Manin’s conjecture for toric varieties

Victor V. Batyrev^{∗}

Universität-GHS-Essen, Fachbereich 6, Mathematik Universitätsstr. 3, 45141 Essen, Germany e-mail: victor.batyrev@aixrs1.hrz.uni-essen.de

and Yuri Tschinkel

Dept. of Mathematics, U.I.C. Chicago, (IL) 60608, U.S.A. e-mail: yuri@math.uic.edu

Abstract

We prove an asymptotic formula conjectured by Manin for the number of K-rational points of bounded height with respect to the anticanonical line bundle for arbitrary smooth projective toric varieties over a number field K.

∗Supported by Deutsche Forschungsgemeinschaft

Contents

1 Algebraic tori and toric varieties 6

2 Tamagawa numbers 11

3 Heights and their Fourier transforms 15

4 Poisson formula 19

5 X -functions of convex cones 23

6 Some technical statements 27

7 Main theorem 38

Introduction

Let X be a projective algebraic variety defined over a number field K, X(K) the set of K-rational points of X. We are interested in relations between the geometry of X and diophantine properties of X(K) in the situation when X(K) is infinite. The main object of our study is the height function on X(K) with respect to a metrized line bundle L. A metrized line bundle is a pair L = (L, ∥ · ∥_{v}), consisting of a line bundle L equipped with a family {∥ · ∥_{v}} of v-adic metrics (v runs over the set Val(K) of all valuation of K), satisfying certain conditions. This defines a height function H_{L} : X(K) → R_{>0} by

H_{L}(x) := ∏_{v∈Val(K)} ∥f(x)∥_{v}^{−1}

where f is a K-rational local section of the line bundle L not vanishing in x ∈ X(K).

Let Pic(X) be the Picard group of X and Λ_{eff} ⊂ Pic(X) the cone of effective divisors. Assume that the class of L is contained in the interior of the cone of effective divisors Λ_{eff} ⊂ Pic(X)_{R}. In this case, some positive tensor power of L defines a birational map of X into some projective space.

We denote by U_{L} ⊂ X the Zariski open subset such that the restriction of the above birational map to U_{L} is an isomorphism on its image.

For any Zariski open subset U ⊂ U_{L} consider the height zeta-function defined by the following series [1, 9]:

Z_{L,U}(s) = ∑_{x∈U(K)} H_{L}(x)^{−s}.

Then Z_{L,U}(s) converges absolutely and uniformly for Re(s) ≫ 0. A Tauberian theorem relates the analytic properties of the zeta-function with the asymptotic behaviour of the number N(U, L, B) of K-rational points x ∈ U(K) with H_{L}(x) ≤ B as B → ∞.

We define

α(L, U) = inf{a ∈ R | Z_{L,U}(s) converges for Re(s) > a}.

We say that U does not contain L-accumulating subvarieties if for any nonempty Zariski open subset U′ ⊂ U we have α(L, U) = α(L, U′).

One expects a good accordance between the geometry of X and diophantine properties of the set of K-rational points of X which are contained in the complement U ⊂ X to some proper closed subvarieties in X. Denote by K^{−1} the metrized anicanonical line bundle. The following conjecture is due to Yu. I. Manin [9]:

Let X be a smooth projective variety over a number field K whose anticanonical line bundle is ample (i.e., X is a Fano variety). Assume that the set X(K) of K-rational points is Zariski dense. Let U ⊂ X be the largest Zariski open subset which doesn’t contain K^{−1}-accumulating subvarieties. Then

N(U, K^{−1}, B) = c(X, K^{−1}, K)B(log B)^{k−1}(1 + o(1)) for B → ∞

where k equals the rank of the Picard group Pic(X) over K and c(X, K^{−1}, K) is some positive constant which depends on X, K and the choice of v-adic metrics on the anticanonical line bundle.

The above conjecture was refined by E. Peyre [21] who defined Tamagawa numbers of Fano varieties and proposed an interpretation of c(X, K^{−1}, K) in terms of these numbers.

Unfortunately, the conjecture of Manin is not true in general (see [3]). It has been proved for some class of Fano varieties, e.g., for generalized flag varieties, complete intersections of small degree and for some blow ups of projective spaces [9, 1, 21].

In this paper we prove Manin’s conjecture and compute the constant c(X, K^{−1}, K) for arbitrary smooth projective equivariant compactifications of algebraic tori over number fields, i.e., for toric varieties P_{Σ} associated with a Galois invariant finite polyhedral fan Σ [27]. We restrict ourselves to the case when U is the dense torus orbit. It is easy to show that U doesn’t contain L-accumulating subvarieties for any metrized line bundle L. On the other hand, it might happen that there is some larger Zariski open subset U′ ⊂ P_{Σ} which contains U and which does not contain L-accumulating subvarieties. It is possible to show that the the asymptotic formula for N(U′, K^{−1}, B) does not depend on the choice of such a Zariski open U′, but we decided to postpone the proof of this fact.

One of our main ideas for the computation of the height zeta-function on a toric variety P_{Σ} is to introduce some canonical simultaneous metrizations on all line bundles and to obtain a pairing

H_{Σ}(x, s) : T (K) × Pic(P_{Σ})_{C} → C

between the set of rational points T (K) ⊂ P_{Σ}(K) in the Zariski open subset T and the complexified Picard group, extending the usual height pairing between T (K) and Pic(P_{Σ}). This allows to extend the one-parameter zeta-function to a function Z_{Σ}(s) defined on the complexified Picard group Pic(P_{Σ})_{C} and holomorphic when the Re(s) is contained in the interior of the cone [K^{−1}] + Λ_{eff}, where Λ_{eff} ⊂ Pic(P_{Σ})_{R} is the cone of effective divisors of P_{Σ}.

The second step is to use the multiplicative group structure on the torus T ⊂ P_{Σ}. With our choice of metrics, the height zeta-function becomes a function on the adelic group T (A_{K}) invariant under the closed subgroup T (K)K_{T}, where K_{T} ⊂ T (A_{K}) is the maximal compact subgroup. The key idea is to use the Poisson formula in order to obtain an integral representation for Z_{Σ}(s).

Our third step is the study of analytic properties of Z_{Σ}(s) using the above integral formula and properties of X -functions of convex cones.

In section 1 we introduce notations and basic notions from the theory of toric varieties over non-closed fields.

In section 2 we recall the definitions of Tamagawa numbers of algebraic tori and Tamagawa numbers of algebraic varieties with metrized anticanonical bundle.

In section 3 we define simultaneous metrizations of all line bundles on toric varieties, introduce the height zeta-function and give formulas for local Fourier transforms of heights.

In section 4 we prove the Poisson formula which yields an integral representation of the height zeta-function.

In section 5 we formulate basic properties of X -functions of convex finitely generated polyhedral cones.

In section 6 we prepare the necessary analytic tools.

