Abstract
We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with m-full norm over a given Galois extension of \(\mathbb {Q}\). We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.
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1 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 [7,8,9,10]. 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, 6, 26, 32, 33], and the former four of these indicate the close relationship between Campana points and m-full solutions of equations. We recall that, given \(m \in \mathbb {Z}_{\ge 2}\), we say that \(n \in \mathbb {Z} \setminus \{0\}\) 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, Sect. 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 \(\left( \mathbb {P}^{d-1}_K,\left( 1-\frac{1}{m}\right) Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \right) \), where \(Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \) is the zero locus of a norm form \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) associated to a K-basis \({{\,\mathrm{\varvec{\omega }}\,}}\) of a Galois extension of number fields E/K of degree \(d \ge 2\) coprime to \(m \in \mathbb {Z}_{\ge 2}\) 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 \(K = \mathbb {Q}\), 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 \(\mathbb {Q}\). We compare the result for Campana points to a conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado [27, Conj. 1.1, p. 3].
1.1 Results
Theorem 1.1
Let E/K be a Galois extension of number fields of degree \(d \ge 2\), and let \(m \ge 2\) be an integer which is coprime to d if d is not prime. Let \({{\,\mathrm{\varvec{\omega }}\,}}\) be a K-basis of E. Denote by \(\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\) the \(\mathbb {Q}\)-divisor \(\left( 1-\frac{1}{m}\right) Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \) of \(\mathbb {P}^{d-1}_K\) for \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) the norm form corresponding to \({{\,\mathrm{\varvec{\omega }}\,}}\). Let H denote the anticanonical height function on \(\mathbb {P}^{d-1}_K\) from Definition 4.5. Then there exists an explicit finite set \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \subset {{\,\mathrm{Val}\,}}\left( K\right) \) such that, for any finite set of places \(S \supset S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), the number \(N\left( \left( \mathbb {P}^{d-1}_{K},\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) ,H,B,S\right) \) of weak Campana \(\mathcal {O}_{K,S}\)-points of height at most \(B \in \mathbb {R}_{\ge 1}\) on the orbifold \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) with respect to the model \(\mathbb {P}^{d-1}_{\mathcal {O}_{K,S}}\) of \(\mathbb {P}^{d-1}_K\) has the asymptotic formula
as \(B \rightarrow \infty \) for some explicit positive constant \(c\left( {{\,\mathrm{\varvec{\omega }}\,}},m,S\right) \), where
Note 1.2
If \({{\,\mathrm{\varvec{\omega }}\,}}\) is a relative integral basis of E/K, then \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) = S_{\infty }\), the set of archimedean places of K, in Theorem 1.1 (see Remark 4.4).
Each rational point \(P \in \mathbb {P}^{d-1}\left( \mathbb {Q}\right) \) possesses precisely two sets of coordinates in \(\mathbb {Z}^d_{\text {prim}} = \{\left( x_0,\dots ,x_{d-1}\right) \in \mathbb {Z}^d : \gcd \left( x_0,\dots ,x_{d-1}\right) = 1\}\). Interpreting H and \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) as functions on this set, we immediately obtain the following result.
Corollary 1.3
Taking \(K=\mathbb {Q}\) and letting \({{\,\mathrm{\varvec{\omega }}\,}}\) be an integral basis with the notation and hypotheses of Theorem 1.1, we have
Arithmetically special (e.g. prime, square-free) values of norm forms are a topic of long-standing interest in number theory (see e.g. [13, 20]).
Campana points are only defined and studied for smooth orbifolds (i.e. smooth varieties for which the orbifold divisor has strict normal crossings support) in [27]. In order to study the Campana points of \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \), which is smooth only when \(d = 2\), we must first generalise the definition of Campana points, which we do in Sect. 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 \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \).
Theorem 1.4
With the notation and hypotheses of Theorem 1.1, denote by \(\widetilde{N}\Big (\big (\mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\big ),H,B,S\Big )\) the number of Campana \(\mathcal {O}_{K,S}\)-points on \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) of height at most \(B \in \mathbb {R}_{\ge 1}\) with respect to H. Then there exists an explicit positive constant \(\widetilde{c}\left( {{\,\mathrm{\varvec{\omega }}\,}},m,S\right) \) such that, as \(B \rightarrow \infty \), we have
Remark 1.5
It is not clear if the exponent of the logarithm in Theorem 1.1 admits a geometric interpretation as it does in Theorem 1.4 (cf. [27, Conj. 1.1, p. 3]).
1.2 Conventions
1.2.1 Algebra
We take \(\mathbb {N} = \mathbb {Z}_{\ge 1}\). 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 \(n \in \mathbb {N}\), we set \(G[n] = \{g \in G: g^n = 1_G\}\). For any perfect field F, we fix an algebraic closure \(\overline{F}\) and set \(G_F = {{\,\mathrm{Gal}\,}}\left( \overline{F}/F\right) \). Given a topological group G, we denote by \(G^{\wedge } = {{\,\mathrm{Hom}\,}}\left( G,S^1\right) \) its group of continuous characters, where \(S^1 = \{z \in \mathbb {C}: z \overline{z} = 1\} \subset \mathbb {C}^*\) is the circle group. A monomial in the variables \(x_1,\dots ,x_n\) is a product \(x_1^{a_1}\cdots x_n^{a_n}\), \(\left( a_1,\dots ,a_n\right) \in \mathbb {Z}_{\ge 0}^n\). For any \(n \in \mathbb {N}\), we denote by \(\mu _n\) the group of nth roots of unity and by \(S_n\) the symmetric group of order n.
1.2.2 Geometry
We denote by \(\mathbb {P}_R^n\) the projective n-space over the ring R. We omit the subscript if the ring R is clear. Given a homogeneous polynomial \(f \in R[x_0,\dots ,x_n]\), we denote by \(Z\left( f\right) = {{\,\mathrm{Proj}\,}}R[x_0,\dots ,x_n]/\left( f\right) \) the zero locus of f viewed as a closed subscheme of \(\mathbb {P}_R^n\). 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 \(X_E= V \times _{{{\,\mathrm{Spec}\,}}F} {{\,\mathrm{Spec}\,}}E\) the base change of X over E, and we write \(\overline{X} = X \times _{{{\,\mathrm{Spec}\,}}F} {{\,\mathrm{Spec}\,}}\overline{F}\). When \(F = K\) and \(E = K_v\) for a number field K and a place v of K, we write \(X_{v} = X_{K_v}\). Given a field F, we define \(\mathbb {G}_{m,F} = {{\,\mathrm{Spec}\,}}F[x_0,x_1]/\left( x_0 x_1-1\right) \). We omit the subscript F if the field is clear.
1.2.3 Number theory
Given an extension of number fields L/K with K-basis \({{\,\mathrm{\varvec{\omega }}\,}}= \{\omega _0,\dots ,\omega _{d-1}\}\), we write \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x_0,\dots ,x_{d-1}\right) = N_{L/K}\left( x_0 \omega _0 + \cdots + x_{d-1}\omega _{d-1}\right) \) for the associated norm form. We denote by \({{\,\mathrm{Val}\,}}\left( K\right) \) the set of valuations of a number field K, and we denote by \(S_{\infty }\) the set of archimedean valuations. For \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), we denote by \(\mathcal {O}_v\) the maximal compact subgroup of \(K_v\). For a finite set of places S containing \(S_{\infty }\), we denote by \(\mathcal {O}_{K,S} = \{\alpha \in K : \alpha \in \mathcal {O}_v \text { for all }v \not \in S\}\) the ring of algebraic S-integers of K. We write \(\mathcal {O}_K = \mathcal {O}_{K,S_{\infty }}\). For \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) non-archimedean, we denote by \(\pi _v\) and \(q_v\) a uniformiser for the residue field of \(K_v\) and the size of the residue field of \(K_v\) respectively. If \(v \mid \infty \), then we set \(\log q_v = 1\). For each \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), we choose the absolute value \(|x|_v = |N_{K_v/\mathbb {Q}_p}\left( x\right) |_p\) for the unique \(p \in {{\,\mathrm{Val}\,}}\left( \mathbb {Q}\right) \) with \(v \mid p\) and the usual absolute value \(|\cdot |_p\) on \(\mathbb {Q}_p\). We denote by \(\mathbb {A}_K =\widehat{\prod }_{v \in {{\,\mathrm{Val}\,}}\left( K\right) }^{\mathcal {O}_v}K_v\) the adele ring of K with the restricted product topology.
2 Background
2.1 Campana points
In this section we define Campana orbifolds, Campana points and weak Campana points, generalising the definitions in [27, Sect. 3.2] in such a way that the exponents in Theorem 1.4 match those in [27, Conj. 1.1, p. 3].
Definition 2.1
A Campana orbifold over a field F is a pair \(\left( X,D_{\epsilon }\right) \) consisting of a proper, normal variety X over F and an effective Cartier \(\mathbb {Q}\)-divisor
on X, where the \(D_{\alpha }\) are prime divisors and \(\epsilon _{\alpha } = 1 - \frac{1}{m_{\alpha }}\) for some \(m_{\alpha } \in \mathbb {Z}_{\ge 2} \cup \{\infty \}\) (by convention, we take \(\frac{1}{\infty } = 0\)). We define the support of the \(\mathbb {Q}\)-divisor \(D_{\epsilon }\) to be
We say that \(\left( X,D_{\epsilon }\right) \) is smooth if X is smooth and \(D_{\text {red}}\) has strict normal crossings (see [31, Sect. 41.21] for the definition of strict normal crossings divisors).
Let \(\left( X,D_{\epsilon }\right) \) be a Campana orbifold over a number field K. Let \(S \subset {{\,\mathrm{Val}\,}}\left( K\right) \) be a finite set containing \(S_{\infty }\).
Definition 2.2
A model of \(\left( X,D_{\epsilon }\right) \) over \(\mathcal {O}_{K,S}\) is a pair \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \), where \(\mathcal {X}\) is a flat proper model of X over \(\mathcal {O}_{K,S}\) (i.e. a flat proper \(\mathcal {O}_{K,S}\)-scheme with \(\mathcal {X}_{\left( 0\right) } \cong X\)) and \(\mathcal {D}_{\epsilon } = \sum _{\alpha \in \mathcal {A}}\epsilon _{\alpha } \mathcal {D}_{\alpha }\) for \(\mathcal {D}_{\alpha }\) the Zariski closure of \(D_{\alpha }\) in \(\mathcal {X}\).
Define \(\mathcal {D}_{\text {red}} = \sum _{\alpha \in \mathcal {A}}\mathcal {D}_{\alpha }\). Denote by \(\mathcal {D}_{\alpha _v}\), \(\alpha _v \in \mathcal {A}_v\) the irreducible components of \(\mathcal {D}_{\text {red}}\) over \({{\,\mathrm{Spec}\,}}\mathcal {O}_v\). We write \(\alpha _v \mid \alpha \) if \(\mathcal {D}_{\alpha _v} \subset \mathcal {D}_{\alpha }\).
Let \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) be a model for \(\left( X,D_{\epsilon }\right) \) over \(\mathcal {O}_{K,S}\). For \(v \not \in S\), any \(P \in X\left( K\right) \) induces some \(\mathcal {P}_v \in \mathcal {X}\left( \mathcal {O}_v\right) \) by the valuative criterion of properness [14, Thm. II.4.7, p. 101].
Definition 2.3
Let \(P \in X\left( K\right) \) and take a place \(v \not \in S\). For each \(\alpha _v \in \mathcal {A}_v\), we define the local intersection multiplicity \(n_v\left( \mathcal {D}_{\alpha _v},P\right) \) of \(\mathcal {D}_{\alpha _v}\) and P at v to be \(\infty \) if \(\mathcal {P}_v \subset \mathcal {D}_{\alpha _v}\), and the colength of the ideal \(\mathcal {P}_v^*\mathcal {D}_{\alpha _v} \subset \mathcal {O}_v\) 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, Sect. 1].
Definition 2.4
We say that \(P \in X\left( K\right) \) is a weak Campana \(\mathcal {O}_{K,S}\)-point of \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) if the following implications hold for all places \(v \not \in S\) of K and for all \(\alpha \in \mathcal {A}\):
-
(1)
If \(\epsilon _\alpha = 1\) (meaning \(m_{\alpha } = \infty \)), then \(n_v\left( \mathcal {D}_{\alpha },P\right) = 0\).
-
(2)
If \(n_v\left( \mathcal {D}_{\epsilon },P\right) > 0\), then
$$\begin{aligned} \sum _{\alpha \in \mathcal {A}} \frac{1}{m_{\alpha }}n_v\left( \mathcal {D}_{\alpha },P\right) \ge 1. \end{aligned}$$
We denote the set of weak Campana \(\mathcal {O}_{K,S}\)-points of \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) by \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) _{\mathbf {w}}\left( \mathcal {O}_{K,S}\right) \).
