f a defines a Pontryagin duality Given an integrable function Φ on G(A F ), we define its Fourier transform by converges absolutely and uniformly when b belongs to a fundamental domain for the quotient G(A F )/G(F ), and that the infinite sum converges absolutely. Then we have

Height zeta functions
In this section, we will establish some basic properties of height zeta functions. Let G = G n a and let X be a smooth equivariant compactification of G defined over a number field F . Applying an equivariant embedded resolution of singularities if necessary, we may assume that the boundary D is a strict normal crossings divisor on X.
4.1. Height functions. We first recall some of the basic properties of height functions, referring to [CLT10a, §2] for more details. Let us consider the decomposition of the boundary into irreducible components: For each line bundle O(D α ), we fix a smooth adelic metrization ( · v ) v∈Ω F . Let f α be the section corresponding to D α . For each place v, we define the local height pairing by This pairing varies linearly on the factor Pic(X) C and continuously on the factor G(F v ). We define the global height pairing H as the product of the local height pairings Again, this pairing varies continuously on the first factor and linearly on the second factor.
The following lemma plays a crucial rôle in the analysis of height zeta functions in general.

Moreover, if
(1) there exists a smooth, projective O v -model X for X, which comes equipped with an action of the O v -group scheme G n a,Ov extending the given action of G on X, and if (2) the unique linearisation on O(D α ) extends to O(D α ) for any α ∈ A, In particular, for all but finitely many places v ∈ Ω F , we may simply take 4.2. Intersection multiplicities. With the notation of the previous paragraph, let S ⊂ Ω F be a finite set containing all archimedean places, such that (X, D) admits a smooth, proper model (X , D) over Spec O F,S , where D = α∈A D α , in the sense of §2.3. Moreover, let ǫ = (ǫ α ) α∈A be a weight vector as in §2.2. Our object of study is . Hence we may define the analogous sets For v ∈ S, we denote by δ ǫ,v the indicator function detecting whether or not a given point in For v / ∈ S, we have the reduction map .
, then Hensel's Lemma implies that D v,β has an F v -point, and therefore it is geometrically irreducible over F v . Using a standard argument in Arakelov geometry (see, e.g., [Sal98, Theorem 2.13] and its proof), we see that there exists analytic local coordinates (z 1 , · · · , z n ) on η −1 v (y) mapping to A n Fv such that • these local coordinates induce an analytic isomorphism If we moreover assume that v satisfies conditions (1) and (2) in Lemma 4.1, then we can take For each non-archimedean place v, we denote by K v a maximal compact open subgroup of G(O v ) satisfying the conclusions of Lemma 4.1 and Lemma 4.2, and we denote Our discussion shows that both H(·, s) and δ ǫ are K-invariant.

4.3.
Height zeta functions. For each α ∈ A, we fix an adelic metrization on the line bundle O(D α ) such that for all v ∈ S, the corresponding metrization · v is induced by the integral model (X , D) chosen previously. This induces an adelic metrization for any line bundle L. To understand the asymptotic formula for the counting function (1.1), we introduce the height zeta function: The proof of [CLT02, Proposition 4.5] shows that Z ǫ (s) is holomorphic when ℜ(s) ≫ 0. The existence of a meromorphic continuation of this zeta function, together with a standard Tauberian theorem, yields a proof of the asymptotic formula for (1.1). We therefore consider the Fourier transform in hopes of using the Poisson summation formula (Theorem 3.5) to obtain the desired meromorphic continuation of Z ǫ (s). The first two of the three conditions in Theorem 3.5 follow from the proof of [CLT02, Lemma 5.2] To verify the third condition, we recall the following result.
Proposition 4.3 ([CLT02, Proposition 5.3]). With the notation introduced above, for all characters ψ a that are non-trivial on K and for all s such that H(·, s) −1 is integrable, we have H ǫ (a, s) = 0.
Let Λ X ⊂ G(F ) be the set of a such that ψ a is trivial on K. Then Λ X is a sub-O F -module of G(F ) of full rank n; proving that the sum is absolutely convergent whenever ℜ(s) ≫ 0 verifies the third condition in Theorem 3.5. This will be done in §7. Once this is established, we obtain (4.1) Interlude I: Dimension 1 Let us first make our analysis explicit for P 1 over Q, considered as the natural equivariant compactification of G = G a = A 1 , with boundary D = (1 : 0). We fix the standard integral models for P 1 as well as D. Given ǫ ∈ W, we consider the problem of counting Campana Z-points on (P 1 Z , D ǫ ). Note that if ǫ < 1, then x ∈ G(Q) = Q is a Campana Z-point if and only if the denominator of x is m-full, where m = 1/(1 − ǫ); this means that any prime dividing the denominator of x occurs with exponent at least m in the prime factorization. If, on the other hand, ǫ = 1, then x is a Campana Z-point if and only if x ∈ Z. Since the latter case is trivial, we will assume from now on that ǫ < 1.
We fix a finite set of places S. Going back to the notation introduced in §4, we see that we can take K = p prime G(Z p ) in this case, so that Λ X = Z. This yields We would like to compute H ǫ (n, s) explicitly.