And finally, in section 7 we prove our main theorem:

Let P_{Σ} be a smooth projective compactification of an algebraic torus T over K. Let k be the rank of Pic(P_{Σ}). Then there is only a finite number N(T, K^{−1}, B) of K-rational points x ∈ T (K) having the anticanonical height H_{K^{−1}}(x) ≤ B. Moreover,

N(T, K^{−1}, B) = \frac{Θ(Σ)}{(k − 1)!} · B(log B)^{k−1}(1 + o(1)), B → ∞,

with the constant Θ(Σ) = α(P_{Σ})β(P_{Σ})τ_{K}(P_{Σ}), where:

1. α(P_{Σ}) is a constant defined by the geometry of the cone of effective divisors Λ_{eff} ⊂ Pic(P_{Σ})_{R};

2. β(P_{Σ}) is the order of the non-trivial part of the Brauer group of P_{Σ};

3. τ_{K}(P_{Σ}) is the Tamagawa number associated with the metrized canonical sheaf on P_{Σ} (as it was defined by E.Peyre in [21]).

Our results provide first examples for asymptotics on unirational varieties which are not rational and on varieties without weak approximation, in general. A new phenomenon is the appearance of the non-trivial part of the Brauer group Br(P_{Σ})/Br(K) in the asymptotic constant. We don’t need to assume that the anticanonical class of P_{Σ} is ample (i.e., P_{Σ} is a toric Fano variety).

This paper is a continuation of our paper [2], where we proved the conjecture of Manin about the distribution of K-rational points of bounded height for the case of projective compactifications of anisotropic tori. An equvariant compactification P_{Σ} of an anisotropic torus T is much simpler in many aspects: the cone of effective divisors Λ_{eff} is always simplicial, all K-rational points of P_{Σ} are contained in T, and the group T (A_{K})/T (K) is compact (this last property significantly simplifies the Poisson formula).

Acknowledgements. We would like to thank our teacher Yu.I. Manin for many years of encouragement and support. We benefited from conversations with W. Hoffmann and B. Z. Moroz. We are grateful to P. Salberger and M. Strauch for their careful reading of our maniscript and for their valuable comments.

The work of the second author was supported by a Junior Fellowship from the Harvard Society of Fellows. The paper was completed while he was visiting the Max-Planck-Arbeitsgruppe Zahlentheorie in Berlin, GHS-Essen and IHES. He would like to thank these institutions for hospitality and perfect working conditions.

1 Algebraic tori and toric varieties

Let X_{K} be an algebraic variety defined over a number field K and E/K a finite extension of number fields. We will denote the set of E-rational points of X_{K} by X(E) and by X_{E} the E-variety obtained from X_{K} by base change. We sometimes omit the subscript in X_{E} if the respective field of definition is clear from the context. Let G_{m,E} = Spec(E[x, x^{−1}]) be the multiplicative group scheme over E.

Definition 1.1 A linear algebraic group T_{K} is called a d-dimensional algebraic torus if there exists a finite extension E/K such that T_{E} is isomorphic to (G_{m,E})^{d}. The field E is called the splitting field of T.

For any field E we denote by T̂_{E} = Hom (T, E^{∗}) the group of regular E-rational characters of T.

Theorem 1.2 [8, 19, 26] There is a contravariant equivalence between the category of algebraic tori defined over a number field K and the category of torsion free Gal(E/K)-modules of finite rank over Z. The functors are given by

M → T = Spec(K[M]); T → T̂_{E}.

The above contravariant equivalence is functorial under field extensions of K.

Let Val(K) be the set of all valuations of a global field K. Denote by S_{∞} the set of archimedian valuations of K. For any v ∈ Val(K), we denote by K_{v} the completion of K with respect to v. Let E be a finite Galois extension of K with the Galois group G. Let V be an extension of v to E, E_{V} the completion of E with respect to V. Then

Gal(E_{V}/K_{v}) ≅ G_{v} ⊂ G,

where G_{v} is the decomposition subgroup of G and K_{v} ⊗_{K} E ≅ ∏_{V|v} E_{V}. Let T be an algebraic torus over K with the splitting field E. Denote by T (K_{v}) = the v-adic completion of T (K) and by T (O_{v}) ⊂ T (K_{v}) its maximal compact subgroup.

Definition 1.3 Denote by T (A_{K}) the adele group of T. Define

T^{1}(A_{K}) = {t ∈ T (A_{K}) : ∏_{v∈Val(K)} | m(t_{v}) |_{v}= 1, for all m ∈ T̂_{K} ⊂ M}.

Let

K_{T} = ∏_{v∈Val(K)} T (O_{v}),

be the maximal compact subgroup of T (A_{K}).

Proposition 1.4 [19] The groups T (A_{K}), T^{1}(A_{K}), T (K), K_{T} have the following properties:

(i) T (A_{K})/T^{1}(A_{K}) ≅ R^{t}, where t is the rank of T̂_{K};

(ii) T^{1}(A_{K})/T (K) is compact;

(iii) T^{1}(A_{K})/T (K) · K_{T} is isomorphic to the direct product of a finite group cl(T_{K}) and a connected compact abelian topological group which dimension equals the rank r′ of the group of O_{K}-units in T (K);

(iv) W (T) = K_{T} ∩T (K) is a finite group of all torsion elements in T (K).

Definition 1.5 We define the following cohomological invariants of the algebraic torus T:

h(T) = Card[H^{1}(G, M)],
III(T) = Ker [H^{1}(G, T (E)) → H^{1}(G_{v}, T (A_{E}))],
i(T) = Card[III(T)].

Definition 1.6 Let \overline{T (K)} be the closure of T (K) in T (A_{K}) in the direct product topology. Define the obstruction group to weak approximation as

A(T) = T (A_{K})/\overline{T (K)}.

Remark 1.7 It is known that over the splitting field E one has A(T_{E}) = 0.

Let T_{K} be a d-dimensional algebraic torus over K with splitting field E and G = Gal(E/K). Denote by M the lattice T̂_{E} and by N = Hom(M, Z) the dual abelian group. Let us recall standard facts about toric varieties over arbitrary fields [5, 6, 10, 18, 2].

Definition 1.8 A finite set Σ consisting of convex rational polyhedral cones in N_{R} = N ⊗ R is called a d-dimensional fan if the following conditions are satisfied:

(i) every cone σ ∈ Σ contains 0 ∈ N_{R};

(ii) every face σ′ of a cone σ ∈ Σ belongs to Σ;

(iii) the intersection of any two cones in Σ is a face of both cones.

Definition 1.9 A d-dimensional fan Σ is called complete and regular if the following additional conditions are satisfied:

(i) N_{R} is the union of cones from Σ;

(ii) every cone σ ∈ Σ is generated by a part of a Z-basis of N. We denote by Σ(j) the set of all j-dimensional cones in Σ. For each cone σ ∈ Σ we denote by N_{σ,R} the minimal linear subspace containing σ.

