Campana points and powerful values of norm forms

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 V\'arilly-Alvarado.


Introduction
The theory of Campana points is of growing interest in arithmetic geometry due to its ability to interpolate between rational and integral points. Two competing notions of Campana points can be found in the literature, both extending a definition of "orbifold rational points" for curves within Campana's theory of "orbifoldes géométriques" in [7], [8], [9] and [10]. 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], [32], [6], [26] and [33], and the former four of these indicate the close relationship between Campana points and m-full solutions of equations. We recall that, given m ∈ Z ≥2 , we say that n ∈ Z \ {0} is m-full if all primes in the prime decomposition of n have multiplicity at least m.
In this paper, we bring together the perspectives in the above papers and provide the first result for Campana points on singular orbifolds. 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 [19] 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 Z(N ω ) is the zero locus of a norm form N ω 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. Although toric varieties also play a prominent role in [26], the tori there are split, whereas we shall work with anisotropic tori. 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. We compare the result for Campana points to a conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado [27,Conj. 1.1,p. 3]. d = 2, we must first generalise the definition of Campana points, which we do in Section 2.1. Using the same strategy employed in the proof of Theorem 1.1, we then derive an asymptotic for the number of Campana points on P d−1 K , ∆ ω m . Theorem 1.4. With the notation and hypotheses of Theorem 1.1, denote by N P Acknowledgements. Parts of this work were completed during the program "Reinventing Rational Points" at the Institut Henri Poincaré and the conference "Topics in Rational and Integral Points" at the Universität Basel; the author thanks the organisers of both events for their hospitality. Thanks go to the authors of [27] for their helpful feedback; to Daniel Loughran for his guidance as an expert and mentor; to Gregory Sankaran and David Bourqui for improvements arising from their examination of this work within the author's PhD thesis; to the anonymous referee for their thoughtful comments and suggestions, and to the Heilbronn Institute for Mathematical Research for their support.