By definition, we have We fix metrizations as follows: if v is archimedean.
The trivial character. Here we compute H ǫ (0, s). For any prime p / ∈ S we have It follows that the highest pole of H ǫ (0, s) is at s = 1 + 1/m = 2 − ǫ, and that it has order 1.
Non-trivial characters. Let n be a non-zero integer. Our aim is to understand where the local factors are given by Suppose first that p / ∈ S and p ∤ n. The local factor then reduces to Let us now assume that p ∈ S and p | n, and let us denote the p-adic valuation of n by k.
In this case, the local factor becomes When p ∈ S, we recover the formula above for ǫ = 0.
Using these explicit formulae, we obtain: Lemma 1. Let p be prime. The function s → H ǫ,p (n, s) is holomorphic everywhere. Moreover, the product p prime H ǫ,p (n, s) is holomorphic for ℜ(s) > 1 − ǫ, and there exists positive constants ℓ and C such that for any s such that ℜ(s) > 1 − ǫ.
Finally we analyze the archimedean place: Lemma 2. The function s → H ǫ,∞ (n, s) is holomorphic everywhere. Moreover, for any integer N, there exists positive constants ℓ and C such that Conclusion. Putting all information together, we obtain that Z ǫ (s) has a unique pole located at s = 1 + 1/m = 2 − ǫ, contributed by the trivial character. Applying a Tauberian theorem (see, e.g., [Ten95, II.7, Theorem 15]), using the line bundle L = O(1) metrized as above, we obtain for some c > 0.

Height integrals I
In this section, we resume our general analysis and study the height integral We begin by setting up some necessary notation. Each c ∈ R gives rise to a tube domain where (ρ α ) α∈A is the integer vector given by where the D v,β are irreducible components, and we write for an analogous decomposition of D α ⊗ F F v into irreducible components. Given β ∈ A v , let us denote by F v,β the field of definition for one of the geometric irreducible components of D v,β , that is, the algebraic closure of F v inside the function field of D v,β , and by f v,β the extension degree [F v,β : Finally, for any subset B ⊆ A v , we define 5.1. Places of good reduction. We will now study the basic properties of To avoid clutter, we first assume that µ v (O v ) = 1. Set ρ = (ρ α ) α∈A . Let ω be a gauge form on G, i.e., a nowhere vanishing differential form of top degree.
Considering ω as a rational section of O(K X ) equipped with the adelic metrization fixed in the previous section, we have the equality Writing dτ = dx v ω v for the corresponding Tamagawa measure, we see that Breaking up this integral over the fibres of the reduction map η v : We now compute the inner integral If B = ∅, then there is a measure preserving analytic isomorphism η −1 v (y) ∼ = m n v . Since any for all such x v , so that (5.1) simply evaluates to 1/q n v . If B = ∅, then every β ∈ B lies above a unique α(β) ∈ A. If D • v,B (k v ) = ∅, then D v,β (k v ) = ∅ for all β ∈ B. Using Hensel's lemma, we deduce that D v,β has an F v -rational point, and hence is geometrically irreducible; in particular, F v,β = F v for all β ∈ B. Writing B = {β 1 , · · · , β ℓ } and α i = α(β i ) for simplicity, we see as in §4.2 that there exist analytic local coordinates (z 1 , · · · , z n ) on η −1 v (y) inducing a measure-preserving analytic isomorphism is given by z i = 0, for i = 1, · · · , ℓ. The integral (5.1) can now be rewritten as where π v denotes a choice of generator for m v . Summing the contributions coming from different subsets of A v , we obtain the equality Here we interpret the term q to be zero whenever ǫ α(β) = 1. This formula resembles Denef's formula in [CLT10a, Proposition 4.5].
When µ v (O v ) = 1, the same arguments show that

Places contained in S.
Assume now that v ∈ S. In this case, δ ǫ,v ≡ 1 by definition. Therefore the local height integral for Campana points coincides with the usual local height integral, so that we do not need to do anything new: admits a holomorphic continuation to the domain ℜ(s) > a − δ for some δ > 0. Moreover, the function s → H v (0, sL) has a pole at s = a of order b.

Euler products.
Given α ∈ A, we denote by F α the field of definition for one of the geometric irreducible components of D α ; in other words, F α is the algebraic closure of F in the function field of D α .
Proof. We may safely assume that µ v (O v ) = 1. We analyze the right hand side of (5.2), separating the analysis into three cases.
Therefore the term corresponding to B in the right hand side of expression (5.2) for H ǫ,v (0, s) is simply equal to 1.
v,B (k v ) = ∅ or ǫ α(β) = 1, then B does not contribute to the right hand side of (5.2). If, on the other hand, for some δ 1 > 0, which may be chosen independently of β. Therefore by choosing δ > 0 sufficiently small and s ∈ T >−δ , the term corresponding to B = {β} contributes to the sum in the right hand side of (5.2) by It follows that if we choose δ sufficiently small and s ∈ T >−δ , then the contribution of the term corresponding to B = {β} can be rewritten as ). Moreover, the product in the term in the right hand side of (5.2) corresponding to B is O(q −(1+δ ′ ) v ), with δ ′ as above, assuming that we have chosen s ∈ T >−δ for δ > 0 sufficiently small. Indeed, each of the factors in the product is bounded from above by q −(1−mδ) v for some m > 0, as s ∈ T >−δ . There are at least two such factors, so the result is bounded from above by q −2(1−mδ) v for some m > 0, and hence certainly by q We conclude that for δ > 0 small enough and s ∈ T >−δ , we have This implies the proposition.