Theorem 1.10 A complete regular d-dimensional fan Σ defines a smooth equivariant compactification P_{Σ,E} of the E-split algebraic torus T_{E}. The toric variety P_{Σ,E} has the following properties:

(i) There is a T_{E}-invariant open covering by affine subsets U_{σ,E}:

P_{Σ,E} = ⋃_{σ∈Σ} U_{σ,E}.

The affine subsets are defined as U_{σ,E} = Spec(E[M ∩σ̌]), where σ̌ is the cone in M_{R} which is dual to σ.

(ii) There is a representation of P_{Σ,E} as a disjoint union of split algebraic tori T_{σ,E} of dimension dim T_{σ,E} = d − dim σ, that is, for all fields F/E we have

P_{Σ,E}(F) = ⋃_{σ∈Σ} T_{σ,E}(F).

For each j-dimensional cone σ ∈ Σ(j) we denote by T_{σ,E} the kernel of a homomorphism T_{E} → (G_{m,E})^{j} defined by a Z-basis of the sublattice N ∩ N_{σ,R} ⊂ N.

To construct compactifications of non-split tori T_{K} over K, we need a complete fan Σ of cones having an additional combinatorial structure: an action of the Galois group G = Gal(E/K) [27]. The lattice M = T̂_{E} is a G-module and we have a representation ρ : G → Aut(M). Denote by ρ^{∗} the induced dual representation of G in Aut(N) ≅ GL(d, Z).

Definition 1.11 A complete fan Σ ⊂ N_{R} is called G-invariant if for any g ∈ G and for any σ ∈ Σ, one has ρ^{∗}(g)(σ) ∈ Σ. Let N^{G} (resp. M^{G}, NR^{G}, MR^{G} and Σ^{G}) be the subset of G-invariant elements in N (resp. in M, N ⊗R, M ⊗R and Σ). Denote by Σ_{G} ⊂ NR^{G} the fan consisting of all possible intersections σ ∩ NR^{G} where σ runs over all cones in Σ.

The following theorem is due to Voskresenskiî [27]:

Theorem 1.12 Let Σ be a complete regular G-invariant fan in N_{R}. Assume that the complete toric variety P_{Σ,E} defined over the splitting field E by the G-invariant fan Σ is projective. Then there exists a unique complete algebraic variety P_{Σ,K} over K such that its base extension P_{Σ,K} ⊗_{Spec(K)} Spec(E) is isomorphic to the toric variety P_{Σ,E}. The above isomorphism respects the natural G-actions on P_{Σ,K} ⊗_{Spec(K)} Spec(E) and P_{Σ,E}.

Remark 1.13 Our definition of heights and the proof of the analytic properties of height zeta functions do not use the projectivity of respective toric varieties. We note that there exist non-projective compactifications of split algebraic tori. We omit the technical discussion of non-projective compactifications of non-split tori.

We proceed to describe the algebraic geometric structure of the variety P_{Σ,K} in terms of the fan with Galois-action. Let Pic(P_{Σ,K}) be the Picard group and Λ_{eff} the cone in Pic(P_{Σ,K}) generated by classes of effective divisors. Let K be the canonical line bundle of P_{Σ,K}.

Definition 1.14 A continuous function φ : N_{R} → R is called Σ-piecewise linear if its restriction to every cone σ ∈ Σ is a linear function; i.e., for every cone σ ∈ Σ there exists an element m_{σ} ∈ M_{R} such that φ|_{σ}(x) =< x, m_{σ} > where < ∗, ∗ >: N_{R} × M_{R} → R is the pairing induced from the duality between N and M. It is called integral if φ(N) ⊂ Z. Denote the group of Σ-piecewise linear integral functions by P L(Σ).

We see that the G-action on M (and N) induces a G-action on the free abelian group P L(Σ). Denote by e_{1}, …, e_{n} the primitive integral generators of all 1-dimensional cones in Σ. A function φ ∈ P L(Σ) is determined by its values on e_{i}, (i = 1, ..., n). Let T_{i} be the (d − 1)-dimensional torus orbit corresponding to the cone R_{≥0}e_{i} ∈ Σ(1) and \overline{T}_{i} the Zariski closure of T_{i} in P_{Σ,E}.

Proposition 1.15 Let P_{Σ,K} be a smooth toric variety over K which is an equivariant compactification of an algebraic torus T_{K} with splitting field E and Σ the corresponding complete regular fan with G = Gal(E/K)-action. Then:

(i) There is an exact sequence

0 → M^{G} → P L(Σ)^{G} → Pic(P_{Σ,K}) → H^{1}(G, M) → 0.

(ii) Let

Σ(1) = Σ_{1}(1) ∪ ... ∪ Σ_{r}(1)

be the decomposition of Σ(1) into a union of G-orbits. The cone of effective divisors Λ_{eff} is generated by classes of G-invariant divisors

D_{j} = ∑_{R_{≥0}e_{i}∈Σ_{j}(1)} \overline{T}_{i} (j = 1, ..., r).

(iii) The class of the anticanonical line bundle K^{−1} ∈ Pic(P_{Σ,K}) is the class of the G-invariant piecewise linear function φ_{Σ} ∈ P L(Σ)^{G} given by φ_{Σ}(e_{j}) = 1 for all j = 1, ..., n.

Proof. (i) It is known that Pic(P_{Σ,K}) = (Pic(P_{Σ,E}))^{G} ([26] Prop. 4.40). It remains to take G-invariants in the standard short exact sequence

0 → M → P L(Σ) → Pic(P_{Σ,E}) → 0

describing the Picard group of a toric variety over a splitting field (see [10] 3.4) and notice that H^{1}(G, P L(Σ)) = 0, because P L(Σ) is a permutational G-module.

(ii) The statement was proved in [2] 1.3.

(iii) The statement is standard [5, 18]

Theorem 1.16 [26, 4] Let T be an algebraic torus over K with splitting field E. Let P_{Σ,K} be a complete smooth equivariant compactification of T. There is an exact sequence:

0 → A(T) → Hom(H^{1}(G, Pic(P_{Σ,E})), Q/Z) → III(T) → 0.

Remark 1.17 The group H^{1}(G, Pic(P_{Σ,E})) is canonically isomorphic to the non-trivial part of the Brauer group Br(P_{Σ,K})/Br(K), where Br(P_{Σ,K}) = H_{et}^{2}(P_{Σ,K}, G_{m}). This group appears as the obstruction group to the Hasse principle and weak approximation in [14, 4].

Corollary 1.18 Let β(P_{Σ}) be the cardinality of H^{1}(G, Pic(P_{Σ,E})). Then

Card[A(T)] = \frac{β(P_{Σ})}{i(T)}.

2 Tamagawa numbers

In this section we recall the definitions of Tamagawa numbers of tori following A. Weil [28] and of algebraic varieties with a metrized canonical line bundle following E. Peyre [21]. The constructions of Tamagawa numbers depend on a choice of a finite set of valuations S ⊂ Val(K) containing archimedian valuations and places of bad reduction, but the Tamagawa numbers themselves do not depend on S.