Definition 2.5
We say that \(P \in X\left( K\right) \) is a Campana \(\mathcal {O}_{K,S}\)-point of \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) if the following implications hold for all places \(v \not \in S\) of K and for all \(\alpha \in \mathcal {A}\).
-
(3)
If \(\epsilon _{\alpha } = 1\) (meaning \(m_{\alpha } = \infty \)), then \(n_v\left( \mathcal {D}_{\alpha },P\right) = 0\).
-
(4)
If \(\epsilon _\alpha \ne 1\) and \(n_v\left( \mathcal {D}_{\alpha _v},P\right) > 0\) for some \(\alpha _v \mid \alpha \), then \( n_v\left( \mathcal {D}_{\alpha _v},P\right) \ge m_{\alpha }. \)
We denote the set of Campana \(\mathcal {O}_{K,S}\)-points of \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) by \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \left( \mathcal {O}_{K,S}\right) \).
Remark 2.6
Informally, weak Campana points are rational points \(P \in X\left( K\right) \) avoiding \(\cup _{\epsilon _\alpha = 1}\mathcal {D}_{\alpha }\) which, upon reduction modulo any place \(v \not \in S\), either do not lie on \(D_{\text {red}}\) or lie on \(D_\alpha \) with multiplicity at least \(m_\alpha \) on average over \(\alpha \). Similarly, Campana points are rational points \(P \in X\left( K\right) \) avoiding \(\cup _{\epsilon _\alpha = 1}\mathcal {D}_{\alpha }\) which, upon reduction modulo any place \(v \not \in S\), either do not lie on \(D_{\text {red}}\) or lie on each v-adic irreducible component of each \(D_{\alpha }\) with multiplicity either 0 or at least \(m_{\alpha }\).
Remark 2.7
Our definition of Campana points differs from the one in [27, Sect. 3.2], in which one requires simply that \(n_v\left( \mathcal {D}_\alpha ,P\right) \ge m_{\alpha }\) instead of \(n_v\left( \mathcal {D}_{\alpha _v},P\right) \ge m_{\alpha }\) in the second implication. If one were to apply this definition to the orbifold \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) of Theorem 1.1, which is singular for all \(d \ge 3\) as \(Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \) 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 \(d \ge 3\) (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 \(\left( X,D_{\epsilon }\right) \) be a smooth Campana orbifold over a number field K which is Kawamata log terminal (i.e. \(\epsilon _{\alpha } < 1\) for all \(\alpha \in \mathcal {A}\)), and let \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) be a model of \(\left( X,D_{\epsilon }\right) \) over \(\mathcal {O}_{K,S}\) with \(\mathcal {X}\) smooth over \(\mathcal {O}_{K,S}\) and \(\mathcal {D}_{\text {red}}\) a relative strict normal crossings divisor in \(\mathcal {X}/\mathcal {O}_{K,S}\) as defined in [17, Sect. 2]. Then the definition of Campana points on \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) above coincides with the one in [27, Sect. 3.2].
Proof
Since \(\mathcal {D}_{\text {red}}\) is a relative strict normal crossings divisor, each irreducible component \(\mathcal {D}_{\alpha }\) is smooth over \(\mathcal {O}_{K,S}\). In particular, its base change over \({{\,\mathrm{Spec}\,}}\mathcal {O}_v\) is smooth for any \(v \not \in S\), so the divisors \(\mathcal {D}_{\alpha _v}\), \(\alpha _v \mid \alpha \) are disjoint. Then, for any rational point \(P \in X\left( K\right) \), the reduction of P at the place v can lie on at most one of the divisors \(\mathcal {D}_{\alpha _v}\), \(\alpha _v \mid \alpha \), so \(n_v\left( \mathcal {D}_{\alpha },P\right) = \sum _{\alpha _v \mid \alpha } n_v\left( \mathcal {D}_{\alpha _v},P\right) \) is either 0 or at least \(m_{\alpha }\) if and only if each \(n_v\left( \mathcal {D}_{\alpha _v},P\right) \), \(\alpha _v \mid \alpha \), is either 0 or at least \(m_{\alpha }\). \(\square \)
2.2 Toric varieties
Definition 2.9
An (algebraic) torus over a field F is an algebraic group T over F such that \(\overline{T} \cong \mathbb {G}_m^n\) for some \(n \in \mathbb {N}\). 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 \(T_E \cong \mathbb {G}_m^n\).
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 \(X^*\left( \overline{T}\right) = {{\,\mathrm{Hom}\,}}\left( \overline{T},\mathbb {G}_m\right) \), and we have \(X^*\left( T\right) = X^*\left( \overline{T}\right) ^{G_F}\). The cocharacter group of T is \(X_*\left( \overline{T}\right) = {{\,\mathrm{Hom}\,}}\left( X^*\left( \overline{T}\right) , \mathbb {Z}\right) \), and we have \(X_*\left( T\right) = X_*\left( \overline{T}\right) ^{G_F}\). We let \(X^*\left( T\right) _{\mathbb {R}} = X^*\left( T\right) \otimes _{\mathbb {Z}} \mathbb {R}\) and \(X_*\left( T\right) _{\mathbb {R}} = X_*\left( T\right) \otimes _{\mathbb {Z}} \mathbb {R}\).
Definition 2.12
An algebraic torus T over a field F is anisotropic if it has trivial character group over F, i.e. \(X^*\left( T\right) = 0.\)
Let T be a torus over a number field K with splitting field E. Set \(T_\infty = \prod _{v \mid \infty }T_v\). For \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), let \(T\left( \mathcal {O}_v\right) \) denote the maximal compact subgroup of \(T\left( K_v\right) \).
Definition 2.13
Let \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) and \(w \in {{\,\mathrm{Val}\,}}\left( E\right) \) with \(w \mid v\).
For \(v \not \mid \infty \) with ramification degree \(e_v\) in E/K, define the maps
and \(\deg _{T,E,v} = e_v \deg _{T,v}\).
For \(v \mid \infty \), define the maps
and \(\deg _{T,E,v} = [E_w:K_v] \deg _{T,v}\).
Finally, define the maps
Lemma 2.14
[4, Sect. 2.2], [19, Sect. 4.2] Let \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), and let f be either \(\deg _{T,v}\) or \(\deg _{T,E,v}\).
-
(i)
If v is non-archimedean, then we have the exact sequence
$$\begin{aligned} 0 \rightarrow T\left( \mathcal {O}_v\right) \rightarrow T\left( K_v\right) \xrightarrow {f} X_*\left( T_v\right) . \end{aligned}$$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
$$\begin{aligned} 0 \rightarrow T\left( \mathcal {O}_v\right) \rightarrow T\left( K_v\right) \xrightarrow {f} X_*\left( T_v\right) _{\mathbb {R}} \rightarrow 0. \end{aligned}$$Further, f admits a canonical section.
-
(iii)
Letting g be either \(\deg _T\) or \(\deg _{T,E}\) and denoting its kernel by \(T\left( \mathbb {A}_K\right) ^1\), we have the split short exact sequence
$$\begin{aligned} 0 \rightarrow T\left( \mathbb {A}_K\right) ^1 \rightarrow T\left( \mathbb {A}_K\right) \xrightarrow {g} X_*\left( T\right) _{\mathbb {R}} \rightarrow 0, \end{aligned}$$hence we have an isomorphism
$$\begin{aligned} T\left( \mathbb {A}_K\right) \cong T\left( \mathbb {A}_K\right) ^1 \times X_*\left( T\right) _{\mathbb {R}}. \end{aligned}$$(2.1)
Definition 2.15
Let \(\chi \) be a character of \(T\left( \mathbb {A}_K\right) \). We say that \(\chi \) is automorphic if it is trivial on \(T\left( K\right) \). We say that \(\chi \) is unramified at \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) if \(\chi _v\) is trivial on \(T\left( \mathcal {O}_v\right) \), and we say that it is unramified if it is unramified at every \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \).
The canonical sections of the maps \(T\left( K_v\right) \xrightarrow {\deg _{T,v}} X_*\left( T_v\right) _{\mathbb {R}}\) for each \(v \mid \infty \) from Lemma 2.14(ii) induce a canonical section of the composition
which in turn induces a “type at infinity map”
Defining \(\mathsf {K}_T = \prod _{v}T\left( \mathcal {O}_v\right) \), the splitting (2.1) for \(g = \deg _T\) induces a map
which has finite kernel and image a lattice of codimension \({{\,\mathrm{rank}\,}}X^*\left( T\right) \) (see [4, Lem. 4.52, p. 96]).
Note 2.16
When T is anisotropic, we have \(T\left( \mathbb {A}_K\right) ^1 = T\left( \mathbb {A}_K\right) \) 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.
2.3 Hecke characters
Definition 2.17
A Hecke character for K is an automorphic character of \(\mathbb {G}_{m,K}\).
Each Hecke character \(\chi \) has a conductor \(q\left( \chi \right) \in \mathbb {N}\) (see [18, Sect. 5.10]), which measures the ramification of \(\chi \) at the non-archimedean places of K.
Definition 2.18
A Hecke character is principal if it is trivial on \(\mathbb {G}_{m,K}\left( \mathbb {A}_K\right) ^1\).
By Lemma 2.14(iii), \(\chi \) is principal if and only if \( \chi = \Vert \cdot \Vert ^{i\theta }\) for some \(\theta \in \mathbb {R}\), where \(\Vert \cdot \Vert \) denotes the adelic norm map, i.e.
Definition 2.19
The (Hecke) L-function \(L\left( \chi ,s\right) \) of a Hecke character \(\chi \) is
where the product is taken over all places \(v \not \mid \infty \) at which \(\chi \) is unramified.
The Dedekind zeta function of K is
Given a Hecke character \(\chi \) for a number field L and \(w \in {{\,\mathrm{Val}\,}}\left( L\right) \), we denote by \(L_w\left( \chi ,s\right) \) the local factor at w for the Euler product of \(L\left( \chi ,s\right) \), i.e.
When working over the field \(L \supset K\), we define \(L_v\left( \chi ,s\right) \) for each \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) by
Theorem 2.20
[15, Sect. 6] The L-function of a Hecke character \(\chi \) admits a meromorphic continuation to \(\mathbb {C}\). If \(\chi = \Vert \cdot \Vert ^{i \theta }\) for some \(\theta \in \mathbb {R}\), then this continuation admits a single pole of order 1 at \(s = 1+ i\theta \). Otherwise, it is holomorphic.
Definition 2.21
Let \(\psi \) be a character of \(\prod _{v | \infty } K_v^*\). The restriction of \(\psi \) to each \(\mathbb {R}_{>0} \subset K_v^*\) is of the form \(x \mapsto |x|^{i\kappa _v}\) for some \(\kappa _v \in \mathbb {R}\). We define
Lemma 2.22
[18, Exercise 3, §5.2, p. 100] Let \(\chi \) be a non-principal Hecke character of K, let C be a compact subset of \({\text {Re}}s \ge 1\) and let \(\varepsilon > 0\). Then
Definition 2.23
Let E/K be Galois, let \(\chi \) be a Hecke character for E and let \(g \in {{\,\mathrm{Gal}\,}}\left( E/K\right) \). We define the (Galois) twist of \(\chi \) by g to be the character
Here, gw denotes the place of E obtained by the action of g on \({{\,\mathrm{Val}\,}}\left( E\right) \), and \(g_w: E_w \rightarrow E_{gw}\) is the induced map on completions. One may easily verify that \(\chi ^g\) is trivial on \(E^*\), hence it is also a Hecke character for E.