Corollary 5.3. The function Proof. This follows immediately from Proposition 5.2, taking into account the fact that for all α ∈ A.

Height integrals II
In this section, we study the height integral v∈Ω F H ǫ,v (a, s).
We introduce some notation. For each a ∈ G(F ) with a = 0, we denote the linear functional x → a · x by f a , where a · x is the standard inner product. Recall from §3 that For any place v ∈ Ω F , we define and for any non-archimedean place v, we take Note that we have H ∞ (a) ≫ H fin (a) −1 . (6.1) 6.1. Places of good reduction. Let us first assume that v / ∈ S. Since H ǫ (a, s) = 0 whenever a / ∈ Λ X by Proposition 4.3, we may safely assume that a ∈ Λ X . We will distinguish two cases, depending on whether j v (a) = 0 or j v (a) = 0; we start with the former case.
To analyze the integral in the domain T >−δ , we begin by stratifying G(F v ) by the fibers of the reduction map: and a ∈ Λ X . • If B = {β}, we define α(β) as in §5.1. Without loss of generality, we may assume that D v,β is geometrically irreducible and that ǫ α(β) = 1. We distinguish two cases: either α(β) ∈ A 0 (a), or α(β) ∈ A ≥1 (a).
• If #B ≥ 2, then arguing as in the proof of Proposition 5.2, one can show that for some δ 5 > 0 assuming that δ > 0 is sufficiently small.
Combining the estimates above, we obtain the following analogue of Proposition 5.2.
This finishes the analysis in the case j v (a) = 0. From now on, we assume that j v (a) = 0.
Proposition 6.2. Suppose that v is a non-archimedean place such that v / ∈ S and j v (a) = 0. There exists a real number δ > 0, independent of a, such that the function is holomorphic on the domain T >−δ .
Proof. As before we use the stratification of G(F v ) by the fibers of the reduction map: • If B = ∅, the inner summation is holomorphic everywhere and equals 1 as in §6.1.
• If B = {β}, we define α(β) as in §5.1. Without loss of generality, we may assume that D v,β is geometrically irreducible and that ǫ α(β) = 1. We again distinguish two cases: either α(β) ∈ A 0 (a) or α(β) ∈ A ≥1 (a). If α(β) ∈ A 0 (a), the character ψ a,v becomes trivial on η −1 v (D • v,B (k v )). Hence, arguing as in the proof of Proposition 5.2, for a sufficiently small δ > 0, the inner summation is holomorphic and bounded by If on the other hand α(β) ∈ A ≥1 (a), we denote d := d α(β) (f a ). If y / ∈ E(f a )(k v ), then we use Lemma 3.4 to compute We note that the implied constant can be taken independent of a; indeed, there are only finitely many possibilities for d α (f a ) by Proposition 3.2. Finally, if y ∈ E(f a )(k v ), then for δ > 0 sufficiently small we have for some δ ′ > 0. Thus, using the Lang-Weil estimates as in §6.1, we obtain • If #B ≥ 2, then as in the proof of Proposition 5.2 we have We conclude as in the proof of Proposition 5.2.
6.2. Places contained in S. We now treat the remaining places.
Proposition 6.3. The following hold whenever v ∈ S.
(1) Let δ > 0 be any positive real number. Then the function is holomorphic in the domain given by ℜ(s α ) > ρ α − 1 + δ for each α ∈ A. Moreover, there exists a real number M N > 0, that does not depend on a, such that in the above domain.
(2) Let L = α∈A λ α D α be a big divisor, and let  We introduce some notation. For every α ∈ A we set Proposition 6.4. Assume that ⌊D ǫ ⌋ = 0. There is a real number δ > 0, independent of a, such that the function is holomorphic on T >−δ .

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Moreover, for any integer N > 0, there exists a real number M N > 0 such that  Proof. This follows from Propositions 6.1, 6.2, and 6.3, together with the estimate (6.1). The implied constant can be chosen independently of a, since a belongs to the O F -module Λ X .

Proof of the main result for klt Campana points
In this section we prove our main result, Theorem 1.1. We work in the setting introduced in §1.3, recalled here for the reader's convenience.
By X we mean a smooth, projective and equivariant compactification of G = G n a , defined over a number field F . We assume that the boundary divisor D = X \ G is a strict normal crossings divisor on X, with irreducible components (D α ) α∈A , so that D = α∈A D α . We denote by F α the field of definition for one of the geometric irreducible components of D α ; in other words, F α is the algebraic closure of F in the function field of D α .