Conventions.
Algebra. We take N = Z ≥1 . We denote by R * the group of units of a ring R. Given a group G, we denote by 1 G the identity element of G, and for any n ∈ N, we set G[n] = {g ∈ G : g n = 1 G }. For any perfect field F , we fix an algebraic closure F and set G F = Gal F /F . Given a topological group G, we denote by G ∧ = Hom(G, S 1 ) its group of continuous characters, where S 1 = {z ∈ C : zz = 1} ⊂ C * is the circle group. A monomial in the variables x 1 , . . . , x n is a product x a 1 1 · · · x an n , (a 1 , . . . , a n ) ∈ Z n ≥0 . For any n ∈ N, we denote by µ n the group of nth roots of unity and by S n the symmetric group of order n. Geometry. We denote by P n R the projective n-space over the ring R. We omit the subscript if the ring R is clear. Given a homogeneous polynomial f ∈ R[x 0 , . . . , x n ], we denote by Z(f ) = Proj R[x 0 , . . . , x n ]/(f ) the zero locus of f viewed as a closed subscheme of P n R . A variety over a field F is a geometrically integral separated scheme of finite type over F . Given a variety X defined over F and an extension E/F , we denote by X E = V × Spec F Spec E the base change of X over E, and we write X = X × Spec F Spec F . When F = K and E = K v for a number field K and a place v of K, we write X v = X Kv . Given a field F , we define G m,F = Spec F [x 0 , x 1 ]/(x 0 x 1 − 1). We omit the subscript F if the field is clear.
Number theory. Given an extension of number fields L/K with K-basis ω = {ω 0 , . . . , ω d−1 }, we write N ω (x 0 , . . . , x d−1 ) = N L/K (x 0 ω 0 + · · · + x d−1 ω d−1 ) for the associated norm form. We denote by Val(K) the set of valuations of a number field K, and we denote by S ∞ the set of archimedean valuations. For v ∈ Val(K), we denote by O v the maximal compact subgroup of K v . For a finite set of places S containing S ∞ , we denote by O K,S = {α ∈ K : α ∈ O v for all v ∈ S} the ring of algebraic S-integers of K. We write O K = O K,S∞ . For v ∈ Val(K) nonarchimedean, we denote by π v and q v a uniformiser for the residue field of K v and the size of the residue field of K v respectively. If v | ∞, then we set log q v = 1. For each v ∈ Val(K), we choose the absolute value |x| v = |N Kv/Qp (x)| p for the unique p ∈ Val(Q) with v | p and the usual absolute value | · | p on Q p . We denote We say that (X, D ǫ ) is smooth if X is smooth and D red has strict normal crossings (see [31, §41.21] for the definition of strict normal crossings divisors).
Let (X, D ǫ ) be a Campana orbifold over a number field K. Let S ⊂ Val(K) be a finite set containing S ∞ . Definition 2.2. A model of (X, D ǫ ) over O K,S is a pair (X , D ǫ ), where X is a flat proper model of X over O K,S (i.e. a flat proper O K,S -scheme with X (0) ∼ = X) and Let (X , D ǫ ) be a model for (X, D ǫ ) over O K,S . For v ∈ S, any P ∈ X(K) induces some P v ∈ X (O v ) by the valuative criterion of properness [14,Thm. II.4.7,p. 101]. Definition 2.3. Let P ∈ X(K) and take a place v ∈ S. For each α v ∈ A v , we define the local intersection multiplicity n v (D αv , P ) of D αv and P at v to be ∞ if P v ⊂ D αv , and the colength of the ideal P * v D αv ⊂ O v otherwise. We then define the quantities We are now ready to define weak Campana points and Campana points. Both notions arise from [1], with the former appearing in its current form in [2, §1].
Definition 2.4. We say that P ∈ X(K) is a weak Campana O K,S -point of (X , D ǫ ) if the following implications hold for all places v ∈ S of K and for all α ∈ A: (1) If ǫ α = 1 (meaning m α = ∞), then n v (D α , P ) = 0.
(2) If n v (D ǫ , P ) > 0, then We denote the set of weak Campana O K,S -points of (X , D ǫ ) by (X , D ǫ ) w (O K,S ).
Definition 2.5. We say that P ∈ X(K) is a Campana O K,S -point of (X , D ǫ ) if the following implications hold for all places v ∈ S of K and for all α ∈ A.
Remark 2.6. Informally, weak Campana points are rational points P ∈ X(K) avoiding ∪ ǫα=1 D α which, upon reduction modulo any place v ∈ S, either do not lie on D red or lie on D α with multiplicity at least m α on average over α. Similarly, Campana points are rational points P ∈ X(K) avoiding ∪ ǫα=1 D α which, upon reduction modulo any place v ∈ S, either do not lie on D red or lie on each v-adic irreducible component of each D α with multiplicity either 0 or at least m α .
Remark 2.7. Our definition of Campana points differs from the one in [27, §3.2], in which one requires simply that n v (D α , P ) ≥ m α instead of n v (D αv , P ) ≥ m α in the second implication. If one were to apply this definition to the orbifold P d−1 K , ∆ ω m of Theorem 1.1, which is singular for all d ≥ 3 as Z(N ω ) is not a strict normal crossings divisor in this case, then the weak Campana points and the Campana points would be the same, but the asymptotic of Theorem 1.1 differs to [27, Conj. 1.1, p. 3] for d ≥ 3 (at least if one takes the thin set there to be the empty set). Using the definitions above, we obtain the asymptotic for Campana points in Theorem 1.4, whose exponents match this conjecture.
Lemma 2.8. Let (X, D ǫ ) be a smooth Campana orbifold over a number field K which is Kawamata log terminal (i.e. ǫ α < 1 for all α ∈ A), and let (X , D ǫ ) be a model of (X, D ǫ ) over O K,S with X smooth over O K,S and D red a relative strict normal crossings divisor in X /O K,S as defined in [17, §2]. Then the definition of Campana points on (X , D ǫ ) above coincides with the one in [27, §3.2].
Proof. Since D red is a relative strict normal crossings divisor, each irreducible component D α is smooth over O K,S . In particular, its base change over Spec O v is smooth for any v ∈ S, so the divisors D αv , α v | α are disjoint. Then, for any rational point P ∈ X(K), the reduction of P at the place v can lie on at most one of the divisors D αv , α v | α, so n v (D α , P ) = αv |α n v (D αv , P ) is either 0 or at least m α if and only if each n v (D αv , P ), α v | α, is either 0 or at least m α .