3 The norm torus
In this section, we fix an extension of number fields L/K of degree \(d \ge 2\) with Galois closure E and a K-basis \({{\,\mathrm{\varvec{\omega }}\,}}= \{\omega _0,\dots ,\omega _{d-1}\}\). We write \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x_0,\dots ,x_{d-1}\right) \) for the norm form corresponding to \({{\,\mathrm{\varvec{\omega }}\,}}\), and \(G = {{\,\mathrm{Gal}\,}}\left( E/K\right) \). From the equality
we see that \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) is irreducible over K and has splitting field E. We denote by T the norm torus \(T_{{{\,\mathrm{\varvec{\omega }}\,}}} = \mathbb {P}^{d-1}_K\setminus Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \). As noted in [19, Sect. 1.2], \(\mathbb {P}^{d-1}_K\) is a toric variety with respect to T, and \(T \cong R_{L/K}\mathbb {G}_m/\mathbb {G}_m\) is anisotropic. Since its boundary is \(Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \), its splitting field is E. We have the short exact sequence
Note 3.1
The isomorphisms \(T\left( \mathbb {A}_K\right) \cong \mathbb {A}_L^*/\mathbb {A}_K^*\) and \(T\left( K\right) \cong L^*/K^*\) follow from Hilbert’s Theorem 90 [22, Prop. IV.3.8, p. 283] by applying Galois cohomology to (3.2). They allow us to interpret an automorphic character \(\chi \) of T as a Hecke character for L, and we will do so frequently. In fact, distinct automorphic characters of T correspond to distinct Hecke characters of L by [3, Cor. 1.4.16, p. 606] and [3, Thm. 3.1.1, p. 619]. Since \(T\left( K_v\right) \cong \left( \prod _{w \mid v}L_w^*\right) /K_v^*\) for each \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), we see that, if \(\chi \) is unramified at v, then it is unramified as a Hecke character at all \(w \mid v\). In particular, if \(\chi \) is unramified at v and v is unramified in L/K, then \(\prod _{w \mid v} \chi _w\left( \pi _w\right) = 1\), since \(\pi _v\) is a uniformiser for \(L_w\) for each \(w \mid v\).
3.1 Geometry
In this section we study fan-theoretic objects related to T. We begin by describing the fan \(\Sigma \subset X_*\left( \overline{T}\right) _{\mathbb {R}}\) associated to the equivariant compactification \(\mathbb {P}^{d-1}_K\) of T and the associated piecewise-linear function \(\varphi _{\Sigma }\) (see [3, Sect. 1.2]) used to define the Batyrev–Tschinkel height function.
Denoting by \(l_0\left( x\right) ,\dots ,l_{d-1}\left( x\right) \in E[x]\) the E-linear factors of \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x\right) \), we have the E-isomorphism
By [16, Sect. 1.1], the fan associated to \(\mathbb {P}^{d-1}_E\) as a compactification of \(\mathbb {G}_{m,E}^{d-1}\) is the fan whose r-dimensional cones are generated by the r-fold subsets of \(\{e_0',\dots ,e_{d-1}'\}\) for \(0 \le r \le d-1\), where \(e_i' \in X_*\left( \mathbb {G}_m^{d-1}\right) \cong {{\,\mathrm{Hom}\,}}\left( \mathbb {G}_m,\mathbb {G}_m^{d-1}\right) \) is defined by
where in turn
Definition 3.2
Set \(e_i = \Phi ^{-1} \circ e'_i\) for \(i=0,\dots ,d-1\), and define \(\Sigma \) to be the fan whose r-dimensional cones are generated by the r-fold subsets of \(\{e_0,\dots ,e_{d-1}\}\) for \(0 \le r \le d-1\).
It follows that \(\Sigma \) is the fan associated to \(\mathbb {P}^{d-1}_E\) as a compactification of \(T_E\). Also, we see that \(\sum _{i=0}^{d-1}e_i = 0\) and that \(\{e_1,\dots ,e_{d-1}\}\) is the dual of the basis \(\{m_1,\dots ,m_{d-1}\}\) of \(X^*\left( \overline{T}\right) \), where \(m_i\left( x\right) = \frac{l_i\left( x\right) }{l_0\left( x\right) }\) for \(i=1,\dots ,d-1\).
We now show that the action of G on \(\Sigma \left( 1\right) \) is compatible with its action on the E-linear factors of \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\). Denote by \(*\) the action of G, and set \(l_{g\left( i\right) } = g * l_i\).
Lemma 3.3
For all \(g \in G\) and \(i=0,\dots ,d-1\), we have
Proof
Let \(g \in G\). It suffices to show that
for all \(i \in \{0,\dots ,d-1\}\) and \(j \in \{1,\dots ,d-1\}\). Note that, for any \(i,j,k \in \{0,\dots ,d-1\}\), we have
where \(\delta _{ij}\) is the Kronecker delta symbol, defined by
Then (3.3) becomes
which clearly holds. \(\square \)
By [16, Thm. 1.22, p. 217], \(\Sigma \) is the fan associated to the compactification \(\mathbb {P}^{d-1}_K = \mathbb {P}^{d-1}_E / G\) of T over K.
By [3, Prop. 1.2.12, p. 597], the line bundle \(L\left( \varphi _{\Sigma }\right) \) associated to the piecewise-linear function \(\varphi _{\Sigma }: X_*\left( \overline{T }\right) _{\mathbb {R}} \rightarrow \mathbb {R}\) (see [3, Prop. 1.2.9, p. 597]) defined by \(\varphi _{\Sigma }\left( e_i\right) = 1\) for all \(i=0,\dots ,d-1\) is the anticanonical bundle \(-K_{\mathbb {P}^{d-1}}\).
It follows from the above that G acts transitively on \(\Sigma \left( 1\right) = \{\langle e_0 \rangle , \dots , \langle e_{d-1} \rangle \}\). For \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) 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 \(\Sigma \left( 1\right) \) are in bijection with the places of L over v, and the length of the \(G_v\)-orbit corresponding to a place \(w \mid v\) is its inertia degree.
Proposition 3.4
Let \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) be non-archimedean with ramification degree \(e_v\) in E/K, and let
denote the decomposition of \(\Sigma \left( 1\right) = \{\langle e_0 \rangle , \dots , \langle e_{d-1} \rangle \}\) into \(G_v\)-orbits. For each \(w \mid v\), let \(n_w\) be the sum of the elements of \(\Sigma _w\left( 1\right) \) and let \(f_w\left( x\right) \) be the product of the linear factors in the \(G_v\)-orbit of \(\{l_0,\dots ,l_{d-1}\}\) corresponding to \(\Sigma _w\left( 1\right) \) by Lemma 3.3. Then the map \(\deg _{T,E,v}: T\left( K_v\right) \rightarrow X_*\left( T_v\right) \) is given by
Proof
The image of \(t_v\) in \(X_*\left( T_v\right) \cong X_*\left( \overline{T}\right) ^{G_v}\) under \(\deg _{T,E,v}\) is the cocharacter
We first show that \(\{n_w : w \mid v\}\) spans \(X_*\left( \overline{T}\right) ^{G_v}\). Given \(g \in G\) and \(\sigma = \sum _{i=0}^{d-1} a_i e_i\), we have \(g * \sigma = \sum _{i=0}^{d-1} a_{g^{-1}\left( i\right) }e_i\), so \(g * \sigma = \sigma \) if and only if there exists \(r_g \in \mathbb {Z}\) such that \(a_i = a_{g^{-1}\left( i\right) } + r_g\) for all \(i \in \{0,\dots ,d-1\}\). Setting \(s = \#G\), we have
hence \(r_g = 0\). We deduce that \(\sigma \in \Sigma ^{G_v}\) 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 \(\sum _{w \mid v}a_w n_w = \sum _{w \mid v}b_w n_w\) if and only if there exists \(r \in \mathbb {Z}\) such that \(b_w = a_w + r\) for all \(w \mid v\), since there is a unique expression for \(\sigma \in X_*\left( \overline{T}\right) \) in the form \(\sigma = \sum _{i=0}^{d-1} c_i e_i\) where \(c_d = 0\).
Now, write
Define \(\mu _i \in X^*\left( \overline{T}\right) \) and \(\lambda _w \in X^*\left( \overline{T}\right) ^{G_v}\) for all \(i \in \{0,\dots ,{d-1}\}\) and all \(w \mid v\) by
By (3.4), we have
Then, setting \(d_w = \deg f_w\), we see that
so we deduce that
for all \(w \mid v\). On the other hand, we have
Set \(\beta _w = d_w \alpha _w - e_v v\left( f_w\left( t_v\right) \right) \). Combining (3.5) and (3.6), we obtain
hence \(\beta _{w'} = \frac{d_{w'}}{d_w}\beta _w\) for all \(w \mid v\), \(w' \mid v\). Since \(K_v \cong E_w^{G_v}\) for any \(w \mid v\), it follows that \(d_w \mid v\left( f_w\left( t_v\right) \right) \), so \(\beta _w \in d_w\mathbb {Z}\) for all \(w \mid v\). We deduce that there exists an integer \(n \in \mathbb {Z}\) such that, for all \(w \mid v\), we have \(\beta _w = d_w n\), hence
Since \(\sum _{w \mid v} n_w = \sum _{i}e_i = 0\), we conclude that
\(\square \)
We now study polynomials introduced by Batyrev and Tschinkel in [3, Sect. 2.2], which play a key role in the analysis of local Fourier transforms in Sect. 5.
Definition 3.5
Let \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) be non-archimedean, and let \(\Sigma \left( 1\right) = \bigcup _{i=1}^l\Sigma _i\left( 1\right) \) be the decomposition of \(\Sigma \left( 1\right) \) into \(G_v\)-orbits. Let \(d_i\) be the cardinality of \(\Sigma _i\left( 1\right) \). For each \(\Sigma _i\left( 1\right) \), define an independent variable \(u_i\). Let \(\sigma \in \Sigma ^{G_v}\), and let \(\Sigma _{i_1}\left( 1\right) \cup \cdots \cup \Sigma _{i_k}\left( 1\right) \) be the set of 1-dimensional faces of \(\sigma \). We define the rational function
and we define the polynomial \(Q_{\Sigma ,v}\left( u_1,\dots ,u_l\right) \) by
Proposition 3.6
For all non-archimedean valuations \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), we have
Proof
Observe that the \(G_v\)-invariant cones in \(\Sigma \) are precisely those cones generated by a set of 1-dimensional cones of the form \(\Sigma _{i_1}\left( 1\right) \cup \cdots \cup \Sigma _{i_k}\left( 1\right) \) for some \(i_1,\dots ,i_k \in \{1,\dots ,l\}\) 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 \(t_2,\dots ,t_l\), we deduce (3.7). \(\square \)
3.2 Haar measures and volume
Let \(\omega \) be an invariant d-form on T. By a classical construction (see [12, Sect. 2.1.7]), \(\omega \) gives rise to a Haar measure \(|\omega |_v\) on \(T\left( K_v\right) \) for each \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \). In [23, Sect. 3.3], Ono constructs the convergence factors
Here, \(L_v\left( X^*\left( \overline{T}\right) ,s\right) \) is the local factor at v of the Artin L-function \(L\left( X^*\left( \overline{T}\right) ,s\right) \). Defining \(\mu _v = c_v^{-1}|\omega |_v\), the product of the \(\mu _v\) converges to give a Haar measure \(\mu \) on \(T\left( \mathbb {A}_K\right) \), which is independent of \(\omega \) by the product formula.
Note 3.7
From the short exact sequence (3.2), we obtain
Lemma 3.8
With respect to the Haar measure \(\mu \), we have
Proof
By [23, Sect. 3.5] and [24, Main Thm., p. 68], we have
where 
is the In this proof, two equations appear in overly large font in the eProof, although they are the correct size in the PDF version. of T, i.e.
By [29, Prop. 8.3, p. 58] and [19, Cor. 4.6, p. 2568], the rationality of T implies that 
is trivial. Further, we have \({{\,\mathrm{Pic}\,}}T \cong \mathbb {Z}/d\mathbb {Z}\) (see [14, Prop. II.6.5(c), p. 133]). Since \(\zeta _K\left( s\right) \) and \(\zeta _L\left( s\right) \) both have a simple pole at \(s=1\), we have \(L\left( X^*\left( \overline{T}\right) ,1\right) = \frac{{{\,\mathrm{Res}\,}}_{s=1}\zeta _L\left( s\right) }{{{\,\mathrm{Res}\,}}_{s=1}\zeta _K\left( s\right) }\). Finally, as T is anisotropic, we have \(T\left( \mathbb {A}_K\right) ^1 = T\left( \mathbb {A}_K\right) \). \(\square \)
4 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 K-basis \({{\,\mathrm{\varvec{\omega }}\,}}= \{\omega _0,\dots ,\omega _{d-1}\}\) and Galois closure E/K. When \(L=E\), for any \(i,j \in \{0,\dots ,d-1\}\) and \(g \in G = {{\,\mathrm{Gal}\,}}\left( E/K\right) \), write
Definition 4.1
When \(L=E\), we define \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) to be the minimal subset of \({{\,\mathrm{Val}\,}}\left( K\right) \) containing \(S_{\infty }\) such that \(a^{ij}_k, b^g_k,c_k \in \mathcal {O}_v\) for all \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), \(i,j,k \in \{0,\dots ,d-1\}\) and \(g \in G\). Otherwise, we define \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) to be the minimal subset of \({{\,\mathrm{Val}\,}}\left( K\right) \) containing \(S_{\infty }\) such that \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) is an irreducible polynomial over \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\).