Let X be a smooth algebraic variety over K, X(K_{v}) the set of K_{v}-rational points of X. Then a choice of local analytic coordinates x_{1}, …, x_{d} on X(K_{v}) defines a homeomorphism ϕ : U → K_{v}^{d} in v-adic topology between an open subset U ⊂ X(K_{v}) and ϕ(U) ⊂ K_{v}^{d}. Let dx_{1} ⋯ dx_{d} be the Haar measure on K_{v}^{d} normalized by the condition

∫_{O_{v}^{d}} dx_{1} ⋯ dx_{d} = \frac{1}{(\sqrt{δ_{v}})^{d}}

where δ_{v} is the absolute different of K_{v}. Denote by dx_{1} ∧ ⋯ ∧ dx_{d} the standard differential form on K_{v}^{d}. Then f = ϕ^{∗}(dx_{1} ∧ ⋯ ∧ dx_{d}) is a local analytic section of the canonical sheaf K. If ∥ · ∥ is a v-adic metric on K, then we obtain the v-adic measure on U by the formula

∫_{U′} ω_{K,v} = ∫_{ϕ(U′)} ∥f(ϕ^{−1}(x))∥_{v}^{−1}dx_{1} ⋯ dx_{d},

where U′ is arbitrary open subset in U. The measure ω_{K,v} does not depend on the choice of local coordinates and extends to a global measure on X(K_{v}) [21].

Definition 2.1 [19] Let T be an algebraic torus defined over a number field K with splitting field E. Denote by

L_{S}(s, T; E/K) = ∏_{v∈Val(K)} L_{v}(s, T; E/K)

the Artin L-function corresponding to the representation

ρ : G = Gal(E/K) → Aut(T̂_{E})

and a finite set S ⊂ Val(K) containing all archimedian valuations and all non-archimedian valuations of K which are ramified in E. By definition, L_{v}(s, T; E/K) ≡ 1 if v ∈ S, L_{v}(s, T; E/K) = det(Id − q_{v}^{−s}F_{v})^{−1} if v ∉ S, where F_{v} ∈ Aut(T̂_{E}) is a representative of the Frobenius automorphism.

Let T be an algebraic torus of dimension d and Ω a T -invariant algebraic K-rational differential d-form. The form Ω defines an isomorphism of the canonical sheaf on T with the structure sheaf on T. Since the structure sheaf has a canonical metrization, using the above construction, we obtain a v-adic measure ω_{Ω,v} on T (K_{v}). Moreover, according to A. Weil [28], we have

∫_{T (O_{v})} ω_{Ω,v} = \frac{Card[T (k_{v})]}{q_{v}^{d}} = L_{v}(1, T; E/K)^{−1}

for all v ∉ S. We put dμ_{v} = L_{v}(1, T; E/K)ω_{Ω,v} for all v ∈ Val(K). Then the local measures dμ_{v} satisfy

∫_{T (O_{v})} dμ_{v} = 1

for all v ∉ S.

Definition 2.2 We define the canonical measure on the adele group T (A_{K})

ω_{Ω,S} = ∏_{v∈Val(K)} L_{v}(1, T; E/K)ω_{Ω,v} = ∏_{v∈Val(K)} dμ_{v}.

By the product formula, ω_{Ω,S} does not depend on the choice of Ω. Let dx be the standard Lebesgue measure on T (A_{K})/T^{1}(A_{K}). There exists a unique Haar measure ω_{Ω,S}^{1} on T^{1}(A_{K}) such that ω_{Ω,S}^{1}dx = ω_{Ω,S}.

We proceed to define Tamagawa measures on algebraic varieties following E. Peyre [21]. Let X be a smooth projective algebraic variety over K with a metrized canonical sheaf K. We assume that X satisfies the conditions h^{1}(X, O_{X}) = h^{2}(X, O_{X}) = 0. Under these assumptions, the Néron-Severi group NS(X) (or, equivalently, the Picard group Pic(X) modulo torsion) over the algebraic closure \overline{K} is a discrete continuous Gal(\overline{K}/K)-module of finite rank over Z. Denote by T_{NS} the corresponding torus under the duality from 1.2 and by E_{NS} a splitting field.

Definition 2.3 [21] The adelic Tamagawa measure ω_{K,S} on X(A_{K}) is defined by

ω_{K,S} = ∏_{v∈Val(K)} L_{v}(1, T_{NS}; E_{NS}/K)^{−1}ω_{K,v}.

Definition 2.4 Let t be the rank of the group of K-rational characters T̂_{K} of T. Then the Tamagawa number of T is defined as

τ(T) = \frac{b_{S}(T)}{l_{S}(T)}

where

b_{S}(T) = ∫_{T^{1}(A_{K})/T (K)} ω_{Ω,S}^{1},

l_{S}(T) = lim_{s→1}(s − 1)^{t}L_{S}(s, T; E/K).

Definition 2.5 [21] Let k be the rank of the Néron-Severi group of X over K, and \overline{X(K)} the closure of X(K) ⊂ X(A_{K}) in the direct product topology. Then the Tamagawa number of X is defined by

τ_{K}(X) = \frac{b_{S}(X)}{l_{S}(X)}

where

b_{S}(X) = ∫\overline{X(K)} ω_{K,S}

whenever the adelic integral converges, and

l_{S}^{−1}(X) = lim_{s→1}(s − 1)^{k}L_{S}(s, T_{NS}; E_{NS}/K).

Remark 2.6 Notice the difference in the choice of convergence factors for the Tamagawa measure associated with a metrized canonical line bundle on a complete algebraic variety X and for the classical Tamagawa measure on an algebraic torus T. In the first case, we choose L_{v}^{−1}(1, T_{NS}; E_{NS}/K) whereas in the second case one uses L_{v}(1, T; E/K). This explains the difference in the definitions of l_{S}(X) and l_{S}(T).

Remark 2.7 For a toric variety P_{Σ} ⊃ T one can take E_{NS} = E, where E is a splitting field of T.

Remark 2.8 It is clear that in both definitions the Tamagawa numbers do not depend on the choice of the finite subset S ⊂ Val(K). E. Peyre ([21]) proves the existence of the Tamagawa number for Fano varieties by using the Weil conjectures. The same method shows the existence of the Tamagawa number for smooth complete varieties X satisfying the conditions h^{1}(X, O_{X}) = h^{2}(X, O_{X}) = 0.

Theorem 2.9 [20] Let T be an algebraic torus defined over K. The Tamagawa number τ(T) doesn’t depend on the choice of a splitting field E/K. We have

τ(T) = h(T)/i(T).

The constants h(T), i(T) were defined in 1.5.

We see that the Tamagawa number of an algebraic torus is a rational number. We have τ(G_{m}(K)) = 1. The Tamagawa number of a Fano variety with a metrized canonical line bundle is certainly not rational in general. For PQ^{1} with our metrization we have τ_{K}(PQ^{1}) = 1/ζ_{Q}(2).