Toric varieties.
Definition 2.9. An (algebraic) torus over a field F is an algebraic group T over F such that T ∼ = G n m for some n ∈ N. The splitting field of a torus T over a field F is defined to be the smallest Galois field extension E of F for which T E ∼ = G n m . Definition 2.10. A toric variety is a smooth projective variety X equipped with a faithful action of an algebraic torus T such that there is an open dense orbit containing a rational point. Definition 2.11. Let T be a torus over a field F . The character group of T is X * T = Hom T , G m , and we have X * (T ) = X * T G F . The cocharacter group of T is X * T = Hom X * T , Z , and we have X * (T ) = X * T G F . We let 12. An algebraic torus T over a field F is anisotropic if it has trivial character group over F , i.e. X * (T ) = 0.
Let T be a torus over a number field K with splitting field E.
For v ∤ ∞ with ramification degree e v in E/K, define the maps Finally, define the maps (i) If v is non-archimedean, then we have the exact sequence The image of f is open and of finite index. Further, if v is unramified in E, then f is surjective. (ii) If v is archimedean, then we have the short exact sequence Further, f admits a canonical section. (iii) Letting g be either deg T or deg T,E and denoting its kernel by T (A K ) 1 , we have the split short exact sequence hence we have an isomorphism Definition 2.15. Let χ be a character of T (A K ). We say that χ is automorphic if it is trivial on T (K). We say that χ is unramified at v ∈ Val(K) if χ v is trivial on T (O v ), and we say that it is unramified if it is unramified at every v ∈ Val(K).
The canonical sections of the maps 14(ii) induce a canonical section of the composition which in turn induces a "type at infinity map" (2.2) which has finite kernel and image a lattice of codimension rank X * (T ) (see [4,Lem. 4.52, p. 96]).
Note 2.16. When T is anisotropic, we have T (A K ) 1 = T (A K ) by Lemma 2.14(iii), and then we see from the above that there is a map with finite kernel and image a lattice of full rank.

Hecke characters.
Definition 2.17. A Hecke character for K is an automorphic character of G m,K .
Definition 2.19. The (Hecke) L-function L(χ, s) of a Hecke character χ is where the product is taken over all places v ∤ ∞ at which χ is unramified. The Dedekind zeta function of K is Given a Hecke character χ for a number field L and w ∈ Val(L), we denote by L w (χ, s) the local factor at w for the Euler product of L(χ, s), i.e.
if w ∤ ∞ and χ is unramified at w, 1 otherwise.
When working over the field L ⊃ K, we define L v (χ, s) for each v ∈ Val(K) by Theorem 2.20. [15, §6] The L-function of a Hecke character χ admits a meromorphic continuation to C. If χ = · iθ for some θ ∈ R, then this continuation admits a single pole of order 1 at s = 1 + iθ. Otherwise, it is holomorphic.
Definition 2.23. Let E/K be Galois, let χ be a Hecke character for E and let g ∈ Gal(E/K). We define the (Galois) twist of χ by g to be the character Here, gw denotes the place of E obtained by the action of g on Val(E), and g w : E w → E gw is the induced map on completions. One may easily verify that χ g is trivial on E * , hence it is also a Hecke character for E.

The norm torus
In this section, we fix an extension of number fields L/K of degree d ≥ 2 with Galois closure E and a K-basis ω = {ω 0 , . . . , ω d−1 }. We write N ω (x 0 , . . . , x d−1 ) for the norm form corresponding to ω, and G = Gal(E/K). From the equality we see that N ω is irreducible over K and has splitting field E. We denote by T the norm torus T ω = P d−1 K \ Z(N ω ). As noted in [19, §1.2], P d−1 K is a toric variety with respect to T , and T ∼ = R L/K G m /G m is anisotropic. Since its boundary is Z(N ω ), its splitting field is E. We have the short exact sequence , we see that, if χ is unramified at v, then it is unramified as a Hecke character at all w | v. In particular, if χ is unramified at v and v is unramified in L/K, then w|v χ w (π w ) = 1, since π v is a uniformiser for L w for each w | v.