Remark 4.2
When \(L=E\), by (3.1), the \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \)-integrality of the \(a^{ij}_k\) and \(b^g_k\) implies that \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) is defined over \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\), while the \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \)-integrality of the \(c_k\) implies that the coefficients of \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) are not all divisible by some \(\alpha \in \mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) } \setminus \mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }^*\). Since \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) is irreducible over K, we deduce that \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) is irreducible over \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\), hence, for any L, the Zariski closure of \(Z\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \) in \(\mathbb {P}^{d-1}_{\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }}\) is \({{\,\mathrm{Proj}\,}}\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }[x_0,\dots ,x_{d-1}]/\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \).
From now on, we fix the model \(\left( \mathbb {P}^{d-1}_{\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }},\mathcal {D}^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) for \(\left( \mathbb {P}^{d-1}_K,\Delta _m^{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \), where \(\mathcal {D}^{{{\,\mathrm{\varvec{\omega }}\,}}}_m = \left( 1-\frac{1}{m}\right) {{\,\mathrm{Proj}\,}}\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }[x_0,\dots ,x_{d-1}]/\left( N_{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \). We denote by \(\mathcal {D}^{{{\,\mathrm{\varvec{\omega }}\,}}}_{\text {red}}\) the support of \(\mathcal {D}^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\).
Note 4.3
In both the Galois and non-Galois cases, the conditions on \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) 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 Sect. 4.2.
Remark 4.4
When \(L=E\) and \({{\,\mathrm{\varvec{\omega }}\,}}\) is a relative integral basis, we get \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) = S_{\infty }\), since every algebraic integer is expressible as an \(\mathcal {O}_K\)-linear combination of elements of a relative integral basis, and \(\mathcal {O}_L\) is closed under multiplication and conjugation.
4.1 Definitions
Definition 4.5
[3, Sect. 2.1] For each place v of K, we define the local height function
We then define the global height function
Definition 4.6
For each place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), define the function
Remark 4.7
Note that \(H'_v\left( x\right) \ge 1\) for all \(x \in T\left( K_v\right) \). Indeed, one may always select v-adic coordinates \(x_i\) such that \(\max \{|x_i|_v\} = 1\), and \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) has coefficients in \(\mathcal {O}_v\) by Remark 4.2, so, by the strong triangle inequality, we have \(|N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x\right) |_v \le 1\).
Lemma 4.8
For all but finitely many places \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have \(H'_v = H_v\).
Proof
Note that \(H_v'\) is the local Weil function associated to the basis of global sections of \(-K_{\mathbb {P}^{d-1}}\) consisting of all monomials of degree d in [3, Def. 2.1.1, p. 606]. It is well-known (see [12, Sect. 2.2.3]) that two height functions corresponding to adelic metrisations of the same line bundle are equal over all but finitely many places. \(\square \)
Definition 4.9
We define the finite set
Definition 4.10
For each place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), define the local indicator function
Setting \(\phi _{m,v} = 1\) for \(v \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we then define the global indicator function
Remark 4.11
Let \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) be a non-archimedean place of K. Since \(H'_v\) is continuous with discrete image in \(\mathbb {R}_{>0}\), its level sets are clopen. It follows that \(\phi _{m,v}\) is continuous for all \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \). Also, since \(\phi _{m,v}\left( T\left( \mathcal {O}_v\right) \right) = 1\) for all \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) by Lemma 2.14(i), we see that \(\phi _m\) is well-defined and continuous on \(T\left( \mathbb {A}_K\right) \).
Lemma 4.12
The weak Campana \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\)-points of \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) are precisely the rational points \(t \in T\left( K\right) \) such that \(\phi _m\left( t\right) = 1\).
Proof
Take \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), and let \(t_0,\dots ,t_{d-1}\) be a set of \(\mathcal {O}_v\)-coordinates for \(t \in T\left( K\right) \) with at least one \(t_i \in \mathcal {O}_v^*\). Then we have
\(\square \)
4.2 Invariant subgroups
For this section, let \(L = E\) be Galois over K.
Lemma 4.13
For all \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) and \(x,y \in T\left( K_v\right) \), we have
Proof
Choose sets of projective coordinates \(\{x_0,\dots ,x_{d-1}\}\) and \(\{y_0,\dots ,y_{d-1}\}\) for x and y respectively. Note that
where, for \(a^{ij}_k \in \mathcal {O}_v\) as in Definition 4.1, we have
Using \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x \cdot y\right) = N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x\right) N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( y\right) \) and the strong triangle inequality, we deduce that
\(\square \)
Lemma 4.14
For any place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), the level set
is a subgroup of \(T\left( \mathcal {O}_v\right) \).
Proof
From Proposition 3.4 and Lemma 2.14(i), it is clear that \(H_v'\left( t_v\right) = 1\) implies \(t_v \in T\left( \mathcal {O}_v\right) \), so \(\mathcal {K}_v \subset T\left( \mathcal {O}_v\right) \). It is also clear that \(H'_v\left( 1\right) = 1\), and closure under multiplication follows from Lemma 4.13 and Remark 4.7. It only remains to verify that \(x \in \mathcal {K}_v\) implies \(x^{-1} \in \mathcal {K}_v\). Let \(x \in \mathcal {K}_v\), and choose coordinates \(x_0,\dots ,x_{d-1}\) with \(\max \{|x_i|_v^d\}\) = 1. Since \(H'_v\left( x\right) = 1\), we must have \(|N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x\right) |_v = 1\). Note that
Recursively applying Lemma 4.13, we obtain
By Remark 4.7, it suffices to show that, for any \(g \in G\), we have \(H_v'\left( g\left( x\right) \right) = 1\). Since \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( g\left( x\right) \right) = N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x\right) \), it suffices by Remark 4.7 to show that \(\max \{|g\left( x\right) _i|_v\} \le 1\). This follows from the fact that \(b^g_k \in \mathcal {O}_v\) since \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), see Definition 4.1. \(\square \)
Corollary 4.15
For every place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), the function \(H'_v\) is \(\mathcal {K}_v\)-invariant.
Proof
Take \(x \in \mathcal {K}_v\), and let \(y \in T\left( K_v\right) \). Then by Lemma 4.13, we have
while on the other hand, since \(x^{-1} \in \mathcal {K}_v\) by Lemma 4.14, we have
so we conclude that \(H'_v \left( x \cdot y\right) = H'_v\left( y\right) \). \(\square \)
Lemma 4.16
For each place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), the functions \(H_v\) and \(\phi _{m,v}\) are both \(\mathcal {K}_v\)-invariant and 1 on \(\mathcal {K}_v\). Further, \(\mathcal {K}_v\) is compact, open and of finite index in \(T\left( \mathcal {O}_v\right) \). Moreover, when \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have \(\mathcal {K}_v = T\left( \mathcal {O}_v\right) \).
Proof
Let \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \). By [3, Thm. 2.1.6(i), p. 608], \(H_v\) is \(T\left( \mathcal {O}_v\right) \)-invariant, hence trivial and invariant on all of \(T\left( \mathcal {O}_v\right) \). By Corollary 4.15, the function \(\phi _{m,v}\) is \(\mathcal {K}_v\)-invariant; since \(\phi _{m,v}\left( 1\right) = 1\), it is also trivial on \(\mathcal {K}_v\).
Now, since \(\mathcal {K}_v = \left( H_v'|_{T\left( \mathcal {O}_v\right) }\right) ^{-1}\left( \{1\}\right) \), it is open. Since the cosets of an open subgroup form an open cover of a topological group, any open subgroup of a compact topological group is closed and of finite index. Then \(\mathcal {K}_v \subset T\left( \mathcal {O}_v\right) \) is closed, hence compact, and of finite index. Finally, we note that, when \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have \(H'_v = H_v\), and \(H_v^{-1}\left( \{1\}\right) = T\left( \mathcal {O}_v\right) \) by Lemma 2.14(i) and the equality \(\varphi ^{-1}_{\Sigma }\left( \{0\}\right) = \{0\}\), so \(\mathcal {K}_v = T\left( \mathcal {O}_v\right) \). \(\square \)
Definition 4.17
For each \(v \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), set \(\mathcal {K}_v = T\left( \mathcal {O}_v\right) \). Let \(\mathcal {K} = \prod _{v \in {{\,\mathrm{Val}\,}}\left( K\right) } \mathcal {K}_v\), and let \(\mathcal {U}\) be the group of automorphic characters of T which are trivial on \(\mathcal {K}\).
4.3 Height zeta function and Fourier transforms
Definition 4.18
For \({\text {Re}}s \gg 0\), we define the height zeta function
Definition 4.19
Let \(\mu _v\) and \(\mu \) be the Haar measures introduced in Sect. 3.2. Let \(f: T\left( \mathbb {A}_K\right) \rightarrow \mathbb {C}\) be a continuous function given as a product of local factors \(f_v: T\left( K_v\right) \rightarrow \mathbb {C}\) such that \(f_v\left( T\left( \mathcal {O}_v\right) \right) = 1\) for all but finitely many places \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \). For each place \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \) and each character \(\chi \) of \(T\left( \mathbb {A}_K\right) \), we define the local Fourier transform of \(\chi _v\) with respect to \(f_v\) to be
for all \(s \in \mathbb {C}\) for which the integral exists. We then define the global Fourier transform of \(\chi \) with respect to f to be
5 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 \({{\,\mathrm{\varvec{\omega }}\,}}\), set \(T = T_{{{\,\mathrm{\varvec{\omega }}\,}}}\) as in Sect. 3 and let \(m \in \mathbb {Z}_{\ge 2}\).
5.1 Strategy
Following [3, 19], we will apply a Tauberian theorem [3, Thm. 3.3.2, p. 624] to our height zeta function \(Z_m\left( s\right) \) in order to find an asymptotic for the number of weak Campana points of bounded height. By loc. cit., it suffices to show that \(Z_m\left( s\right) \) is absolutely convergent for \({\text {Re}}s > \frac{1}{m}\) and that \(Z_m\left( s\right) \left( s-\frac{1}{m}\right) ^{b\left( d,m\right) }\) admits an extension to a holomorphic function on \({\text {Re}}s \ge \frac{1}{m}\) which is not zero at \(s = \frac{1}{m}\). In order to do this, we will apply the version of the Poisson summation formula given by Bourqui [4, Thm. 3.35, p. 64]. Formally applying this version with \(\mathcal {G} = T\left( \mathbb {A}_K\right) \), \(\mathcal {H} = T\left( K\right) \), \(dg = d\mu \), dh the discrete measure on \(T\left( K\right) \) and \(F\left( t\right) = \frac{\phi _m\left( t\right) }{H\left( t\right) ^s}\) for some \(s \in \mathbb {C}\) with \({\text {Re}}s > \frac{1}{m}\) gives
5.2 Analytic properties of Fourier transforms
Lemma 5.1
For any place \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), any character \(\chi _v\) of \(T\left( K_v\right) \) and any \(\epsilon > 0\), the local Fourier transform \(\widehat{H}_v\left( \phi _{m,v},\chi _v;-s\right) \) is absolutely convergent and is bounded uniformly (in terms of \(\epsilon \) and v) on \({\text {Re}}s \ge \epsilon \).
Proof
Let \({\text {Re}}s \ge \epsilon \). Since
it suffices to prove that \(\widehat{H}_v\left( 1,1;-\epsilon \right) \) is convergent. For \(v \mid \infty \), this follows from [3, Prop. 2.3.2, p. 614], so assume that \(v \not \mid \infty \). The following argument is essentially the one in [3, Rem. 2.2.8, p. 613], but we fill in the details for the sake of clarity.
Since \(H_v\) and \(d\mu _v\) are \(T\left( \mathcal {O}_v\right) \)-invariant and \(\int _{T\left( \mathcal {O}_v\right) }d\mu _v = 1\), we have
Now, by Lemma 2.14(i), \(T\left( K_v\right) /T\left( \mathcal {O}_v\right) \) can be identified with a sublattice of finite index in \(X_*\left( T_v\right) \), and this sublattice coincides with \(X_*\left( T_v\right) \) when v is unramified in L/K. Then we see that, interpreting \(H_v\) as a function on \(X_*\left( T_v\right) \), we have
and the proof of [3, Thm. 2.2.6, p. 611] and Proposition 3.6 give
so we deduce that \(\widehat{H}_v\left( 1,1;-\epsilon \right) \) is convergent, and this concludes the proof. \(\square \)
Lemma 5.2
For any \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), the local Fourier transform \(\widehat{H}_v\left( \phi _{m,v},1;-s\right) \) is non-zero for all \(s \in \mathbb {R}_{> 0}\).