Let S ⊆ Ω F be a finite set containing Ω ∞ F , such that there exists a smooth, proper model (X , D) of (X, D) over Spec O F,S as in §2.3, and let D = α∈A D α . Having fixed ǫ α ∈ W for each α ∈ A, we let D ǫ = α∈A ǫ α D α and D ǫ = α∈A ǫ α D α . In this section we assume that the pair (X, D ǫ ) is Kawamata log terminal (klt for short), that is, ǫ α < 1 for all α ∈ A.
Let L denote a big line bundle L on X, equipped with an adelic metrization induced by the integral model X for all v ∈ S. Our goal is to understand the asymptotic behavior of the counting function N(G(F ) ǫ , L, T ), which records the number of points of L-height at most To do this, we apply a Tauberian theorem to the height zeta function 3. This function is holomorphic function when ℜ(s) ≫ 0; our first goal is to establish a meromorphic continuation of this function. Subsequently, knowledge of the location of the rightmost pole of Z ǫ (sL) along ℜ(s), its order, and its residue will serve as inputs to the Tauberian theorem that establishes the asymptotic formula we seek.
Recall that for any real number c we defined where the ρ α are integers satisfying −K X ∼ α∈A ρ α D α .
Proposition 7.1. The function is holomorphic in the region T ≥0 .
Proof. We begin by verifying that the Poisson summation formula holds for ℜ(s) ≫ 0. The discussion in §4.3 shows that all that remains to be done is checking that the right hand side converges absolutely. This follows from Proposition 6.4, as a∈Λ X 1 (1 + H ∞ (a)) N converges for sufficiently large N. The result now follows from an application of Propositions 5.1 and 6.4 and Corollary 5.3 to the summands of the right hand side of (7.1).
Remark 7.2. It is important to note that the local height integrals studied in § §5-6 have poles along s α = ρ α − 1; however, it follows from Proposition 7.1 that the rightmost pole of Z ǫ (s) occurs along some s α = ρ α − ǫ α > ρ α − 1, because of the klt condition.
With a meromorphic continuation of Z ǫ (s) in hand, we turn to the case where s = sL. We may write L = α∈A λ α D α , where λ α > 0 for all α ∈ A, because L is big. Then s α = sλ α . Proposition 7.1 suggests that the rightmost pole along ℜ(s) of the zeta function Z ǫ (sL) is the order of this pole should be b = b(F, (X, D ǫ ), L) := #A ǫ (L); see Remark 7.2. We shall establish these statements, separating our analysis into two cases, according to the Iitaka dimension of the adjoint divisor 7.1. Rigid case. In this subsection we assume that the adjoint divisor aL + K X + D ǫ has Iitaka dimension equal to zero; we say that aL + K X + D ǫ is rigid. Recall that Λ X ⊂ G(F ) is the set of a such that the character ψ a is trivial on the compact open K defined in §4.2. By the Poisson summation formula, we have We study the poles of Z ǫ (sL) by looking at the individual terms of the right hand side. When a = 0, it follows from Corollary 5.3 that H ǫ (0, sL) has a pole at s = a of order b, provided we show that the corresponding residue is not zero (we verify this last claim presently). On the other hand, Proposition 6.4 shows that the term H ǫ (a, sL) has a pole of the highest order equal to that of H ǫ (0, sL) if and only if This condition means that whenever (ρ α − ǫ α )/λ α = a, we must have d α (f a ) = 0. Since it follows that E(f α ) is equivalent to a boundary divisor whose support is contained in that of the adjoint divisor aL + K X + D ǫ . This is not possible as aL + K X + D ǫ is rigid. Our analysis shows that the main term of Z ǫ (sL) is furnished by H ǫ (0, sL), provided is non-zero. Recall that Setting Γ = Gal(F /F ) and Γ Fα = Gal(F /F α ), we construct the virtual Artin representation We denote the corresponding virtual Artin L-function by This function has a pole of order #A ǫ (L) at s = 1 by [IK04,Corollary 5.47]. For v ∈ S we define L v (P (X • ), s) = 1. Using this we define the Tamagawa measure where L S * (P (X • ), 1) is the leading constant of L S (P (X • ), s). We also define Lemma 7.3. With notation as above, we have Proof. First, we note that For each α ∈ A ǫ (L), we have a = (ρ α − ǫ α )/λ α , where ǫ α = 1 − 1/m α . Each of the b-many local zeta factors ζ Fα,S c (m α (λ α s − ρ α + 1)) has a simple pole at s = a, so that the limit is equal to the residue at s = a for the local zeta factor corresponding to α, which we denote by ζ * Fα,S c (1)/m α λ α , where ζ * Fα,S c (1) is the residue of ζ Fα,S c (s) at s = 1, the normalization 1/m α λ α being a consequence of the chain rule. With the notation each of which is regular at s = a (note that τ v and τ X 0 ,v coincide on G). We obtain Using the equality we may simplify the above expression for c to Finally, (7.2) allows us to conclude that Applying a Tauberian theorem (see, e.g., [Ten95, II.7, Theorem 15]), we obtain: Theorem 7.4. Let X , L, D and ǫ be as above. Assume that (X, D ǫ ) is klt and set 7.2. Non-rigid case. The analysis in this subsection is modeled on [Tsc03]. With notation as above, we now assume that the divisor E := aL + K X + D ǫ is not rigid, i.e., that its Iitaka dimension is positive. Then some multiple mE defines the Iitaka fibration φ m : X Y m . Since mE admits a G-linearization, Y m admits a natural G-action, and φ m is G-equivariant. For the sake of simplicity, we assume that φ m is a morphism. The variety Y m contains an open orbit of the G-action, so it has the structure of an equivariant compactification of the quotient vector space G/G L , where G L ⊂ G is a linear subspace of G.