3.1.
Geometry. In this section we study fan-theoretic objects related to T . We begin by describing the fan Σ ⊂ X * T R associated to the equivariant compactification P d−1 K of T and the associated piecewise-linear function ϕ Σ (see [3, §1.2]) used to define the Batyrev-Tschinkel height function.
Denoting by l 0 (x), . . . , By [16, §1.1], the fan associated to P d−1 E as a compactification of G d−1 m,E is the fan whose r-dimensional cones are generated by the r-fold subsets of and define Σ to be the fan whose r-dimensional cones are generated by the r-fold subsets of It follows that Σ is the fan associated to P d−1 E as a compactification of T E . Also, we see that d−1 i=0 e i = 0 and that {e 1 , . . . , e d−1 } is the dual of the basis We now show that the action of G on Σ(1) is compatible with its action on the E-linear factors of N ω . Denote by * the action of G, and set l g(i) = g * l i . Proof. Let g ∈ G. It suffices to show that where δ ij is the Kronecker delta symbol, defined by It follows from the above that G acts transitively on Σ(1) = { e 0 , . . . , e d−1 }. For v ∈ Val(K) non-archimedean, let G v denote the associated decomposition subgroup of G. By the proof of [3, Thm. 3.1.3, p. 619], the G v -orbits of Σ(1) are in bijection with the places of L over v, and the length of the G v -orbit corresponding to a place w | v is its inertia degree.
We first show that {n w : w | v} spans X * T Gv . Given g ∈ G and σ = d−1 i=0 a i e i , we have g * σ = d−1 i=0 a g −1 (i) e i , so g * σ = σ if and only if there exists r g ∈ Z such that a i = a g −1 (i) + r g for all i ∈ {0, . . . , d − 1}. Setting s = #G, we have a i = a g s (i) = a g s−1 (i) + r g = · · · = a i + sr g , hence r g = 0. We deduce that σ ∈ Σ Gv if and only if a i = a j for all e i , e j in the same G v -orbit, so the result follows. Moreover, we observe that w|v a w n w = w|v b w n w if and only if there exists r ∈ Z such that b w = a w + r for all w | v, since there is a unique expression for σ ∈ X * T in the form

By (3.4), we have
Then, setting d w = deg f w , we see that On the other hand, we have ). Combining (3.5) and (3.6), we obtain Since w|v n w = i e i = 0, we conclude that We now study polynomials introduced by Batyrev and Tschinkel in [3, §2.2], which play a key role in the analysis of local Fourier transforms in Section 5.
and we define the polynomial Q Σ,v (u 1 , . . . , u l ) by Proposition 3.6. For all non-archimedean valuations v ∈ Val(K), we have Proof. Observe that the G v -invariant cones in Σ are precisely those cones generated by a set of 1-dimensional cones of the form Σ i 1 (1)∪· · ·∪Σ i k (1) for some i 1 , . . . , i k ∈ {1, . . . , l} pairwise distinct with k < l. From this observation, we deduce that In particular, we see that so it suffices to prove that Splitting the sum into two smaller sums for t 1 = 0 and t 1 = 1, we obtain Repeating this process for each variable t 2 , . . . , t l , we deduce (3.7).
Here, L v X * T , s is the local factor at v of the Artin L-function L X * T , s .
the product of the µ v converges to give a Haar measure µ on T (A K ), which is independent of ω by the product formula.
By Since ζ K (s) and ζ L (s) both have a simple pole at s = 1, we have L X * T , 1 =