Proof
The proof is analogous to the proof of [19, Lem. 5.1, p. 2575]. \(\square \)
Lemma 5.3
Let \(L = E\) be a Galois extension of K. For any place \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), let \(\chi _v\) be a character of \(T\left( K_v\right) \) which is non-trivial on \(\mathcal {K}_v\). Then
Proof
Since \(\phi _{m,v}\) and \(H_v\) are \(\mathcal {K}_v\)-invariant, the result follows by character orthogonality. \(\square \)
Corollary 5.4
Let \(L = E\) be a Galois extension of K, and let \(\chi \) be an automorphic character of T. If \(\chi \not \in \mathcal {U}\) for \(\mathcal {U}\) as in Definition 4.17, then
Lemma 5.5
Let \(v \not \mid \infty \) be a non-archimedean place of K unramified in L/K, and let \(\chi \) 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. \(\square \)
Definition 5.6
Given a vector \(u = \left( u_1,\dots ,u_r\right) \in \mathbb {N}^r\), define \(f_{r,n,u}\left( x_1\dots ,x_r\right) \) to be the sum of all degree-n monomials in \(x_1,\dots ,x_r\) weighted via u, i.e.
Set
Proposition 5.7
Let \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) be a non-archimedean place of K, and let \(\chi \in \mathcal {U}\). Let \(w_1,\dots ,w_r \in {{\,\mathrm{Val}\,}}\left( L\right) \) be the places of L over v. Let \(u_i\) be the inertia degree of \(w_i\) over v for each \(i=1,\dots ,r\). Set
Then, for \({\text {Re}}s > 0\), we have
Proof
Let \(s \in \mathbb {C}\) with \({\text {Re}}s > 0\). As \(\chi \in \mathcal {U}\) and \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), it follows that \(\chi \) 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 \(n \ge 1\) that
Since \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have \(\phi _{m,v}\left( t_v\right) = 1\) if and only if \(H_v\left( t_v\right) = 1\) or \(H_v\left( t_v\right) \ge q_v^m\), 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, 27], we will construct our regularisations from L-functions.
Proposition 5.8
Let G be a subgroup of \(S_d\) acting freely and transitively on \(\{1,\dots ,d\}\), and let \(m \ge 2\) be a positive integer. Let \(S_m\) act upon \(G^m\) by permutation of coordinates, and let G act on \(G^m/S_m\) by right multiplication of every element of a representative m-tuple. Set \(S\left( G,m\right) = \left( G^m/S_m\right) /G\).
-
(i)
If \(\overline{\left( g_1,\dots ,g_m\right) } = \overline{\left( h_1,\dots ,h_m\right) }\) in \(S\left( G,m\right) \), then
$$\begin{aligned} \sum _{i=1}^d x_{g_1\left( i\right) }\cdots x_{g_m\left( i\right) } = \sum _{i=1}^d x_{h_1\left( i\right) }\cdots x_{h_m\left( i\right) }, \end{aligned}$$(5.2)hence, for \(\overline{\left( g_1,\dots ,g_m\right) } \in S\left( G,m\right) \), we may define the sum
$$\begin{aligned} \phi _{\overline{\left( g_1,\dots ,g_m\right) }}\left( x_1,\dots ,x_d\right) = \sum _{i=1}^d x_{g_1\left( i\right) }\cdots x_{g_m\left( i\right) }. \end{aligned}$$ -
(ii)
If \(\overline{\left( g_1,\dots ,g_m\right) } \ne \overline{\left( h_1,\dots ,h_m\right) }\), then the sums \(\phi _{\overline{\left( g_1,\dots ,g_m\right) }}\left( x_1,\dots ,x_d\right) \) and \(\phi _{\overline{\left( h_1,\dots ,h_m\right) }}\left( x_1,\dots ,x_d\right) \) share no summands. Further, a monomial appears twice in \(\phi _{\overline{\left( g_1,\dots ,g_m\right) }}\left( x_1,\dots ,x_d\right) \) if and only if \(\left( g_1,\dots ,g_m\right) = \left( g_1 g_r,\dots ,g_m g_r\right) \) for some \(r \in \{1,\dots ,m\}\) with \(g_r \ne 1_G\).
-
(iii)
If d is coprime to m, then we have
$$\begin{aligned} f_{d,m}\left( x_1,\dots ,x_d\right) = \sum _{\overline{\left( g_1,\dots ,g_m\right) } \in S\left( G,m\right) } \phi _{\left( g_1,\dots ,g_m\right) }\left( x_1,\dots ,x_d\right) . \end{aligned}$$ -
(iv)
If d is prime and \(m = kd\), then we have
$$\begin{aligned} f_{d,m}\left( x_1,\dots ,x_d\right) + \left( d-1\right) x_1^k \cdots x_d^k = \sum _{\overline{\left( g_1,\dots ,g_m\right) } \in S\left( G,m\right) } \phi _{\left( g_1,\dots ,g_m\right) }\left( x_1,\dots ,x_d\right) . \end{aligned}$$
Proof
First, we prove (i). Note that \(\overline{\left( g_1,\dots ,g_m\right) } = \overline{\left( h_1,\dots ,h_m\right) }\) if and only if \(\{h_1,\dots ,h_m\} = \{g_1g,\dots ,g_m g\}\) as multisets for some \(g \in G\). If the coordinates of \(\left( h_1,\dots ,h_m\right) \in G^m\) are a permutation of those of \(\left( g_1,\dots ,g_m\right) \in G^m\), then
for any \(i \in \{1,\dots ,d\}\), while if \(\left( h_1,\dots ,h_m\right) = \left( g_1g,\dots ,g_m g\right) \) for some \(g \in G\), then for any \(i \in \{1,\dots ,d\}\), we have
for the unique j with \(g\left( i\right) = j\). In either case, we obtain (5.2). The claim follows.
Next we prove (ii). Note that for \(\overline{\left( g_1,g_2,\dots ,g_m\right) } \in S\left( G,m\right) \), we may take \(g_1 = 1_G\) without loss of generality. Suppose that
for some \(i,j \in \{1,\dots ,d\}\). This is equivalent to the equality of multisets
If \(i=j\), we have
and by the freeness of the action of G on \(\{1,\dots , d\}\), we have \(\left( g_2,\dots ,g_m\right) = \left( h_2,\dots ,h_m\right) \) up to reordering, i.e. the m-tuple \(\left( 1_G,h_2,\dots ,h_m\right) \) is a permutation of \(\left( 1_G,g_2,\dots ,g_m\right) \). If \(i \ne j\), we may take \(g_2\left( i\right) = j\) without loss of generality (note that \(g_2 \ne 1_G\) in this case), and we obtain the equality of multisets
Once again, by freeness of the action of G on \(\{1,\dots ,d\}\), we get that
as multisets, from which both claims follow.
Before proceeding to the proofs of (iii) and (iv), we make the following observation: if a multiset \(M = \{g_1,\dots ,g_m\}\) of elements of G is closed under right multiplication by some \(g \in G\) of order \(n \ge 2\), then n divides m. Indeed, M must contain the n distinct elements \(g_1, g_1 g, g_1 g^2,\dots ,g_1 g^{n-1}\). Without loss of generality, we may take \(g_l = g_1 g^{l-1}\) for \(l =1,\dots ,n\). Since \(\{g_1,\dots ,g_n\} = \{g_1,g_1 g,\dots , g_1 g^{n-1}\}\) is clearly closed under right multiplication by g, so is the sub-multiset \(\{g_{n+1},\dots ,g_m\}\). 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 \(x_1,\dots ,x_d\) appears in a unique summand of \(\sum _{\overline{\left( g_1,\dots ,g_m\right) } \in S\left( G,m\right) } \phi _{\left( g_1,\dots ,g_m\right) }\left( x_1,\dots ,x_d\right) \), and a monomial appears twice in \(\phi _{\overline{\left( g_1,\dots ,g_m\right) }}\left( x_1,\dots ,x_d\right) \) if and only if the multiset \(\{g_1,\dots ,g_m\}\) is closed under right multiplication by some \(g_r \ne 1_G\). 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
Letting g be a generator of G, we see that \(\phi _{\overline{\left( 1_G,g,\dots ,g^{p-1},\dots ,1_G,g,\dots ,g^{p-1}\right) }}\left( x_1,\dots ,x_p\right) = px_1^k \cdots x_p^k\) is the only one of the polynomials \(\phi _{\overline{\left( g_1,\dots ,g_{kp}\right) }}\left( x_1,\dots ,x_p\right) \), \(\overline{\left( g_1,\dots ,g_{kp}\right) } \in S\left( G,kp\right) \) in which a monomial appears more than once, and so we obtain (iv). \(\square \)
Remark 5.9
It follows immediately from Proposition 5.8 that
since the number of monomials of degree m in d variables is \({{d+m-1}\atopwithdelims (){d-1}}\).
Remark 5.10
As we shall shortly see, it is the ability to partition (or in the case where d is a prime dividing m, nearly partition) the sum of all degree-m monomials in d variables as in Proposition 5.8 that allows us to construct well-behaved regularisations.
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 \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) be a non-archimedean place which is totally split in E/K and let \(\chi \in \mathcal {U}\). Then
where
Proof
Let \(G=\{g_1,\dots ,g_d\}\). First, we show that every factor of the denominator of the left-hand side appears on the numerator. Let \(\overline{\left( h_1,\dots ,h_{m-d}\right) } \in S\left( G,m-d\right) \). Then we claim that
Since G acts freely and transitively on the places \(w_1,\dots ,w_d\) of E over v, we have
for any \(i=1,\dots ,d\). Since \(\chi _v\) is trivial on \(\mathcal {K}_v = T\left( \mathcal {O}_v\right) \), we have from Note 3.1 that \(\prod _{i=1}^d\chi _{w_i}\left( \pi _{w_i}\right) = \chi _v\left( 1\right) = 1\), so \(\left( \chi ^{g_1}\cdots \chi ^{g_d}\right) _w\left( \pi _w\right) =1\) for all \(w \mid v\). Then the equality follows.
It now suffices to show that, for \(\overline{\left( h_1,\dots ,h_{m-d}\right) } \ne \overline{\left( h'_1,\dots ,h'_{m-d}\right) }\), we have
If not, then \(\{h'_1,\dots ,h'_{m-d},g_1,\dots ,g_d\} = \{gh_1,\dots ,gh_{m-d},gg_1,\dots ,gg_d\}\) as multisets for some \(g \in G\). Since we have \(\{gg_1,\dots ,gg_d\} = \{g_1,\dots ,g_d\}\), this implies that \(\{h'_1,\dots ,h'_{m-d}\} = \{gh_1,\dots ,gh_{m-d}\}\) as multisets, but then we have \(\overline{\left( h_1,\dots ,h_{m-d}\right) } = \overline{\left( h'_1,\dots ,h'_{m-d}\right) }\), which is false. \(\square \)
Remark 5.12
It follows from the proof of Lemma 5.11 that \(\#S'\left( G,m\right) = S\left( G,m\right) - S\left( G,m-d\right) \). Combining this with Remark 5.9, we obtain
Note that the term \(\frac{1}{d}{{m-1}\atopwithdelims (){d-1}}\) only appears when \(d \le m\).
Definition 5.13
For all \(\chi \in \mathcal {U}\), \({\text {Re}}s > 0\) and non-archimedean places \(v \not \mid \infty \), set
and define
For any non-archimedean place \(v \not \mid \infty \), write
where \(a_{\chi ,v,n} = \int _{H_v\left( t_v\right) = q_v^n}\phi _{m,v}\left( t_v\right) \chi _v\left( t_v\right) d\mu _v\), and write
for the expansion of \(F_{m,\chi ,v}\left( s\right) \) as a multidimensional geometric series in \(q_v^{ms}\), so
where \(d_{\chi ,v,n}\) is defined for all \(n \ge 0\) by the iterative formula
In particular, we have \(d_{\chi ,v,n} = a_{\chi ,v,n}\) for \(0 \le n \le m-1\).
Corollary 5.14
For \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) a non-archimedean place, we have \(d_{\chi ,v,0} = 1\) and \(d_{\chi ,v,n} = 0\) for all \(n \in \{1,\dots ,m\}\).
Proof
Let \(c_{\chi ,v,n}\) be as defined in Proposition 5.7. Since \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have from loc. cit. that \(a_{\chi ,v,0} =1\), \(a_{\chi ,v,n} = 0\) for \(n \in \{1,\dots ,m-1\}\) and \(a_{\chi ,v,m} = c_{\chi ,v,m} - c_{\chi ,v,m-d}\). Then, by (5.4), we see that \(d_{\chi ,v,0} = a_{\chi ,v,0} = 1\) and \(d_{\chi ,v,n} = a_{\chi ,v,n} = 0\) for \(1 \le n \le m-1\). Further, we obtain
so, to complete the proof, it suffices to show that \(b_{\chi ,v,m} = c_{\chi ,v,m} - c_{\chi ,v,m-d}\).