Proposition 2.10 [2] One has

∫\overline{T (K)} ω_{K,S} = ∫\overline{P_{Σ}(K)} ω_{K,S}.

3 Heights and their Fourier transforms

Let φ ∈ P L(Σ)C^{G}. Using the decomposition of Σ(1) into a union of G-orbits

Σ(1) = Σ_{1}(1) ∪ ... ∪ Σ_{r}(1),

we can identify φ with a T -invariant divisor with complex coefficients

D_{φ} = s_{1}D_{1} + ⋯ + s_{r}D_{r}

where s_{j} = φ(e_{j}) ∈ C and e_{j} is a primitive lattice generator of some cone σ ∈ Σ_{j}(1) (j = 1, …, r). It will be convenient to identify an element φ = φ_{s} ∈ P L(Σ)C^{G} with the vector s = (s_{1}, …, s_{r}) of its complex coordinates.

Let us recall the definition of heights on toric varieties from [2]. For our purposes it will be sufficient to describe the restrictions of heights to the Zariski open subset T ⊂ P_{Σ,K}.

Proposition 3.1 Let v ∈ Val(K) be a valuation and G_{v} ⊂ G the decomposition group of v. There is an injective homomorphism

π_{v} : T (K_{v})/T (O_{v}) ↪ N_{v},

which is an isomorphism for all but finitely many v ∈ Val(K). Here N_{v} = N^{G_{v}} ⊂ N for non-archimedian v and N_{v} = NR^{G_{v}} for archimedian valuations v. For every non-archimedian valuation we can identify the image of π_{v} with a sublattice of finite index in N_{v}.

Definition 3.2 Let s ∈ C^{r} be a complex vector defining a complex piecewise linear G-invariant function φ ∈ P L(Σ)C^{G}. For any point x_{v} ∈ T (K_{v}) ⊂ P_{Σ}(K_{v}), denote by \overline{x}_{v} the image of x_{v} in N_{v}, where N_{v} is considered as a canonical lattice in the real space NR^{G_{v}} for non-archimedian valuations (resp. as the real Lie-algebra N_{R,v} of T (K_{v}) for archimedian valuations). Define the complexified local Weil function H_{Σ,v}(x_{v}, s) by the formula

H_{Σ,v}(x_{v}, s) = e^{φ(\overline{x}_{v}) log q_{v}}

where q_{v} is the cardinality of the residue field k_{v} of K_{v} if v is non-archimedian and log q_{v} = 1 if v is archimedian.

Theorem 3.3 [2] The complexified local Weil function H_{Σ,v}(x_{v}, s) satisfies the following properties:

(i) H_{Σ,v}(x_{v}, s) is T (O_{v})-invariant.

(ii) If s = 0, then H_{Σ,v}(x_{v}, s) = 1 for all x_{v} ∈ T (K_{v}).

(iii) H_{Σ,v}(x_{v}, s + s′) = H_{Σ,v}(x_{v}, s)H_{Σ,v}(x_{v}, s′).

(iv) If s = (s_{1}, ..., s_{r}) ∈ Z^{r}, then H_{Σ,v}(x_{v}, s) is a classical local Weil function corresponding to a Cartier divisor D_{s} = s_{1}D_{1} + ⋯ + s_{r}D_{r} on P_{Σ,K}.

Definition 3.4 For a piecewise linear function φ_{s} ∈ P L(Σ)C^{G} we define the complexified height function on T (K) ⊂ P_{Σ,K}(K) by

H_{Σ}(x, s) = ∏_{v∈Val(K)} H_{Σ,v}(x_{v}, s).

Remark 3.5 Although the local heights are defined only as functions on P L(Σ)C^{G} ≅ C^{r}, the product formula implies that for x ∈ T (K) the global complexified height function descends to the Picard group Pic(P_{Σ,K})_{C}. Moreover, since H_{Σ}(x, s) is the product of local complex Weil functions H_{Σ,v}(x, s) and since for all x_{v} ∈ T (O_{v}) we have H_{Σ,v}(x_{v}, s) = 1 for all v, we can immediately extend H_{Σ}(x, s) to a function on T (A_{K}) × P L(Σ)C^{G}.

Definition 3.6 Let Σ(1) = Σ_{1}(1) ∪ ⋯ ∪ Σ_{l}(1) be the decomposition of Σ(1) into a disjoint union of G_{v}-orbits. Denote by d_{j} the length of the G_{v}-orbit Σ_{j}(1) (d_{1} + ⋯ + d_{l} = n). We establish a 1-to-1 correspondence Σ_{j}(1) ↔ u_{j} between the G_{v}-orbits Σ_{1}(1), …, Σ_{l}(1) and independent variables u_{1}, …, u_{l}.

Let σ ∈ Σ^{G_{v}} be any G_{v}-invariant cone and Σ_{j_{1}}(1) ∪ ⋯ ∪ Σ_{j_{k}}(1) the set of all 1-dimensional faces of σ. We define the rational function R_{σ}(u_{1}, …, u_{l}) corresponding to σ as follows:

R_{σ}(u_{1}, …, u_{l}) := \frac{u_{j_{1}}^{d_{j_{1}}} ⋯ u_{j_{k}}^{d_{j_{k}}}}{(1 − u_{j_{1}}^{d_{j_{1}}}) ⋯ (1 − u_{j_{k}}^{d_{j_{k}}})}.

Define the polynomial Q_{Σ}(u_{1}, …, u_{l}) by the formula

∑_{σ∈Σ^{Gv}} R_{σ}(u_{1}, …, u_{l}) = \frac{Q_{Σ}(u_{1}, …, u_{l})}{(1 − u_{1}^{d_{1}}) ⋯ (1 − u_{l}^{d_{l}})}.

Proposition 3.7 [2] Let Σ be a complete regular G_{v}-invariant fan. Then the polynomial

Q_{Σ}(u_{1}, …, u_{l}) − 1

contains only monomials of degree ≥ 2.

Let χ be a topological character of T (A_{K}) such that its v-component χ_{v} : T (K_{v}) → S^{1} ⊂ C^{∗} is trivial on T (O_{v}). For each j ∈ {1, …, l}, we denote by n_{j} the sum of d_{j} generators of all 1-dimensional cones if the G_{v}- orbit Σ_{j}(1). By (3.1) we know that the homomorphism

π_{v} : T (K_{v})/T (O_{v}) → N_{v}

is an isomorphism for almost all v. We call these valuations good. Hence, for good non-archimedian valuations, n_{j} represents an element of T (K_{v}) modulo T (O_{v}). Therefore, χ_{v}(n_{j}) is well-defined.

Definition 3.8 Denote by

Ĥ_{Σ,v}(χ_{v}, −s) = ∫_{T (K_{v})} H_{Σ,v}(x_{v}, −s)χ_{v}(x_{v})dμ_{v}

the value at χ_{v} of the multiplicative Fourier transform of the local Weil function H_{Σ,v}(x_{v}, −s) with respect to the v-adic Haar measure dμ_{v} on T (K_{v}) normalized by ∫_{T (O_{v})} dμ_{v} = 1.