As in §7.1, the term H ǫ (a, sL) has a pole of highest order equal to that of H ǫ (0, sL) if and only if A 0 (a) ⊃ A ǫ (L). This condition is equivalent to having f a = 0 on G L . Therefore the rightmost pole of Z ǫ (sL) is furnished by the sum where the last equality follows from the Poisson summation formula. Note that the equality holds for any s with ℜ(s) > a by the monotone and dominated convergence theorems. Let X y be the fiber of φ m above y. It is a smooth equivariant compactification of G L , with boundary divisor D| Xy . Let X y be the closure of X y inside X . The restriction (aL + K X + D ǫ )| Xy is rigid, since φ m is an Iitaka fibration. Applying the analysis of §7.1, we conclude that the inner integral has a pole at s = a((X y , D ǫ | Xy ), L) of order b(F, (X y , D ǫ | Xy ), L). Now [HTT15, Lemma 5.2] yields a((X, D ǫ ), L) = a((X y , D ǫ | Xy ), L), and b(F, (X, D ǫ ), L) = b(F, (X y , D ǫ | Xy ), L).
We claim that All we need to do is justify the interchange of limits: the right hand side converges by Fatou's lemma, and the claim then follows from the Poisson summation formula (Theorem 3.5).
As before, applying a Tauberian theorem ([Ten95, II.7, Theorem 15]), we obtain: Theorem 7.5. Let X, L, D and ǫ be as above. Assume that (X, D ǫ ) is klt, and that m is an integer such that the Iitaka fibration φ m : X Y m defined by mE is a morphism. Set Then

Interlude II: Examples
Blow-ups of P n . Let f ∈ Z[x 0 , . . . , x n ] be a homogeneous polynomial of degree d such that the subvariety {x 0 = f = 0} of P n Q is smooth. Let S be a finite set of places of Q that contains the infinite place and the primes of bad reduction of {x 0 = f = 0}. Let Z S be the ring of S-integers of Q. Let ϕ : X → P n Z S be the blow up with center {x 0 = f = 0}. Let D 1 be the exceptional divisor and D 2 the strict transform of {x 0 = 0}. Fix positive integers m 1 and m 2 , and let ǫ i = 1 − 1/m i for i ∈ {1, 2}. Then (X , D ǫ ) is a smooth proper model of a klt Campana orbifold in the sense of §2. By definition of Campana points, the restriction of the morphism ϕ to the set of Z S -Campana points (X , D ǫ )(Z S ) is injective. Thus, ϕ induces a bijection between (X , D ǫ )(Z S ) and the set A of (n + 1)-tuples (x 0 , . . . , x n ) ∈ Z n S such that gcd(x 0 , . . . , x n ) = 1, x 0 > 0, gcd(x 0 , f (x 0 , . . . , x n )) is m 1 -full, Then an application of Theorem 7.4 with L = π * O P n (1) shows that #{(x 0 , . . . , x n ) ∈ A : max{|x 0 |, . . . , |x n |} ≤ T } ∼ cT n+1/m 2 as T → ∞, for some c > 0.
A singular del Pezzo surface. Let X be the minimal desingularization of a split quartic del Pezzo surface of type D 5 over Q. Then X is an equivariant compactification of G 2 a by [DL10, Lemmas 4 and 6]. The irreducible components of the boundary on X are the divisors E 1 , . . . , E 6 from [Der14, §3.4 Type D 5 ]. We fix coordinates (x 0 : x 1 : x 2 ) on P 2 Q and we denote by ϕ : X → P 2 the morphism from [Der14, §3.4 Type D 5 ] that contracts E 1 , E 2 , E 4 , E 5 , E 6 to the point (0 : 0 : 1) and maps E 3 onto {x 0 = 0}. The morphism ϕ is a sequence of successive blow ups at Q-points. Performing the same sequence of blow ups over Z (with centers at the corresponding Z-points) yields a smooth projective Z-model X for X. For every i ∈ {1, . . . , 6}, we fix a positive integer m i , we define ǫ i = 1 − 1 m i , and we denote by E i the closure of E i in X . Then (X , 6 i=1 ǫ i E i ) is a smooth proper model for the klt Campana orbifold (X, 6 i=1 ǫ i E i ). We use the notation f (·) := ·/ gcd(·, x 1 ) and g(·) := x 1 / gcd(·, x 1 ), and we denote by f (n) the n-th composition of f with itself. The set of Z-Campana points on (X , 6 i=1 ǫ i E i ) is in bijection, via ϕ, with the set A of triples (x 0 , x 1 , x 2 ) ∈ Z 3 such that gcd(x 0 , x 1 , x 2 ) = 1, Then an application of Theorem 7.4 with L = ϕ * O P 2 (1) shows that #{(x 0 , . . . , x n ) ∈ A : max{|x 0 |, . . . , |x n |} ≤ T } ∼ cT 2+1/m 3 as T → ∞, for some c > 0.