Heights and indicator functions
In this section we define functions which allow us to use harmonic analysis to study weak Campana points. Let L/K be an extension of number fields with . . , d−1} and g ∈ G. Otherwise, we define S(ω) to be the minimal subset of Val(K) containing S ∞ such that N ω is an irreducible polynomial over O K,S(ω) .
Remark 4.2. When L = E, by (3.1), the S(ω)-integrality of the a ij k and b g k implies that N ω is defined over O K,S(ω) , while the S(ω)-integrality of the c k implies that the coefficients of N ω are not all divisible by some From now on, we fix the model In both the Galois and non-Galois cases, the conditions on S(ω) ensure that we may take the "obvious" model above. The potentially stronger conditions in the Galois case (in which we obtain our results) ensure compatibility between intersection multiplicity and toric multiplication, as we shall see in Section 4.2.
Remark 4.4. When L = E and ω is a relative integral basis, we get S(ω) = S ∞ , since every algebraic integer is expressible as an O K -linear combination of elements of a relative integral basis, and O L is closed under multiplication and conjugation.
We then define the global height function Definition 4.6. For each place v ∈ S(ω), define the function It is well-known (see [12, §2.2.3]) that two height functions corresponding to adelic metrisations of the same line bundle are equal over all but finitely many places. Definition 4.9. We define the finite set Setting φ m,v = 1 for v ∈ S(ω), we then define the global indicator function Remark 4.11. Let v ∈ S(ω) be a non-archimedean place of K. Since H ′ v is continuous with discrete image in R >0 , its level sets are clopen. It follows that φ m,v is continuous for all v ∈ Val(K). Also, since φ m,v (T (O v )) = 1 for all v ∈ S ′ (ω) by Lemma 2.14(i), we see that φ m is well-defined and continuous on T (A K ). Proof. Take v ∈ S(ω), and let t 0 , . . . , t d−1 be a set of O v -coordinates for t ∈ T (K) with at least one t i ∈ O * v . Then we have

Invariant subgroups.
For this section, let L = E be Galois over K. Lemma 4.13. For all v ∈ S(ω) and x, y ∈ T (K v ), we have Proof. Choose sets of projective coordinates {x 0 , . . . , x d−1 } and {y 0 , . . . , y d−1 } for x and y respectively. Note that where, for a ij k ∈ O v as in Definition 4.1, we have Using N ω (x · y) = N ω (x)N ω (y) and the strong triangle inequality, we deduce that Lemma 4.14. For any place v ∈ S(ω), the level set

Proof. From Proposition 3.4 and Lemma 2.14(i), it is clear that H
. It is also clear that H ′ v (1) = 1, and closure under multiplication follows from Lemma 4.13 and Remark 4.7. It only remains to verify that x ∈ K v implies x −1 ∈ K v . Let x ∈ K v , and choose coordinates x 0 , . . . , Recursively applying Lemma 4.13, we obtain By Remark 4.7, it suffices to show that, for any g ∈ G, we have H ′ v (g(x)) = 1. Since N ω (g(x)) = N ω (x), it suffices by Remark 4.7 to show that max{|g(x) i | v } ≤ 1. This follows from the fact that b g k ∈ O v since v ∈ S(ω), see Definition 4.1. Corollary 4.15. For every place v ∈ S(ω), the function H ′ v is K v -invariant. Proof. Take x ∈ K v , and let y ∈ T (K v ). Then by Lemma 4.13, we have , while on the other hand, since x −1 ∈ K v by Lemma 4.14, we have by Lemma 2.14(i) and the equality for all s ∈ C for which the integral exists. We then define the global Fourier transform of χ with respect to f to be

Weak Campana points
In this section we prove Theorem 1.1. Fix an extension of number fields L/K of degree d with K-basis ω, set T = T ω as in Section 3 and let m ∈ Z ≥2 . 5.1. Strategy. Following [19] and [3], we will apply a Tauberian theorem Proof. Let Re s ≥ ǫ. Since Now, by Lemma 2.14(i), T (K v )/T (O v ) can be identified with a sublattice of finite index in X * (T v ), and this sublattice coincides with X * (T v ) when v is unramified in L/K. Then we see that, interpreting H v as a function on X * (T v ), we have , so we deduce that H v (1, 1; −ǫ) is convergent, and this concludes the proof. Proof. Since φ m,v and H v are K v -invariant, the result follows by character orthogonality. Lemma 5.5. Let v ∤ ∞ be a non-archimedean place of K unramified in L/K, and let χ be an automorphic character of T which is unramified at v. Then we have Proof. The result follows from [3, Thm. 2.2.6, p. 611] and Proposition 3.6.
Then, for Re s > 0, we have Proof. Let s ∈ C with Re s > 0. As χ ∈ U and v ∈ S ′ (ω), it follows that χ is unramified at v. Then, expanding geometric series, we have so, by Lemma 5.5, we obtain On the other hand, we may write so, comparing these expressions, we see for n ≥ 1 that Since v ∈ S ′ (ω), we have φ m,v (t v ) = 1 if and only if H v (t v ) = 1 or H v (t v ) ≥ q m v , so the result follows.