Proposition 3.9 [2] Let v be a good non-archimedian valuation of K . For any topological character χ_{v} of T (K_{v}) which is trivial on the subgroup T (O_{v}) and a piecewise linear function φ = φ_{s} ∈ P L(Σ)C^{G} one has

Ĥ_{Σ,v}(χ_{v}, −s) = \frac{Q_{Σ} (\frac{χ_{v}(n_{1})}{q_{v}^{φ(n_{1})}} , …, \frac{χ_{v}(n_{l})}{q_{v}^{φ(n_{l})}} )}{(1 − \frac{χ_{v}(n_{1})}{q_{v}^{φ(n_{1})}} ) ⋯ (1 − \frac{χ_{v}(n_{l})}{q_{v}^{φ(n_{l})}} )}.

Corollary 3.10 [2] Let v be a good non-archimedian valuation of K. The restriction of

∫_{T (K_{v})} H_{Σ,v}(x_{v}, −s)dμ_{v}

to the line s_{1} = ⋯ = s_{r} = s is equal to

L_{v}(s, T; E/K) · L_{v}(s, T_{NS}, E/K) · Q_{Σ}(q_{v}^{−s}, …, q_{v}^{−s}).

Remark 3.11 It is difficult to calculate the Fourier transforms of local heights for the finitely many ”bad” non-archimedian valuations v, because there is only an embedding of finite index

T (K_{v})/T (O_{v}) ↪ N_{v}.

However, for our purposes it will be sufficient to use upper estimates for these local Fourier transforms. One immediately sees that for all non-archimedian valuations v the local Fourier transforms of H_{Σ,v}(x_{v}, −s) can be bounded absolutely and uniformly in all characters by a finite combination of multidimensional geometric series in q_{v}^{−1/2} in the domain Re(s) ∈ R_{>1/2}.

Now we assume that v is an archimedian valuation. By (3.1), we have T (K_{v})/T (O_{v}) = NR^{G_{v}} ⊂ N_{R} where G_{v} is the trivial group for the case K_{v} = C, and G_{v} = Gal(C/R) ≅ Z/2Z for the case K_{v} = R. Let ⟨·, ·⟩ be the pairing between N_{R} and M_{R} induced from the duality between N and M. Let y be an arbitrary element of the dual R-space MR^{G_{v}} = Hom(T (K_{v})/T (O_{v}), R). Then χ_{y}(x_{v}) = e^{−i⟨\overline{x}_{v},y⟩} is a topological character of T (K_{v}) which is trivial on T (O_{v}). We choose the Haar measure dμ_{v} on T (K_{v}) as the product of the Haar measure dμ_{v}^{0} on T (O_{v}) and the Haar measure d\overline{x}_{v} on T (K_{v})/T (O_{v}). We normalize the measures such that the dμ_{v}^{0}-volume of T (O_{v}) equals 1 and d\overline{x}_{v} is the standard Lebesgue measure on NR^{G_{v}} normalized by the full sublattice N^{G_{v}}.

Proposition 3.12 [2] Let v be an archimedian valuation of K. The Fourier transform Ĥ_{Σ,v}(χ_{y}, −s) of a local archimedian Weil function

H_{Σ,v}(x_{v}, −s) = e^{−φs(\overline{x}_{v})}

is a rational function in variables s_{j} = φ_{s}(e_{j}) for Re(s) ∈ R_{>0}.

Proof. Let us consider the case K_{v} = C. One has a decomposition of the space N_{R} into a union of d-dimensional cones N_{R} = ⋃_{σ∈Σ(d)} σ. We calculate the Fourier transform as follows:

Ĥ_{Σ,v}(χ_{y}, −s) = ∫_{NR} e^{−φs(\overline{x}_{v})−i⟨\overline{x}_{v},y⟩}d\overline{x}_{v} =

= ∑_{σ∈Σ(d)} ∫_{σ} e^{−φs(\overline{x}_{v})−i⟨\overline{x}_{v},y⟩}d\overline{x}_{v} = ∑_{σ∈Σ(d)} \frac{1}{∏_{e_{j}∈σ}(s_{j} + i⟨e_{j}, y⟩)}.

The case K_{v} = R can be reduced to the above situation. □

4 Poisson formula

Let P_{Σ} be a toric variety and H_{Σ}(x, s) the height function constructed above.

Definition 4.1 We define the zeta-function of the complex height-function H_{Σ}(x, s) as

Z_{Σ}(s) = ∑_{x∈T (K)} H_{Σ}(x, −s).

Theorem 4.2 The series Z_{Σ}(s) converges absolutely and uniformly for s contained in any compact in the domain Re(s) ∈ R_{>1}^{r}.

Proof. It was proved in [19] that we can always choose a finite set S such that the natural map

π_{S} : T (K) → ⨁_{v∉S} T (K_{v})/T (O_{v}) = ⨁_{v∉S} N_{v}

is surjective. Denote by T (O_{S}) the kernel of π_{S} consisting of all S-units in T (K). Let W (T) ⊂ T (O_{S}) the subgroup of torsion elements in T (O_{S}). Then T (O_{S})/W (T) has a natural embedding into the finite-dimensional logarithmic space

N_{R,S} = ⨁_{v∈S} T (K_{v})/T (O_{v}) ⊗ R

as a sublattice of codimension t. Let Γ be a full sublattice in N_{R,S} containing the image of T (O_{S})/W (T). Denote by Δ a bounded fundamental domain of Γ in N_{R,S}. For any x ∈ T (K) we denote by \overline{x}_{S} the image of x in N_{R,S}. Define ϕ(x) to be the element of Γ such that \overline{x}_{S} − ϕ(x) ∈ Δ. Thus, we have obtained the mapping

ϕ : T (K) → Γ.

Define a new function H̃_{Σ}(x, s) on T (K) by

H̃_{Σ}(x, s) = ∏_{v∈S} H_{Σ,v}(ϕ(x)_{v}, s) ∏_{v∉S} H_{Σ,v}(x_{v}, s).

If K ⊂ C^{r} is a compact in the domain Re(s) ∈ R_{>1}^{r}, then there exist two positive constants C_{1}(K) < C_{2}(K) such that

0 < C_{1}(K) < \frac{H̃_{Σ}(x, s)}{H_{Σ}(x, s)} < C_{2}(K) for s ∈ K, x ∈ T (K),

since \overline{x}_{v}−ϕ(x)_{v} belongs to some bounded subset Δ_{v} in N_{R,v} for any x ∈ T (K) and v ∈ S. Therefore, it is sufficient to prove that the series

Z̃_{Σ}(s) = ∑_{x∈T (K)} H̃_{Σ}(x, −s)

is absolutely converent for s ∈ K. Notice that Z̃_{Σ}(s) can be estimated from above by the the following Euler product

(∑_{γ∈Γ} ∏_{v∈S} H_{Σ,v}(γ_{v}, −s)) ∏_{v∉S} ( ∑_{z∈N_{v}} H_{Σ,v}(z, −s)) .