Proof of the main result for dlt Campana points
In this section we sketch the proof of Theorem 1.3. We use the notation of §7, but this time we assume that ⌊D ǫ ⌋ = 0, so that (X, D ǫ ) is not a klt pair, but only a dlt pair. We set Let L = −(K X + D ǫ ). Arguing as in the proof of Proposition 7.1, we obtain: is holomorphic in the region ℜ(s) ≥ 1.
This implies that the zeta function Z ǫ (sL) possibly has a pole at s = 1. Then there exists a constant c > 0 that depends on F, S, (X , D ǫ ), and L, such that

A Manin-type conjecture for klt Campana points on Fano orbifolds
In this section we postulate an asymptotic formula for the counting function of Campana points of klt type on Fano orbifolds. We expect that it is necessary to remove a thin set of Campana points from the count in order to obtain a formula that reflects the global geometry of the Campana orbifold; indeed, already for rational points it has been understood for quite some time that a version of Manin's conjecture with only a closed -rather than thin -exceptional set admits counterexamples, see [BT96b,LR14,BL17]. Meanwhile, several authors have recently built up evidence towards a version of Manin's conjecture with a thin exceptional set, see [LT17,Pey17,Sen17,LST18]. While we do believe that the set of klt Campana points is itself not thin, we are unable at present to show this; however, we propose a problem we hope will ameliorate this circumstance. Finally, we discuss a recent example of Browning and Yamagishi [BY19], and its compatibility with our proposed asymptotic.
Let (X, D ǫ ) be a Campana orbifold over a number field F , where D ǫ = α∈A ǫ α D α . Assume moreover that −(K X + D ǫ ) is ample; a pair (X, D ǫ ) with this additional property is called a Fano orbifold. Fix a finite set S ⊂ Ω F containing all archimedean places of F , as well as a smooth, proper model (X , D) of (X, D) over Spec O F,S , as in §2.2. Write (X , D ǫ )(O F,S ) for the set of O F,S -Campana points of (X , D ǫ ) (Definition 2.4), and assume that ⌊D ǫ ⌋ = 0, so that the Campana points considered in this section are of klt type. 9.1. Thin exceptional sets. We keep the notation introduced above.
Definition 9.1. A thin subset of (X , D ǫ )(O F,S ) is a subset of a finite union of (1) type I sets: those of the form Z ∩ (X , D ǫ )(O F,S ) for a proper Zariski closed subset Z ⊂ X;

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(2) type II sets: those of the form f (Y (F )) ∩ (X , D ǫ )(O F,S ), where f : Y → X is a generically finite cover of degree at least 2, with Y a projective, integral F -variety.
It is natural to ask whether (X , D ǫ )(O F,S ) is itself not thin, possibly after a finite extension of the ground field. After all, if a version of Manin's conjecture with a thin exceptional set is to hold for Campana points on Fano orbifolds, we would like to have something left to count after the removal of a thin subset. We are thus forced to make what we hope is a superfluous hypothesis in Conjecture 9.5 below, namely, that (X , D ǫ )(O F,S ) itself is not thin in our setting.
This shortcoming is already present in the traditional case of rational points on smooth Fano varieties, where we expect the set of rational points to be not thin if it is non-empty. This is known conditionally on Colliot-Thélène's conjecture predicting that the Brauer-Manin obstruction controls all failures of weak approximation on rationally connected varieties [CT03]. Indeed, this conjecture implies that smooth Fano varieties satisfy "weak weak approximation", which in turn implies that the set of rational points is not thin [Ser92, Theorem 3.5.7].
On a positive note, Serre has shown that P n (F ) is not thin [Ser92,§3.4]. This prompts us to ask: Question 9.2. Let F be a number field and let D = α∈A D α be a divisor on P n F with strict normal crossings. For each α ∈ A, pick ǫ α ∈ W with ǫ α < 1 and set D ǫ = α∈A ǫ α D α , so that the Campana orbifold (P n , D ǫ ) is klt. Assume moreover that −(K P n + D ǫ ) is ample. Fix an integral model (P n , D ǫ ) of (P n , D ǫ ), and a finite set S of places of F that includes all the archimedean places. Is the set (P n , D ǫ )(O F,S ) of klt Campana points non-thin?
For some partial results, we refer to the recent paper of Browning-Yamagishi [BY19, §4]. A version of this question for integral points is addressed in [Coc19].
In a different direction, if the set of Campana points (X , D ǫ )(O F,S ) were thin, then there would exist a set of places T such that the image of this set in v∈T X(F v ) is not dense, by [Ser92, Theorem 3.5.3]. Since we expect (X , D ǫ )(O F,S ) to be not thin, we ask: In other words, does the set of Campana points satisfy weak weak approximation?