Since E/K is Galois, all of the places \(w_1,\dots ,w_r\) of E over v share a common inertia degree \(d_v\). Since \(\chi _v\left( T\left( \mathcal {O}_v\right) \right) = 1\), it is unramified as a Hecke character at all of the \(w_i\) (see Note 3.1), and for any \(g_1,\dots ,g_m \in G\), so is \(\chi ^{g_1}\cdots \chi ^{g_m}\). 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 5.8 and Lemma 5.11 that \(b_{\chi ,v,m} = c_{\chi ,v,m} - c_{\chi ,v,m-d}\), and so \(d_{\chi ,v,m} = 0\).
Now assume that v is not totally split in E/K. If \(\gcd \left( d,m\right) = 1\), then \(c_{\chi ,v,m} = c_{\chi ,v,m-d} = 0\), as \(c_{\chi ,v,n} = 0\) whenever \(d_v \not \mid n\). If d is prime, then v is inert and we have \(\widehat{H}_v\left( \phi _{m,v},\chi _v;-s\right) = 1\) since \(T\left( K_v\right) = T\left( \mathcal {O}_v\right) \). Then, in either case, we have \(c_{\chi ,v,m} - c_{\chi ,v,m-d} = 0\), and (5.5) implies that \(b_{\chi ,v,m} = 0\), hence \(d_{\chi ,v,m} = 0\). \(\square \)
Corollary 5.15
For any \(\chi \in \mathcal {U}\), we have
where \(G_{m,\chi }\left( s\right) \) is holomorphic and uniformly bounded with respect to \(\chi \) for \({\text {Re}}s \ge \frac{1}{m}\) and \(G_{m,1}\left( \frac{1}{m}\right) \ne 0\). In particular, \(\widehat{H} \left( \phi _m,\chi ;-s\right) \) possesses a holomorphic continuation to the line \({\text {Re}}s = \frac{1}{m}\), apart from possibly at \(s = \frac{1}{m}\). When \(\chi =1\), the right-hand side has a pole of order \(b\left( d,m\right) \) at \(s=\frac{1}{m}\).
Proof
By construction, \(\widehat{H}\left( \phi _{m};\chi ;-s\right) = \prod _{v \mid \infty }\widehat{H}_v\left( 1,\chi _v;-s\right) F_{m,\chi }\left( s\right) G_{m,\chi }\left( s\right) \). We will prove the stronger result that \(G_{m,\chi }\left( s\right) \) is holomorphic on \({\text {Re}}s > \frac{1}{m+1}\) and uniformly bounded with respect to both \(\chi \) and \(\epsilon \) on \({\text {Re}}s \ge \frac{1}{m+1}+\epsilon \) for all \(\epsilon > 0\).
For a place \(v \not \mid \infty \) and \(s \in \mathbb {C}\) with \({\text {Re}}s = \sigma \ge \epsilon \) for some \(\epsilon > 0\), we have
so, by Lemma 5.1, the series \(\sum _{n=0}^{\infty }\frac{a_{\chi ,v,n}}{q_v^{ns}}\) is absolutely convergent and bounded by a constant depending only on \(\epsilon \) and v. Now, for any \(N \in \mathbb {N}\), we have
from which it follows that \(\sum _{n=0}^{\infty }\frac{a_{\chi ,v,n}}{q_v^{ns}}\) is also uniformly convergent, hence the function \(\widehat{H}_v\left( \phi _{m,v},\chi _v;-s\right) \) is holomorphic on \({\text {Re}}s > 0\). Then, we note that \(F_{m,\chi ,v}\left( s\right) \) is clearly holomorphic on \({\text {Re}}s > 0\), and we have
hence \(G_{m,\chi ,v}\left( s\right) \) is holomorphic on \({\text {Re}}s > 0\) and is bounded uniformly in terms of \(\epsilon \) and v on \({\text {Re}}s \ge \epsilon \).
To conclude the result, it suffices to prove that there exists \(N \in \mathbb {N}\) such that
is holomorphic and uniformly bounded with respect to \(\chi \) on \({\text {Re}}s \ge \frac{1}{m+1} + \epsilon \) for all \(\epsilon > 0\). Let \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) be non-archimedean, and let \({\text {Re}}s = \sigma \ge \frac{1}{m+1} + \epsilon \). From
and the definition of \(F_{m,\chi ,v}\left( s\right) \), we have
Then, by (5.4), it follows inductively that we have
Choose \(N > \left( 2b\left( d,m\right) d\right) ^{\frac{1}{\sigma }}\) so that, for all places \(v \not \mid \infty \) with \(q_v > N\), we have \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \). Now, any normally convergent infinite product is holomorphic (see [28, Sect. 2]), and \(\prod _{q_v > N}G_{m,\chi ,v}\left( s\right) \) converges normally if and only if
converges. By (5.6) and our condition on N, we need only check convergence of
which is clear. Then \(G_{m,\chi }\left( s\right) \) is holomorphic on \({\text {Re}}s > \frac{1}{m+1}\). Further, for \({\text {Re}}s \ge \frac{1}{m+1}+\epsilon \), we have the bound
which is uniform with respect to \(\chi \). Now, as a convergent infinite product, \(G_{m,1}\left( \frac{1}{m}\right) \) is zero if and only if \(G_{m,1,v}\left( \frac{1}{m}\right) = \frac{\widehat{H}_v\left( \phi _{m,v},1;-\frac{1}{m}\right) }{F_{m,1,v}\left( \frac{1}{m}\right) } = 0\) for some place \(v \not \mid \infty \). However, \(\widehat{H}_v\left( \phi _{m,v},1;-\frac{1}{m}\right) \ne 0\) by Lemma 5.2, so we conclude that \(G_{m,1}\left( \frac{1}{m}\right) \ne 0\). The order of the pole of the right-hand side being \(b\left( d,m\right) \) follows from Theorem 2.20, since
\(\square \)
Note 5.16
In constructing the regularisation \(F_{m,\chi }\left( s\right) \), 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 \(S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \). 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.
Lemma 5.17
[19, Lem. 5.9, p. 2577] Choose an \(\mathbb {R}\)-vector space norm \(\Vert \cdot \Vert \) on \(X^*\left( T_{\infty }\right) _{\mathbb {R}}\) and let \(\mathcal {L} \subset X^*\left( T_{\infty }\right) _{\mathbb {R}}\) be a lattice. Let C be a compact subset of \({\text {Re}}s \ge \frac{1}{m}\) and let \(g: X^*\left( T_{\infty }\right) _{\mathbb {R}} \times C \rightarrow \mathbb {C}\) be a function. If there exists \(0 \le \delta < \frac{1}{d-1}\) such that
for all \(\psi \in X^*\left( T_{\infty }\right) _{\mathbb {R}}\) and \(s \in C\), then the sum
is absolutely and uniformly convergent on C.
Theorem 5.18
Let
Then \(\Omega _{m}\left( s\right) \) admits an extension to a holomorphic function on \({\text {Re}}s \ge \frac{1}{m}\).
Proof
Let \(s \in \mathbb {C}\) with \({\text {Re}}s > \frac{1}{m}\). 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 \(s \mapsto \frac{\phi _m\left( t\right) }{H\left( t\right) ^s}\) is \(L^1\) for \({\text {Re}}s > \frac{1}{m}\). 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, \(s \mapsto \frac{\phi _m\left( t\right) }{H\left( t\right) ^s}\) is continuous and there exists an open neighbourhood \(U \subset T\left( \mathbb {A}_K\right) \) of the origin and strictly positive constants \(C_1\) and \(C_2\) such that for all \(u \in U\) and all \(t \in T\left( \mathbb {A}_K\right) \), we have
We may take \(U = \mathcal {K}\) 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 \({\text {Re}}s \ge \frac{1}{m}\), which will both verify validity of the application and prove the theorem.
Since \(\mathcal {K} \subset \mathsf {K}_T\) is of finite index, the map (2.2) yields a homomorphism
with finite kernel \(\mathcal {N}\) and image \(\mathcal {L}\) a lattice of full rank. We obtain
where the inner sum is finite. Then, for \(s \in C\), we have
Now, for \(\chi \in \mathcal {U}\), 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 \(\psi \in \mathcal {L}\) and some constant \(0 \le \delta < \frac{1}{d-1}\), we have
for \(\Vert \cdot \Vert \) as in Definition 2.21. As \(\mathcal {K} \subset \mathsf {K}_T\) is of finite index, there exists a constant \(Q>0\) such that \(q\left( \chi \right) < Q\) for all \(\chi \in \mathcal {U}\) (cf. [19, Proof of Thm. 5.12, p. 2579]). Since \(F_{m,\chi }\left( s\right) \) is a product of \(b\left( d,m\right) \) L-functions of Hecke characters evaluated at ms, it follows from Lemma 2.22 that
for all for all \(\varepsilon > 0\) and \(s \in C\). The result now follows from Lemma 5.17. \(\square \)
5.4 The leading constant
In order to apply [3, Thm. 3.3.2, p. 624] and deduce Theorem 1.1 from Theorem 5.18, it only remains to show that \(\Omega _m\left( \frac{1}{m}\right) \ne 0\).
Definition 5.19
Let \(\mathcal {U}^G\) be the subgroup of G-invariant elements of \(\mathcal {U}\), and set
Lemma 5.20
For any Galois extension of number fields E/K, the subgroup \(\mathcal {U}[m] \le \left( T\left( \mathbb {A}_K\right) /T\left( K\right) \right) ^\wedge \) is finite. In particular, \(\mathcal {U}_0\) is a finite subgroup of \(\mathcal {U}\).
Proof
By class field theory [22, Ch. VI, Sect. 6; Ch. VII, Sect. 6], \(\mathcal {U}\) may be interpreted as a subset of \({{\,\mathrm{Gal}\,}}\left( E^{{{\,\mathrm{ab}\,}}}_{S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }/E\right) ^\wedge \) for \(E^{{{\,\mathrm{ab}\,}}}_{S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\) the maximal \(S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \)-unramified abelian extension of E, hence \(\mathcal {U}[m]\) is in bijection with a subset of \({{\,\mathrm{Hom}\,}}\left( {{\,\mathrm{Gal}\,}}\left( E^{\text {ab}}_{S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }/E\right) ,\mu _m\right) \), a finite set. \(\square \)
Lemma 5.21
-
(i)
The characters \(\chi \in \mathcal {U}\) contributing to the pole of \(Z_m\left( s\right) \) of order \(b\left( d,m\right) \) at \(s=\frac{1}{m}\) are precisely those \(\chi \in \mathcal {U}_0\) such that \(\prod _{v \mid \infty }\widehat{H}_v\left( 1,\chi _v;-\frac{1}{m}\right) G_{m,\chi }\left( \frac{1}{m}\right) \ne 0\).
-
(i)
Suppose that \(d \ne 2\). If d and m are coprime, then \(\mathcal {U}_0 = \{1\}\). If d is prime and m is a multiple of d, then \(\mathcal {U}_0 = \{\chi \in \mathcal {U}: \chi ^d = 1\}\).
Proof
From Theorem 2.20 and Corollary 5.15, \(\chi \in \mathcal {U}\) contributes to the pole of \(Z_m\left( s\right) \) at \(s = \frac{1}{m}\) if and only if each factor of \(F_{m,\chi }\left( s\right) \) in (5.3) equals \(\zeta _E\left( ms\right) \) and \(\prod _{v \mid \infty }\widehat{H}_v\left( 1,\chi _v;-\frac{1}{m}\right) G_{m,\chi }\left( \frac{1}{m}\right) \ne 0\). Denoting by \(\psi \) the Hecke character associated to \(\chi \), this means precisely that \(\prod _{v \mid \infty }\widehat{H}_v\left( 1,\chi _v;-\frac{1}{m}\right) G_{m,\chi }\left( \frac{1}{m}\right) \ne 0\) and, for each \(\overline{\left( g_1,\dots ,g_m\right) } \in S'\left( G,m\right) \), we have \(\left( \psi ^{g_1}\dots \psi ^{g_m}\right) _v = 1\) for all \(v \not \mid \infty \), which is equivalent by strong approximation [11, Thm., p. 67] to \(\psi ^{g_1}\dots \psi ^{g_m}= 1\). By Note 3.1, this holds if and only if
To conclude the first part, it only remains to show that (5.8) holds if and only if \(\chi \in \mathcal {U}_0\). Taking \(\overline{\left( g_1,\dots ,g_m\right) } = \overline{\left( 1,\dots ,1\right) }\) in (5.8), we obtain \(\chi ^m =1\). If \(d=2\), then \(S'\left( G,m\right) = \left\{ \overline{\left( 1,\dots ,1\right) }\right\} \), and we are done. Otherwise, taking \(\overline{\left( g_1,\dots ,g_m\right) } = \overline{\left( g,1,\dots ,1\right) }\) for any \(g \in G\), we obtain \(\chi ^{m-1}\chi ^g = 1\), so \(\chi ^m = 1\) and \(\chi =\chi ^g\) for all \(g \in G\). Conversely, if \(\chi ^m = 1\) and \(\chi = \chi ^g\) for all \(g \in G\), then (5.8) holds.