The sum

∑_{γ∈Γ} ∏_{v∈S} H_{Σ,v}(γ_{v}, −s)

is an absolutely convergent geometric series for Re(s) ∈ R_{>1}^{r}. On the other hand, the Euler product

∏_{v∉S} ( ∑_{z∈N_{v}} H_{Σ,v}(z, −s))

can be estimated from above by the product of zeta-functions

C_{3}(K) ∏_{j=1}^{r} ζ_{K_{j}}(s_{j}),

where C_{3}(K) is some constant depending on K. Since each ζ_{K_{j}}(s_{j}) is absolutely convergent for Re(s_{j}) > 1, we obtain the statement. □

We need the Poisson formula in the following form:

Theorem 4.3 ([11], 31.46 e) Let G be a locally compact abelian group with Haar measure dg, H ⊂ G a closed subgroup with Haar measure dh. The factor group G/H has a unique Haar measure dx normalized by the condition dg = dx · dh. Let F : G → R be an L^{1}-function on G and F̂ its Fourier transform with respect to dg. Suppose that F̂ is also an L^{1}-function on H^{⊥}, where H^{⊥} is the group of topological characters χ : G → S^{1} which are trivial on H. Then

∫_{H} F (x)dh = ∫_{H^{⊥}} F̂(χ)dχ,

where dχ is the orthogonal Haar measure on H^{⊥} with respect to the Haar measure dx on G/H.

We will apply this theorem in the case when G = T (A_{K}) and H = T (K), dg = ω_{Ω,S}, and dh is the discrete measure on T (K).

Theorem 4.4 (Poisson formula) For all s with Re(s) ∈ R_{>1}^{r} we have the following formula:

Z_{Σ}(s) = \frac{1}{(2π)^{t}b_{S}(T)} ∫_{(T (A_{K})/T (K))^{∗}} (∫_{T (A_{F})} H_{Σ}(x, −s)χ(x)ω_{Ω,S}) dχ,

where χ ∈ (T (A_{K})/T (K))^{∗} is a topological character of T (A_{K}), trivial on the closed subgroup T (K) and dχ is the orthogonal Haar measure on (T (A_{K})/T (K))^{∗}. The integral converges absolutely and uniformly to a holomorphic function in s in any compact in the domain Re(s) ∈ R_{>1}^{r}.

Proof. Because of 4.2 we only need to show that the Fourier transform Ĥ_{Σ}(χ, −s) of the height function is an L^{1}-function on (T (A_{K})/T (K))^{∗}. By 3.9 and uniform estimates at places of bad reduction 3.11, we know that the Euler product

∏_{v∉S_{∞}} Ĥ_{Σ,v}(χ_{v}, −s)

converges absolutely and is uniformly bounded by a constant c(K) for all characters χ and all s ∈ K, where K is some compact in the domain Re(s) ∈ R_{>1}^{r}.

Since the height function H_{Σ,v}(x, −s) is invariant under T (O_{v}) for all v, the Fourier transform of Ĥ_{Σ}(χ, −s) equals zero for characters χ which are non-trivial on the maximal compact subgroup K_{T}. Denote by P the set of all such characters χ ∈ (T (A_{K})/T (K))^{∗}.

We have a non-canonical splitting of characters χ = χ_{l} · χ_{y}, where χ_{l} ∈ (T^{1}(A_{K})/T (K))^{∗} and χ_{y} ∈ (T (A_{K})/T^{1}(A_{K}))^{∗}. Let us consider the logarithmic space

N_{R,∞} = ⨁_{v∈S_{∞}} T (K_{v})/T (O_{v}) = ⨁_{v∈S_{∞}} N_{R,v}.

It contains the lattice T (O_{K})/W (T) of O_{K}-integral points of T (K) modulo torsion. Denote by M_{R,∞} = ⨁_{v∈S_{∞}} M_{R,v} the dual space. It has a decomposition as a direct sum of vector spaces M_{R,∞} = M_{L} ⊕ M_{Y}, such that the space M_{L} contains the dual lattice L := (T (O_{K})/W (T))^{∗} as a full sublattice and the space M_{Y} is isomorphic to (T (A_{K})/T^{1}(A_{K}))^{∗} = T̂_{K} ⊗ R.

By 1.4, we have an exact sequence

0 → cl^{∗}(T) → P → M → 0,

where M is the image of the projection of P to M_{R,∞} and cl^{∗}(T) is a finite group. We see that the character χ ∈ P is determined by its archimedian component up to a finite choice. Denote by y(χ) ∈ M ⊂ M_{R,∞} the image of χ ∈ P.

The following lemmas will provide the necessary estimates of the Fourier transform of local heights at archimedian places. This allows to apply the Poisson formula 4.3. □

Lemma 4.5 [2] Let Σ ⊂ N_{R} be a complete fan in a real vector space of dimension d. Denote by M_{R} the dual space. For all m ∈ M_{R} and all s with 0 < δ_{1} < Re(s_{j}) < δ_{2} there exists a constant c = c(δ_{1}, δ_{2}, Σ) such that we have the following estimate

| ∑_{σ∈Σ(d)} \frac{1}{∏_{e_{j}∈σ}(s_{j} + i < e_{j}, m >)}| ≤ c ∑_{σ∈Σ(d)} \frac{1}{∏_{e_{j}∈σ}(1 + | < e_{j}, m > |)^{1+1/d}} .

Corollary 4.6 Consider

Ĥ_{Σ,∞}(χ, −s) = ∏_{v∈S_{∞}} Ĥ_{Σ,v}(χ, −s)

as a function on

M ⊂ M_{R,∞} = ⨁_{v∈S_{∞}} M_{R,v}.

Let d′ be the dimension of M_{R,∞}. We have a direct sum decomposition M_{R,∞} = M_{L} ⊕ M_{Y} of real vector spaces. Let M′_{Y} ⊂ M_{Y} be any affine subspace, dy′ the Lebesgue measure on M′_{Y} and L′ ⊂ M_{L} any lattice. Let g(y, −s) be a function on M_{R,∞} satisfying |g(y, −s)| ≤ c(1 + ∥y∥)^{δ} for all y ∈ M_{R,∞}, all s contained in some compact K ⊂ C^{r} in the domain R_{>1/2}^{r}, some 0 < δ < 1/d′ and some constant c > 0. Then the series

∑_{y(χ)∈L} ∫_{M′_{Y}} g(y, −s)Ĥ_{Σ,∞}(y(χ), −s)dy′

is absolutely and uniformly convergent to a holomorphic function in s in this domain.