5.3.
Regularisation. Now that we have expressions for the local Fourier transforms at all but finitely many places, our goal is to find "regularisations" for the global Fourier transforms, i.e. functions expressible as Euler products whose convergence is well-understood and whose local factors approximate the local Fourier transforms well (as expansions in q v ) at all but finitely many places. As in [3], [19] and [27], we will construct our regularisations from L-functions.
Next we prove (ii). Note that for (g 1 , g 2 , . . . , g m ) ∈ S(G, m), we may take g 1 = 1 G without loss of generality. Suppose that for some i, j ∈ {1, . . . , d}. This is equivalent to the equality of multisets  m-tuple (1 G , h 2 , . . . , h m ) is a permutation of (1 G , g 2 , . . . , g m ). If i = j, we may take g 2 (i) = j without loss of generality (note that g 2 = 1 G in this case), and we obtain the equality of multisets Once again, by freeness of the action of G on {1, . . . , d}, we get that {1 G , g 2 , . . . , g m } = {g 2 , h 2 g 2 , . . . , h m g 2 } as multisets, from which both claims follow.
Before proceeding to the proofs of (iii) and (iv), we make the following observation: if a multiset M = {g 1 , . . . , g m } of elements of G is closed under right multiplication by some g ∈ G of order n ≥ 2, then n divides m. Indeed, M must contain the n distinct elements g 1 , g 1 g, g 1 g 2 , . . . , g 1 g n−1 . Without loss of generality, we may take g l = g 1 g l−1 for l = 1, . . . , n. Since {g 1 , . . . , g n } = {g 1 , g 1 g, . . . , g 1 g n−1 } is clearly closed under right multiplication by g, so is the sub-multiset {g n+1 , . . . , g m }.
Iterating this process, we deduce that n divides the cardinality of M, as otherwise we would eventually obtain a non-empty sub-multiset closed under right multiplication by g with fewer than n elements, which is clearly impossible.
We now prove (iii) and (iv). By (i) and (ii), every degree-m monomial in x 1 , . . . , x d appears in a unique summand of (g 1 ,...,gm)∈S(G,m) φ (g 1 ,...,gm) (x 1 , . . . , x d ), and a monomial appears twice in φ (g 1 ,...,gm) (x 1 , . . . , x d ) if and only if the multiset {g 1 , . . . , g m } is closed under right multiplication by some g r = 1 G . By the above, the latter is possible only if the order of g r divides m, while by Lagrange's theorem, the order of g r divides d. If d is coprime to m, then we deduce that no such g r exists, thus we obtain (iii). If d = p is prime and m = kp, then g r is necessarily a generator of the cyclic group G. It then follows as in the above observation that (g 1 , . . . , g kp ) = 1 G , g r , . . . , g p−1 r , . . . , 1 G , g r , . . . , g p−1 r .
Letting g be a generator of G, we see that φ (1 G ,g,...,g p−1 ,...,1 G ,g,...,g p−1 ) (x 1 , . . . , x p ) = px k 1 · · · x k p is the only one of the polynomials φ (g1,...,gkp) (x 1 , . . . , x p ), (g 1 , . . . , g kp ) ∈ S(G, kp) in which a monomial appears more than once, and so we obtain (iv). For the rest of this section, let L = E be Galois over K with Galois group G, and assume that m is coprime to d if d is not prime.
Lemma 5.11. Let v ∈ S ′ (ω) be a non-archimedean place which is totally split in E/K and let χ ∈ U. Then (g 1 ,...,gm)∈S(G,m) L v (χ g 1 · · · χ gm , ms)  Then we claim that Since G acts freely and transitively on the places w 1 , . . . , w d of E over v, we have Note that the term 1 and define For any non-archimedean place v ∤ ∞, write for the expansion of F m,χ,v (s) as a multidimensional geometric series in q ms v , so where d χ,v,n is defined for all n ≥ 0 by the iterative formula In particular, we have d χ,v,n = a χ,v,n for 0 ≤ n ≤ m − 1.
Since E/K is Galois, all of the places w 1 , . . . , w r of E over v share a common inertia degree d v . Since χ v (T (O v )) = 1, it is unramified as a Hecke character at all of the w i (see Note 3.1), and for any g 1 , . . . , g m ∈ G, so is χ g 1 · · · χ gm . Then First, suppose that v is totally split in E/K. Then (5.5) gives Since G acts freely and transitively on the w i , it follows from Proposition 5.
where G m,χ (s) is holomorphic and uniformly bounded with respect to χ for Re s ≥ We will prove the stronger result that G m,χ (s) is holomorphic on Re s > 1 m+1 and uniformly bounded with respect to both χ and ǫ on Re s ≥ 1 m+1 + ǫ for all ǫ > 0. For a place v ∤ ∞ and s ∈ C with Re s = σ ≥ ǫ for some ǫ > 0, we have so, by Lemma 5.1, the series ∞ L v (χ g 1 · · · χ gm , ms , hence G m,χ,v (s) is holomorphic on Re s > 0 and is bounded uniformly in terms of ǫ and v on Re s ≥ ǫ.
To conclude the result, it suffices to prove that there exists N ∈ N such that qv>N G m,χ,v (s) is holomorphic and uniformly bounded with respect to χ on Re s ≥ 1 m+1 + ǫ for all ǫ > 0. Let v ∈ S ′ (ω) be non-archimedean, and let Re s = σ ≥ 1 m+1 + ǫ. From and the definition of F m,χ,v (s), we have Then, by (5.4), it follows inductively that we have σ so that, for all places v ∤ ∞ with q v > N, we have v ∈ S ′ (ω). Now, any normally convergent infinite product is holomorphic (see [28, §2]  Note 5.16. In constructing the regularisation F m,χ (s), one must ensure that for all non-archimedean places v with q v is sufficiently large. As seen above, the restrictions on d, m and E ensure that this is automatic for all such places which are not totally split, i.e. we only need to approximate the local Fourier transform at totally split places not in S ′ (ω). Without these restrictions, one might have to approximate the local Fourier transform at places of more than one splitting type simultaneously, and to do this would require a new approach.
Before applying our key theorems, we give one more result, which will be used in order to move from the Poisson summation formula to the Tauberian theorem. is absolutely and uniformly convergent on C.
Then Ω m (s) admits an extension to a holomorphic function on Re s ≥ 1 m .
Proof. Let s ∈ C with Re s > 1 m . Combining the formal application (5.1) of the Poisson summation formula with Lemma 3.8 and Corollary 5.4 gives By Corollary 5.15, the function s → φm(t) H(t) s is L 1 for Re s > 1 m . To show that this application is valid, we apply Bourqui's criterion [4,Cor. 3.36, p. 64], by which it suffices to show that the right-hand side of (5.7) is absolutely convergent, s → φm(t) H(t) s is continuous and there exists an open neighbourhood U ⊂ T (A K ) of the origin and strictly positive constants C 1 and C 2 such that for all u ∈ U and all t ∈ T (A K ), we have We may take U = K by Lemma 4.16, and continuity is clear. It only remains to prove the absolute convergence. We will prove the stronger result that is absolutely and uniformly convergent on any compact subset C of the half-plane Re s ≥ 1 m , which will both verify validity of the application and prove the theorem. Since K ⊂ K T is of finite index, the map (2.2) yields a homomorphism where the inner sum is finite. Then, for s ∈ C, we have . Now, for χ ∈ U, we deduce from the proof of Corollary 5.15 that .
In order to deduce the result from Lemma 5.17, it suffices to prove that, for each ψ ∈ L and some constant 0 for · as in Definition 2.21. As K ⊂ K T is of finite index, there exists a constant Q > 0 such that q(χ) < Q for all χ ∈ U (cf. [   (g, 1, . . . , 1) for any g ∈ G, we obtain χ m−1 χ g = 1, so χ m = 1 and χ = χ g for all g ∈ G. Conversely, if χ m = 1 and χ = χ g for all g ∈ G, then (5.8) holds. Let now d = 2, χ ∈ U 0 , v ∈ S ′ (ω) and w | v. We have ψ w (π w ) = ψ g w (π w ) = ψ gw (π gw ) for all g ∈ G. Since w|v ψ w (π w ) = 1 (see Note 3.1) and G acts transitively on the places of E over v, we obtain ψ d w (π w ) = 1, hence χ d = 1 by strong approximation. On the other hand, χ m = 1. For d and m coprime, we conclude that χ = 1. is non-zero.
Proof. We have Let t ∈ T (A K ). Note that, if there exists χ ′ ∈ U 0 with χ ′ (t) = 1, then χ∈U 0 so χ∈U 0 χ(t) = 0. Then we have For any χ ∈ U 0 and non-archimedean place v ∈ S ′ (ω), comparing the series expressions of H v (φ m,v , χ v ; −s) and F m,χ,v (s) = F m,1,v (s) in Definition 5.13, we see that Letting θ m,v be the indicator function of K v for v ∈ S ′ (ω), we define the function By the above, we deduce that Then, by comparing limits along the real line, we see that it suffices to prove that It is easily seen that for any non-archimedean place v ∈ S ′ (ω), we have The result for S ⊃ S(ω) follows analogously upon redefining φ m,v to be identically 1 for each v ∈ S \ S(ω) in Definition 4.10.