Let now \(d \ne 2\), \(\chi \in \mathcal {U}_0\), \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) and \(w \mid v\). We have \(\psi _w\left( \pi _w\right) = \psi _w^g\left( \pi _w\right) = \psi _{gw}\left( \pi _{gw}\right) \) for all \(g \in G\). Since \(\prod _{w \mid v} \psi _w\left( \pi _w\right) = 1\) (see Note 3.1) and G acts transitively on the places of E over v, we obtain \(\psi _w^d\left( \pi _w\right) = 1\), hence \(\chi ^d = 1\) by strong approximation. On the other hand, \(\chi ^m = 1\). For d and m coprime, we conclude that \(\chi = 1\). \(\square \)
Proposition 5.22
The limit
is non-zero.
Proof
We have
Let \(t \in T\left( \mathbb {A}_K\right) \). Note that, if there exists \(\chi ' \in \mathcal {U}_0\) with \(\chi '\left( t\right) \ne 1\), then
so \(\sum _{\chi \in \mathcal {U}_0}\chi \left( t\right) = 0\). Then we have
where
For any \(\chi \in \mathcal {U}_0\) and non-archimedean place \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), comparing the series expressions of \(\widehat{H}_v\left( \phi _{m,v},\chi _v;-s\right) \) and \(F_{m,\chi ,v}\left( s\right) = F_{m,1,v}\left( s\right) \) in Definition 5.13, we see that
so \(\chi _v\left( t_v\right) = 1\) for all \(\chi \in \mathcal {U}_0\) whenever \(H_v\left( t_v\right) = 1\) or \(H_v\left( t_v\right) = q_v^m\).
For each place \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), define the continuous function
Letting \(\theta _{m,v}\) be the indicator function of \(\mathcal {K}_v\) for \(v \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we define the function
By the above, we deduce that \(T\left( \mathbb {A}_K\right) ^{\theta _m} \subset T\left( \mathbb {A}_K\right) ^{\mathcal {U}_0, \phi _m}\), where
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 \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have
for \(a_{\chi ,v,n}\) as in Definition 5.13, so, as in Corollary 5.15, we may deduce that
for \(G_m\left( s\right) \) a function holomorphic on \({\text {Re}}s \ge \frac{1}{m}\). It also follows that \(G_m\left( \frac{1}{m}\right) \ne 0\), since \(\widehat{H}_v\left( \theta _{m,v},1;-\frac{1}{m}\right) \ne 0\) analogously to Lemma 5.2. Then the result follows. \(\square \)
Corollary 5.23
We have
Proof of Theorem 1.1
Since \(\Omega _m\left( \frac{1}{m}\right) \ne 0\), the result for \(S = S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) follows from [3, Thm. 3.3.2, p. 624] and Theorem 5.18, taking \(c\left( {{\,\mathrm{\varvec{\omega }}\,}},m,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \right) \) to be
The result for \(S \supset S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) follows analogously upon redefining \(\phi _{m,v}\) to be identically 1 for each \(v \in S \setminus S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) in Definition 4.10. \(\square \)
6 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 \({{\,\mathrm{\varvec{\omega }}\,}}= \{\omega _0,\dots ,\omega _{d-1}\}\), let \(m \in \mathbb {Z}_{\ge 2}\) and set \(T = T_{{{\,\mathrm{\varvec{\omega }}\,}}}\) as in Sect. 3.
Definition 6.1
For each non-archimedean place \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), let
denote the v-adic decomposition of the norm form \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\) associated to \({{\,\mathrm{\varvec{\omega }}\,}}\) into irreducible polynomials \(f_w\left( x\right) \in \mathcal {O}_v[x]\). For each \(w \mid v\), we define the functions
We then define the Campana local indicator function
Setting \(\psi _{m,v} = 1\) for \(v \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we then define the Campana indicator function
If \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), then for each \(w \mid v\), we also define the function
and we define
Letting \(\sigma _{m,v}\) be the indicator function for \(\mathcal {K}_v\) for \(v \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we define the function
Lemma 6.2
The Campana \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\)-points of \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) are precisely the rational points \(t \in T\left( K\right) \) such that \(\psi _m\left( t\right) = 1\).
Proof
Taking coordinates \(t_0,\dots ,t_{d-1}\) as in the proof of Lemma 4.12, we have
for all non-archimedean places \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) and places \(w \mid v\), where \(\mathcal {Z}\left( f_w\right) \) denotes the Zariski closure of \(Z\left( f_w\right) \) in \(\mathbb {P}^{d-1}_{\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }}\). \(\square \)
Lemma 6.3
For all \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \), the function \(\psi _{m,v}\) is \(\mathcal {K}_v\)-invariant and 1 on \(\mathcal {K}_v\).
Proof
For \(v \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) the result is trivial, so let \(v \not \in S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) and \(w \mid v\). Since \(f_w\left( x \cdot y\right) = f_w\left( x\right) f_w\left( y\right) \) for \(x,y \in L\), it follows as in the proof of Lemma 4.13 that \(\widetilde{H}_w\left( x \cdot y\right) \le \widetilde{H}_w\left( x\right) \widetilde{H}_w\left( y\right) \) for all \(x,y \in T\left( K_v\right) \). Since \(H_v' = \prod _{w \mid v}\widetilde{H}_w\), we have \(\widetilde{H}_w\left( \mathcal {K}_v\right) = 1\) for all \(w \mid v\), hence it follows as in the proof of Corollary 4.15 that \(\widetilde{H}_w\) and \(\psi _{m,w}\) are \(\mathcal {K}_v\)-invariant. Since \(\psi _{m,v}\left( 1\right) = 1\), we deduce that \(\psi _{m,v}\left( \mathcal {K}_v\right) = 1\). \(\square \)
Proposition 6.4
Given \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) and \(t_v \in T\left( K_v\right) \), the image of \(t_v\) in \(X_*\left( T_v\right) \) is
with \(n_w\) defined as in Proposition 3.4.
Proof
Follows from Proposition 3.4. \(\square \)
Corollary 6.5
For \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) non-archimedean with \(q_v\) sufficiently large, \(\chi \) an automorphic character of T unramified at v and \(s \in \mathbb {C}\) with \({\text {Re}}s > 0\), we have
Proof
Since \(\chi _v\), \(H_v\) and \(\psi _{m,v}\) are \(T\left( \mathcal {O}_v\right) \)-invariant and \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \), we have
where
Put \(d_w = {{\,\mathrm{deg}\,}}f_w\) and let \(n_v = \sum _{w \mid v} \alpha _w n_w \in X_*\left( T_v\right) \) with \(\min _w\{\alpha _w\} = 0\). By Proposition 3.4 and Note 3.1, we have
In particular, \(\psi _{m,v}\left( n_v\right) = 0\) whenever \(q_v \le H_v\left( n_v\right) \le q_v^{m-1}\), hence \(\gamma _{\chi ,v,r} = 0\) for \(1 \le r \le m-1\). Further, we see that \(\psi _{m,v}\left( n_v\right) =1\) and \(H_v\left( n_v\right) = q_v^m\) if and only if there is exactly one place \(w_0 \mid v\) such that \(\alpha _{w_0} = \frac{m}{d_{w_0}}\) and \(\alpha _w = 0\) for \(w \ne w_0\), so
Since \(|\psi _{m,v}\left( n_v\right) \chi _v\left( n_v\right) | \le 1\), we deduce that
Analogously to the proof of Corollary 5.15, we deduce for \(q_v\) sufficiently large that
\(\square \)
Proposition 6.6
For all places \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) with \(q_v\) sufficiently large, we have
Proof
Let \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) 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 \(w \mid v\), so Corollary 6.5 gives
and so we deduce the equality, since
Now let v have inertia degree \(d_v > 1\) in E/K. Then \({{\,\mathrm{deg}\,}}f_w = d_v \mid d\) for all \(w \mid v\). If d and m are coprime, then \(d_v \not \mid m\), hence \(\gamma _{\chi ,v,m} = 0\) and the result follows from
If d is prime, then v is inert, so \(T\left( \mathcal {O}_v\right) = T\left( K_v\right) \), \(\widehat{H}_v\left( \psi _{m,v},\chi _v;-s\right) = 1\), and
\(\square \)
Proposition 6.7
For any \(\chi \in \mathcal {U}\), we have
where \(\widetilde{G}_{m,\chi }\left( s\right) \) is a function which is holomorphic on \({\text {Re}}s \ge \frac{1}{m}\), \(\widetilde{G}_{m,1}\left( \frac{1}{m}\right) \ne 0\) and \(\widehat{H} \left( \psi _m,1;-s\right) \) has a simple pole at \(s = \frac{1}{m}\).
Proof
Defining \(\widetilde{G}_{m,\chi ,v}\left( s\right) = \frac{\widehat{H}_v \left( \psi _{m,v},\chi _v;-s\right) }{L_v\left( \chi ^m,ms\right) }\) for each place \(v \not \mid \infty \), it follows as in the proof of Corollary 5.15 that \(\widetilde{G}_{m,\chi ,v}\left( s\right) \) is holomorphic and bounded uniformly in terms of \(\epsilon \) and v on \({\text {Re}}s \ge \epsilon \) for all \(\epsilon > 0\). Since Proposition 6.6 gives
it follows as in the proof of Corollary 5.15 that \(\widetilde{G}_{m,\chi }\left( s\right) \) is holomorphic and uniformly bounded with respect to \(\chi \) for \({\text {Re}}s \ge \frac{1}{m}\) with \(\widetilde{G}_{m,1}\left( \frac{1}{m}\right) \ne 0\). Then, since
we conclude from Theorem 2.20 that \(\widehat{H} \left( \psi _m,1;-s\right) \) has a simple pole at \(s = \frac{1}{m}\). \(\square \)
Definition 6.8
For \({\text {Re}}s \gg 0 \), define the functions
The proofs of the following two results are analogous to those of their weak Campana counterparts, namely Theorem 5.18 and Lemma 5.21 respectively.
Theorem 6.9
The function \(\widetilde{\Omega }_m\left( s\right) \) admits a holomorphic extension to \({\text {Re}}s \ge \frac{1}{m}\).
Lemma 6.10
The characters \(\chi \in \mathcal {U}\) contributing to the simple pole of \(\widetilde{Z}_m\left( s\right) \) at \(s = \frac{1}{m}\) are precisely the characters \(\chi \in \mathcal {U}[m]\) such that \(\widetilde{G}_{m,\chi }\left( \frac{1}{m}\right) \ne 0\).
Proposition 6.11
The limit
is non-zero.
Proof
By the same reasoning as in the proof of Proposition 5.22, we have
where
Now, take \(\chi \in \mathcal {U}[m]\), \(v \not \in S'\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) non-archimedean. If \(\sigma _v\left( t_v\right) = 1\) for some \(t_v \in T\left( K_v\right) \), then the image of \(t_v\) in \(X_*\left( T_v\right) \) is of the form \(\sum _{w \mid v} \alpha _w n_w\), where each \(\alpha _w\) is either 0 or \(\frac{m}{d_v}\) for \(d_v\) the common inertia degree of the places of E over v, so
since each \(d_v \alpha _w\) is 0 or m and \(\chi ^0 = \chi ^m = 1\). Then \(\chi _v\left( t_v\right) = 1\) for all \(\chi \in \mathcal {U}[m]\). In particular, we deduce that \(T\left( \mathbb {A}_K\right) ^{\sigma _m} \subset T\left( \mathbb {A}_K\right) ^{\mathcal {U}[m], \psi _m}\), where
Then it suffices to prove that
Analogously to the proof of Proposition 6.7, we may deduce that
for \(\widetilde{G}_m\left( s\right) \) a function holomorphic on \({\text {Re}}s \ge \frac{1}{m}\) with \(\widetilde{G}_m\left( \frac{1}{m}\right) \ne 0\), so the result follows. \(\square \)
Proof of Theorem 1.4
Since \(\widetilde{\Omega }_m\left( \frac{1}{m}\right) \ne 0\) by Proposition 6.11, the result for \(S = S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) now follows from [3, Thm. 3.3.2, p. 624] and Theorem 6.9, taking
The result for \(S \supset S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \) follows analogously upon redefining \(\psi _{m,v}\) to be identically 1 for each \(v \in S \setminus S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) \). \(\square \)
7 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 \(\left( X,D_{\epsilon }\right) \) be a smooth Campana orbifold over a number field K which is klt (i.e. \(\epsilon _{\alpha } < 1\) for all \(\alpha \in \mathcal {A}\)) and Fano (i.e. \(-\left( K_X + D_{\epsilon }\right) \) is ample). Let \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \) be a regular \(\mathcal {O}_{K,S}\)-model of \(\left( X,D_{\epsilon }\right) \) for some finite set \(S \subset {{\,\mathrm{Val}\,}}\left( K\right) \) containing \(S_{\infty }\) (i.e. \(\mathcal {X}\) regular over \(\mathcal {O}_{K,S}\)). Let \(\mathcal {L} = \left( L,\Vert \cdot \Vert \right) \) be an adelically metrised big line bundle with associated height function \(H_{\mathcal {L}}: X\left( K\right) \rightarrow \mathbb {R}_{>0}\) (see [25, Sect. 1.3]). For any subset \(U \subset X\left( K\right) \) and any \(B \in \mathbb {R}_{>0}\), we define
Definition 7.1
Let V be a variety over a field k of characteristic zero, and let \(A \subset V\left( k\right) \). We say that A is of type I if there is a proper Zariski closed subset \(W \subset V\) with \(A \subset W\left( k\right) \). We say that A is of type II if there is a normal geometrically irreducible variety \(V'\) with \(\text {dim}V' = \text {dim}V\) and a finite surjective morphism \(\phi : V' \rightarrow V\) of degree \(\ge 2\) with \(A \subset \phi \left( V'\left( k\right) \right) \). We say that A is thin if it is contained in a finite union of subsets of \(V\left( k\right) \) of types I and II.