9.2. The asymptotic formula. We keep the notation from §9.1; in particular, (X, D ǫ ) is both a Fano orbifold and a klt pair. Recall that the effective cone Eff 1 (X) is finitely generated by [BCHM10]. Let L = (L, · ) be an adelically metrized big divisor on X.
The b-invariant of (X, D ǫ ) with respect to L, denoted by b(F, (X, D ǫ ), L), is the codimension of the minimal supported face of Eff 1 (X) which contains the class a((X, D ǫ ), L)[L] + [K X + D ǫ ] (see [HTT15,Definition 2.1] for the definition of supported faces).
The metrized divisor L on X gives rise to a height function [Pey95, §1.3] For any subset U ⊂ X(F ), we consider the counting function where c(F, S, (X , D ǫ ), L, Z) is a Tamagawa constant, described below.
Remark 9.6. We note that for the example in §2.3.1, in the case m 0 = m 1 = m 2 = 2, and L = O(1), we have a((X, D ǫ ), L) = 3/2 and b = b(F, (X, D ǫ ), L) = 1, so Conjecture 9.5 predicts a counting formula for Campana points of bounded height that grows like cT 3/2 as T → ∞, which is correct. Forthcoming work of Damaris Schindler and the first author show that the upper bound in §2.3.1 is in fact sharp.
Lehmann, Sengupta, and the third author have proposed an explicit construction of the exceptional set in Manin's conjecture in the traditional case of rational points, in their recent paper [LST18]. They prove that it yields a thin set of rational points, via the minimal model program and Hilbert's irreducibility theorem. It would be interesting to have an analogous description in the case of klt Campana points. 9.3. The leading constant. In this section, we define the leading constant that appears in Conjecture 9.5, in the case when the divisor a((X, D ǫ ), L)L + K X + D ǫ ∼ E ≥ 0 is rigid. The construction here is analogous to [Pey95] and [BT98b]. For simplicity, we assume that the boundary divisor D contains all components of E; we denote the by A(L) the set of irreducible components of D that are not contained in the support of E.
The claim then follows from the observation that the outer terms are finite groups, since Γ is profinite and both Pic(X) and Q are free of finite rank; indeed, X is a Fano orbifold and Q is generated by the classes of the geometric irreducible components of E.
The open set U can be endowed with a Tamagawa measure τ U [CLT10a, Definition 2.8]; fixing an adelic metrization on each component of D and on K X , we let τ U,Dǫ = H Dǫ τ U , where H Dǫ is the height function associated to the divisor D ǫ . We define the Tamagawa constant by We observe that by the results in §7.1, our Theorem 1.1 agrees with Conjecture 9.5, including the prediction for the constant.
9.4. Browning-Yamagishi's example. In [BY19, Theorem 1.2], Browning and Yamagishi presented an illuminating example, which illustrates in particular that in the formulation of Conjecture 9.5, it is important to exclude a thin set to obtain the expected growth rate. We briefly recall the construction. Define divisors on P 2 Q = Proj Q[x 0 , x 1 , x 2 ] by D i = {x i = 0} for i = 0, 1, 2, and D 3 = {x 0 + x 1 + x 2 = 0}, and let H denote the hyperplane class. Consider the Campana orbifold (P 2 Q , D ǫ = 1 2 3 i=0 D i ); and extend it to the obvious smooth proper model (P 2 Z , D ǫ ) over Spec(Z). A computation shows that a((P 2 , D ǫ ), H) = 1, b(Q, (P 2 , D ǫ ), H) = 1.
On the other hand, Browning and Yamagishi show that N((P 2 Z , D ǫ )(Z), H, T ) ≫ T log T, a computation at odds with a closed-set version of Conjecture 9.5. We show that a type II thin set already explains the unexpected rapid growth of the counting function. Let Q ⊂ P 3 = Proj Q[w 0 , w 1 , w 2 , w 3 ] be the smooth quadric defined by w 2 0 − w 2 1 + w 2 2 = w 2 3 and consider the finite morphism of degree 8 given by f : Q → P 2 Q (w 0 : w 1 : w 2 : w 3 ) → (w 2 0 : −w 2 1 : w 2 2 ) Note that f (Q(Q)) ⊂ (P 2 Z , D ǫ )(Z), and that, by the ramification formula, we have From this, it follows that a(Q, f * H) = 1, b(Q, Q, f * H) = 2.
Therefore the number of rational points on Q grows more quickly than the expected growth rate on (P 2 Z , D ǫ ). There are in fact infinitely many twists Q σ /P 2 Q such that a(Q σ , H) = 1, b(Q, Q σ , H) = 2, so it is a priori unclear whether the combined images of their rational points on P 2 Q form a thin set. This type of problem is already addressed in [LST18], using Hilbert's irreducibility theorem. We obtain the following auxiliary result: Lemma 9.7. The set Z = σ f σ (Q σ (Q)), where the union is taken over all σ ∈ H 1 (Gal(Q/Q), Aut(Q/P 2 Q )) with the property that b(Q, Q σ , f σ * H) = 2, is thin.
Hilbert's irreducibility theorem shows there is a thin set Z ′ ⊂ P 2 (Q) such that, for any p ∈ P 2 (Q) \ Z ′ , the fiber f −1 (p) is irreducible and the Galois group of its residue field is G. We claim that Z ⊂ Z ′ .