Campana points
In this section we prove Theorem 1.4. We will be brief when the argument is largely similar to the case of weak Campana points, emphasising only the key differences. Fix a Galois extension E/K of number fields with K-basis ω = {ω 0 , . . . , ω d−1 }, let m ∈ Z ≥2 and set T = T ω as in Section 3. Definition 6.1. For each non-archimedean place v ∈ S(ω), let denote the v-adic decomposition of the norm form N ω associated to ω into irre- . For each w | v, we define the functions We then define the Campana local indicator function Setting ψ m,v = 1 for v ∈ S(ω), we then define the Campana indicator function If v ∈ S ′ (ω), then for each w | v, we also define the function Letting σ m,v be the indicator function for K v for v ∈ S ′ (ω), we define the function Lemma 6.2. The Campana O K,S(ω) -points of P d−1 K , ∆ ω m are precisely the rational points t ∈ T (K) such that ψ m (t) = 1.
Proof. Taking coordinates t 0 , . . . , t d−1 as in the proof of Lemma 4.12, we have for all non-archimedean places v ∈ S(ω) and places w | v, where Z(f w ) denotes the Zariski closure of Z(f w ) in P d−1 O K,S(ω) . Since |ψ m,v (n v )χ v (n v )| ≤ 1, we deduce that Analogously to the proof of Corollary 5.15, we deduce for q v sufficiently large that Proposition 6.6. For all places v ∈ S ′ (ω) with q v sufficiently large, we have Proof. Let v ∈ S ′ (ω) with q v sufficiently large as in Corollary 6.5. If v is totally split in E/K, then deg f w = 1 for all w | v, so Corollary 6.5 gives and so we deduce the equality, since Proposition 6.7. For any χ ∈ U, we have H(ψ m , χ; −s).
The result for S ⊃ S(ω) follows analogously upon redefining ψ m,v to be identically 1 for each v ∈ S \ S(ω).