We are now ready to give the statement of the conjecture.
Conjecture 7.2
[27, Conj. 1.1, p. 3] Suppose that L is nef and \(\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \left( \mathcal {O}_{K,S}\right) \) is not thin. Then there exists a thin set \(Z \subset \left( \mathcal {X},\mathcal {D}_{\epsilon }\right) \left( \mathcal {O}_{K,S}\right) \) and explicit positive constants \(a = a\left( \left( X,D_{\epsilon }\right) ,L\right) \), \(b = b\left( K,\left( X,D_{\epsilon }\right) ,L\right) \) and \(c = c\left( K,S,\left( \mathcal {X},\mathcal {D}_{\epsilon }\right) ,\mathcal {L},Z\right) \) such that, as \(B \rightarrow \infty \), we have
7.2 Interpretation for norm orbifolds
The orbifold \(\left( \mathbb {P}^{d-1}_K,\Delta ^{{{\,\mathrm{\varvec{\omega }}\,}}}_m\right) \) in Theorem 1.4 is klt and Fano. It is smooth precisely when \(d=2\). The \(\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }\)-model \(\left( \mathbb {P}^{d-1}_{\mathcal {O}_{K,S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) }},\mathcal {D}_m^{{{\,\mathrm{\varvec{\omega }}\,}}}\right) \) is regular. The Batyrev–Tschinkel height arises from an adelic metrisation \(\mathcal {L}\) of \(-K_{\mathbb {P}^{d-1}} = \mathcal {O}\left( d\right) \). According to [27, Sect. 3.3], we have
where
Here, \(\tau _{\mathbb {P}^{d-1}}\) is the Tamagawa measure defined in [19, Def. 2.8, p. 372], and \(\overline{\mathbb {P}^{d-1}\left( K\right) }_\epsilon \) denotes the topological closure of the Campana points inside \(\mathbb {P}^{d-1}\left( \mathbb {A}_K\right) \). If one assumes that weak approximation for Campana points holds for this orbifold (see [27, Question 3.9, p. 13]), it follows from the definition of \(\tau _{\mathbb {P}^{d-1}}\) (cf. [27, Sect. 9]) that
Given the assumption on weak approximation, the conjectural leading constant is
On the other hand, the leading constant given by Theorem 1.4 in this case is
We observe that our constant differs from the conjectural one in the potential inclusion of non-trivial characters in the limit.
7.3 The quadratic case
We now consider the case \(d=2\), in which the orbifold in Theorem 1.4 is smooth. Here, work of Nakahara and the author [21] shows that weak approximation for Campana points holds for any \(m \in \mathbb {Z}_{\ge 2}\).
In Theorem 1.4, it is not clear that there are non-trivial characters contributing to the leading constant and whether their contribution is positive if so. However, we now exhibit an extension for which a non-trivial character contributes positively to the leading constant, and all contributing characters do so positively.
Proposition 7.3
Let \(K = \mathbb {Q}\left( \sqrt{-39}\right) \), \(E = \mathbb {Q}\left( \sqrt{-3},\sqrt{13}\right) \) and \(m=2\). Choose the K-basis \({{\,\mathrm{\varvec{\omega }}\,}}= \{1,\frac{1+\sqrt{-3}}{2}\}\) of E. Then \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) = S_{\infty }\), \(\#\mathcal {U}[2] > 1\), and for every \(\chi \in \mathcal {U}[2]\), we have \(\lim _{s\rightarrow \frac{1}{2}}\left( s-\frac{1}{2}\right) \widehat{H}\left( \psi _2,\chi ;-s\right) > 0\).
Proof
Writing \(a \cdot 1 + b \cdot \frac{1 + \sqrt{-3}}{2}\) as \(\left( a,b\right) \) and \(G = {{\,\mathrm{Gal}\,}}\left( E/K\right) \) as \(\{1_G,g\}\), we have
and clearly \(1 = \left( 1,0\right) \), hence \(S\left( {{\,\mathrm{\varvec{\omega }}\,}}\right) = S_{\infty }\). Note that \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x,y\right) = x^2 + xy + y^2\).
Since \({{\,\mathrm{Cl}\,}}\left( E\right) \cong \mathbb {Z}/2\mathbb {Z}\), the Hilbert class field M of E is quadratic over E. We obtain the unramified Hecke character \(\chi _M\) of E, which is defined for all split \(w \in {{\,\mathrm{Val}\,}}\left( E\right) \) by \(\chi _{M,w}\left( \pi _w\right) = -1\) and is trivial at all other places. Since \(\chi _M\) is trivial on \(\mathbb {A}_K^*\), it may be viewed inside \(\mathcal {U}[2]\), hence \(\#\mathcal {U}[2] > 1\).
Let \(\chi \in \mathcal {U}[2]\). To show that \(\lim _{s\rightarrow \frac{1}{2}}\left( s-\frac{1}{2}\right) \widehat{H}\left( \psi _2,\chi ;-s\right) > 0\), it suffices to show that \(\widehat{H}_v\left( \psi _{2,v},\chi _v;-\frac{1}{2}\right) > 0\) for all \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \). If \(v \mid \infty \), then \(\widehat{H}_v\left( \psi _{2,v},\chi _v;-\frac{1}{2}\right) = \widehat{H}_v\left( 1,1;-\frac{1}{2}\right) > 0\), as \(\chi _v\) gives a continuous homomorphism from \(T\left( K_v\right) /T\left( \mathcal {O}_v\right) \cong \mathbb {R}_{>0}\) to \(\mu _2\), and \(\mathbb {R}_{>0}\) has no proper open subgroups. If v is inert, then we have \(\widehat{H}_v\left( \psi _{2,v},\chi _v;-\frac{1}{2}\right) = 1\). If v is not inert, then \(N_{{{\,\mathrm{\varvec{\omega }}\,}}}\left( x,y\right) = \left( x+\theta _1 y\right) \left( x+\theta _2 y\right) \) for \(\theta _1,\theta _2 \in \mathcal {O}_v\) roots of \(z^2 - z + 1\). By Proposition 6.4, we have \(H_v = H_v'\) if and only if there are no \(\left( a,b\right) \in \left( K_v^*\right) ^2\) with \(\min \{v\left( a\right) ,v\left( b\right) \} = 0\) such that \(v\left( a+\theta _1 b\right) , v\left( a + \theta _2 b\right) \ge 1\). If \(v\left( a+\theta _1 b\right) , v\left( a + \theta _2 b\right) \ge 1\), then we deduce from the equalities
that \(v\left( a\right) , v\left( b\right) \ge 1 - v\left( \theta _1 - \theta _2\right) \). Since \(\min \{v\left( a\right) ,v\left( b\right) \} = 0\), we have \(H_v' \ne H_v\) if and only if \(v\left( \theta _1 - \theta _2\right) \ge 1\). Since \(\left( \theta _1 - \theta _2\right) ^2 = -3\), the only such place is the unique place \(v_0\) of K above 3, and \(v_0\left( \theta _1 - \theta _2\right) = 1\).
For any split place \(v \ne v_0\), we have \(\mathcal {K}_v = T\left( \mathcal {O}_v\right) \) and \(\psi _{2,v} = \phi _{2,v}\), so
by Proposition 5.7. In fact, for \(w_1\) and \(w_2\) the places of E over v, we have \(\chi _{w_2}\left( \pi _{w_2}\right) = \chi _{w_1}\left( \pi _{w_1}\right) ^{-1} \in \{1,-1\}\), hence \(c_{\chi ,v,n} - c_{\chi ,v,n-2} = 2\chi _{w_1}\left( \pi _{w_1}\right) ^n\), so
It only remains to check that \(\widehat{H}_{v_0}\left( \psi _{2,{v_0}},\chi _{v_0};-\frac{1}{2}\right) > 0\). We will make use of the following property of valuations:
Assume that, for \(a,b \in \left( K_{v_0}^*\right) ^2\) as above, we have \(v_0\left( a+\theta _2 b\right) \ge 2\). We claim that \(v_0\left( a+\theta _1 b\right) = 1\). First, we deduce from (7.1) that
with equality if \(v_0\left( a + \theta _1 b\right) \ne v_0\left( \left( \theta _1 - \theta _2\right) b\right) \). Since \(v_0\left( \left( 2\theta _i - 1\right) ^2\right) = v_0\left( -3\right) = 2\), we have \(v_0\left( 2\theta _i - 1 \right) = 1\), so \(v_0 \left( \theta _i\right) = v_0\left( 2\theta _i\right) = v_0\left( 1\right) =0\) by (7.1), hence \(\min \{v_0\left( a\right) ,v_0\left( \theta _2 b\right) \} = \min \{v_0\left( a\right) ,v_0\left( b\right) \} = 0\). Then, since \(v_0\left( a+\theta _2 b\right) \ge 2\), it follows that \(v_0\left( a\right) = v_0\left( \theta _2 b\right) \). We deduce that \(v_0\left( a\right) = v_0\left( \theta _2 b\right) = v_0\left( b\right) \), so \(v_0\left( a\right) = v_0\left( b\right) = \min \{v_0\left( a\right) ,v_0\left( b\right) \} = 0\). Since \(v_0\left( \theta _1 - \theta _2\right) = 1\), we have \(v_0\left( \left( \theta _1 - \theta _2\right) b\right) = 1\). It follows that \(v_0\left( a + \theta _1 b\right) = 1\) by (7.1).
We deduce that \(\psi _{2,v_0}\left( t_{v_0}\right) = 1\) if and only if \(t_{v_0} \in \mathcal {K}_{v_0}\), hence
positivity follows since \(\mathcal {K}_v \subset T\left( \mathcal {O}_v\right) \) is of finite index for all \(v \in {{\,\mathrm{Val}\,}}\left( K\right) \). \(\square \)
7.4 Possible thin sets
Assuming the truth of Conjecture 7.2, the question arises of which thin set Z should be removed in the setting of Proposition 7.3. Informally, its removal should remove the contribution of all non-trivial characters \(\chi \in \mathcal {U}[2]\). One might therefore postulate that, for each non-trivial character \(\chi \in \mathcal {U}[2]\), there is a finite morphism \(\varphi _{\chi }: C_{\chi } \rightarrow \mathbb {P}^1_K\), where \(C_{\chi }\) is a smooth projective curve, and \(Z = \bigcup _{\chi \in \mathcal {U}[2]}\varphi _{\chi }\left( C_\chi \left( K\right) \right) \). By the height bounds in [30, Sect. 9.7], we would have \(C_\chi \cong \mathbb {P}^1_K\) and \({{\,\mathrm{deg}\,}}\left( \varphi _\chi \right) =2\), making the morphisms \(\varphi _{\chi }\) degree-two endomorphisms of \(\mathbb {P}^1_K\). However, it is not clear how one should construct such endomorphisms. This may be an interesting direction to pursue in future work.
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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.
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Streeter, S. Campana points and powerful values of norm forms. Math. Z. 301, 627–664 (2022). https://doi.org/10.1007/s00209-021-02922-4
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DOI: https://doi.org/10.1007/s00209-021-02922-4