Suppose there exists a twist Q σ together with a point q ∈ Q σ (Q) such that p = f σ (q) ∈ Z ′ . We claim that this forces b(Q, Q σ , f σ * H) = 1. Indeed, since p ∈ Z ′ by assumption, the fiber f −1 (p) is irreducible over Q, and its residue field has Galois group G. Hence any σ ∈ H 1 (Gal(Q/Q), Aut(Q/P 2 Q )) such that the corresponding G-torsor (f σ ) −1 (p) → p is trivial induces a surjective homomorphism σ : Gal(Q/Q) → G. Let F/Q be a Galois extension such that Q × Q F ∼ = Q σ × Q F, and let σ be represented by a 1-cocyle Gal(F/Q) ∋ s → σ s ∈ G = Aut(Q F /P 2 F ). Then Q σ is the quotient of Q × Q F by Gal(F/Q), where s ∈ Gal(F/Q) acts on Q × Q F via the composition σ s • (id × s). In particular, the Galois action on N 1 (Q σ ) factors through G.
9.5. Birational invariance and functoriality. We conclude our investigation into Campana points by exploring functoriality properties of sets of Campana points under birational morphisms.
9.5.1. An instructive example (continued). To motivate our discussion, we appeal to the example of §2.3.1: we recall that X = P 2 Q with coordinates (x 0 : x 1 : x 2 ), D i = {x i = 0} for i ∈ {0, 1, 2}, and consider the Campana orbifold (X, 2 i=0 (1 − 1 m i )D i ) with Z-model X = P 2 Z . Let Y → X be the blow-up with center the intersection point of D 1 and D 2 . Denote by E the exceptional divisor and by D i the strict transform of D i for i ∈ {0, 1, 2}. The blow-up Y of X at the subvariety defined by {x 1 = x 2 = 0} yields a smooth proper Z-model of Y .
We observe that the sets of (weak) Campana points on (X , 2 i=0 (1− 1 m i )D i ) do not intersect the center of the blow-up and hence can be identified with subsets of Y (Q). In particular, a point P ∈ Y(Z) \ (E(Q) ∪ D 0 (Q) ∪ D 1 (Q) ∪ D 2 (Q)) is • a weak Campana Z-point on (X , 2 i=0 (1 − 1 m i )D i ) if for every prime p, the sum 1 m 0 n p ( D 0 , P ) + 1 m 1 n p ( D 1 , P ) + 1 m 2 n p ( D 2 , P ) + 1 m 1 + 1 m 2 n p (E, P ) is either 0 or at least 1; • a Campana Z-point on (X , 2 i=0 (1 − 1 m i )D i ) if for every prime p, the numbers 1 m 0 n p ( D 0 , P ), 1 m 1 (n p ( D 1 , P ) + n p (E, P )), 1 m 2 (n p ( D 2 , P ) + n p (E, P )) are either 0 or at least 1.
This description clearly shows that the set of (weak) Campana points is not invariant under birational morphisms, i.e., for a general choice of m 0 , m 1 , m 2 , there is no choice of Campana weights for a boundary on Y such that the blow-up would induce a bijection between the set of (weak) Campana points on Y and the set of (weak) Campana points on (X , 2 i=0 (1− 1 m i )D i ).
Not all is lost, however. Observe that if we assign multiplicities m i to D i for i ∈ {0, 1, 2} and max{m 1 , m 2 } to E, the set of (weak) Campana points on the resulting Campana orbifold on Y is mapped into a subset of the set of (weak) Campana points on (X , 2 i=0 (1 − 1 m i )D i ).
9.5.2. The general picture. Let X be a rationally connected smooth projective variety defined over a number field F and let D = α∈A D α be a strict normal crossings divisor on X. Fix a weight vector ǫ = (ǫ α ) α∈A where ǫ α ∈ W with ǫ α = 1 − 1/m α < 1. Set D ǫ = α∈A ǫ α D α and consider the Campana orbifold (X, D ǫ ), which is a klt pair. Let ϕ : X → X, be a birational morphism from a smooth projective variety X, such that D = (ϕ * D) red is a strict normal crossings divisor. We assume for simplicity that ϕ is an isomorphism outside of D and that both ( X, D) and (X, D) admit smooth, proper integral models ( X , D) and (X , D) that are compatible. We assign a vectorǫ for D as follows. For the strict transform of a component D α of D, we setǫ α = ǫ α . If E β is an exceptional divisor and if e β,α denotes the coefficient of E β in ϕ * D α , then we definẽ m β = max{⌈m α /e β,α ⌉ | e β,α > 0} andǫ β = 1 − 1/m β . We end by remarking that τ (F, S, ( X , D ǫ ), L) and τ (F, S, (X , D ǫ ), L) will be different in general because (X , D ǫ )(O F,S ) and ( X , Dǫ)(O F,S ) are different. Our overall conclusion is that Manin's conjecture for klt Campana points is quite sensitive to birational modifications. In particular, proving the asymptotic formula for the counting function after a birational modification need not easily yield an asymptotic formula for the original variety.