Comparison to Manin-type conjecture
In this section we compare the leading constant in Theorem 1.4 with the Manin-Peyre constant in the conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.
7.1. Statement of the conjecture. Let (X, D ǫ ) be a smooth Campana orbifold over a number field K which is klt (i.e. ǫ α < 1 for all α ∈ A) and Fano (i.e. −(K X + D ǫ ) is ample). Let (X , D ǫ ) be a regular O K,S -model of (X, D ǫ ) for some finite set S ⊂ Val(K) containing S ∞ (i.e. X regular over O K,S ). Let L = (L, · ) be an adelically metrised big line bundle with associated height function H L : X(K) → R >0 (see [25, §1.3]). For any subset U ⊂ X(K) and any B ∈ R >0 , we define N(U, L, B) = #{P ∈ U : H L (P ) ≤ B}.
Definition 7.1. Let V be a variety over a field k of characteristic zero, and let A ⊂ V (k). We say that A is of type I if there is a proper Zariski closed subset W ⊂ V with A ⊂ W (k). We say that A is of type II if there is a normal geometrically irreducible variety V ′ with dimV ′ = dimV and a finite surjective morphism φ : V ′ → V of degree ≥ 2 with A ⊂ φ(V ′ (k)). We say that A is thin if it is contained in a finite union of subsets of V (k) of types I and II.