Proof. We apply 3.12 and observe that on the space N_{R,∞} we have a fan Σ_{∞} obtained as the direct product of fans Σ^{G_{v}} for v ∈ S_{∞} (i.e., every cone in Σ_{∞} is a direct product of cones in Σ^{G_{v}}). □

5 X -functions of convex cones

Let (A, A_{R}, Λ) be a triple consisting of a free abelian group A of rank k, a k-dimensional real vector space A_{R} = A ⊗ R containing A as a sublattice of maximal rank, and a convex k-dimensional cone Λ ⊂ A_{R} such that Λ ∩ −Λ = 0 ∈ A_{R}. Denote by Λ^{∘} the interior of Λ and by ΛC^{∘} = Λ^{∘} + iA_{R} the complex tube domain over Λ^{∘}. Let (A^{∗}, AR^{∗}, Λ^{∗}) be the triple consisting of the dual abelian group A^{∗} = Hom(A, Z), the dual real vector space AR^{∗} = Hom(A_{R}, R), and the dual cone Λ^{∗} ⊂ AR^{∗}. We normalize the Haar measure dy on AR^{∗} by the condition: vol(AR^{∗}/A^{∗}) = 1.

Definition 5.1 The X -function of Λ is defined as the integral

X_{Λ}(s) = ∫_{Λ^{∗}} e^{−⟨s,y⟩}dy,

where s ∈ ΛC^{∘}.

Remark 5.2 X -functions of convex cones have been investigated in the theory of homogeneous cones by M. Köcher, O.S. Rothaus, and E.B. Vinberg [12, 23, 25]. In these papers X -functions were called characteristic functions of cones, but we find such a notion rather misleading in view of the fact that X_{Λ}(s) is the Fourier-Laplace transform of the standard set-theoretic characteristic function of the dual cone Λ^{∗}.

The function X_{Λ}(s) has the following properties [23, 25]:

Proposition 5.3 (i) If A is any invertible linear operator on R^{k}, then

X_{Λ}(As) = \frac{X_{Λ}(s)}{detA};

(ii) If Λ = R_{≥0}^{k}, then

X_{Λ}(s) = (s_{1} ⋯ s_{k})^{−1}, for Re(s_{i}) > 0;

(iii) If s ∈ Λ^{∘}, then

lim_{s→∂Λ} X_{Λ}(s) = ∞;

(iv) X_{Λ}(s) ≠ 0 for all s ∈ ΛC^{∘}.

Proposition 5.4 If Λ is a k-dimensional finitely generated polyhedral cone, then X_{Λ}(s) is a rational function

X_{Λ}(s) = \frac{P (s)}{Q(s)},

where P is a homogeneous polynomial, Q is a product of all linear homogeneous forms defining the codimension 1 faces of Λ, and deg P − deg Q = −k.

Proof. We subdivide the dual cone Λ^{∗} into a finite union of simplicial subcones Λ_{j}^{∗} (j ∈ J). Let Λ_{j} ⊂ A_{R} be the dual cone to Λ_{j}^{∗}. Then

X_{Λ}(s) = ∑_{j∈J} X_{Λ_{j}}(s).

By 5.3(i) and (ii),

X_{Λ_{j}}(s) = \frac{P_{j}(s)}{Q_{j}(s)} (j ∈ J),

where P_{j} is a homogeneous polynomial of degree 0 and Q_{j} is the product of k homogeneous linear forms defining the codimension 1 faces of Λ_{j}. Therefore, X_{Λ}(s) can be uniquely represented up to constants as a ratio of two homogeneous polynomials P (s)/Q(s) with g.c.d.(P, Q) = 1 where Q is a product of linear homogeneous forms defining some faces of Λ_{j} of codimension 1. Since Q does not depend on a choice of a subdivision of Λ^{∗} into a finite union of simplicial cones Λ_{j}^{∗}, only linear homogeneous forms which vanish on codimension 1 faces of Λ can be factors of Q. On the other hand, by 5.3(iii), every linear homogeneous form vanishing on a face of Λ of codimension 1 must divide Q. □

Theorem 5.5 Let (A, A_{R}, Λ) and (Ã, Ã_{R}, Λ̃) be two triples as above, k = rk A and k̃ = rk Ã, and ψ : A → Ã a homomorphism of free abelian groups with a finite cokernel A′ (i.e., the corresponding linear mapping of real vector spaces ψ : A_{R} → Ã_{R} is surjective), and ψ(Λ) = Λ̃. Let B = Ker ψ ⊂ A, db the Haar measure on B_{R} = B ⊗ R normalized by the condition vol(B_{R}/B) = 1. Then for all s with Re(s) ∈ Λ^{∘} the following formula holds:

XΛ̃(ψ(s)) = \frac{1}{(2π)^{k−k̃}|A′|} ∫_{BR} X_{Λ}(s + ib)db,

where |A′| is the order of the finite abelian group A′.

Proof. We have the dual injective homomorphisms of free abelian groups ψ^{∗} : Ã^{∗} → A^{∗} and of the corresponding real vector spaces ψ^{∗} : ÃR^{∗} → AR^{∗}. Moreover, Λ̃^{∗} = Λ^{∗} ∩ ÃR^{∗}. Let C_{Λ^{∗}}(y) be the set-theoretic characteristic function of the cone Λ^{∗} ⊂ AR^{∗} and C_{Λ^{∗}}(ỹ) the restriction of C_{Λ^{∗}}(y) to ÃR^{∗} which is the set-theoretic characteristic function of Λ̃^{∗} ⊂ ÃR^{∗}. Then

XΛ̃(ψ(s)) = ∫ÃR^{∗} C_{Λ^{∗}}(ỹ)e^{−⟨ψ(s),y^{}˜ ⟩}dỹ.

Now we apply the Poisson formula to the last integral. For this purpose we notice that any additive topological character of AR^{∗} which vanishes on the subgroup ÃR^{∗} ⊂ AR^{∗} has the form

e^{−i⟨b,y⟩}, where b ∈ B_{R}.

Moreover,

\frac{db}{(2π)^{k−k̃}|A′|}

is the orthogonal Haar measure on B_{R} with respect to the Haar measures dỹ and dy on ÃR^{∗} and AR^{∗} respectively. It remains to notice that

X_{Λ}(s + ib) = ∫_{AR^{∗}} C_{Λ^{∗}}(y)e^{−⟨s+ib,y⟩}dy

is the value of the Fourier transform of C_{Λ^{∗}}(y)e^{−⟨s,y⟩} on the topological character of AR^{∗}/ÃR^{∗} corresponding to an element b ∈ B_{R} ⊂ A_{R}. □

Corollary 5.6 Assume that in the above situation rk = k − k̃ = 1 and Ã = A/B. Denote by γ a generator of B. Then

XΛ̃(ψ(s)) = \frac{1}{2πi} ∫_{Re(z)=0} X_{Λ}(s + z · γ)dz,

where z = x + iy ∈ C.

Corollary 5.7 Assume that a k̃-dimensional rational finite polyhedral cone Λ̃ ⊂ Ã_{R} contains exactly r one-dimensional faces with primitive lattice generators a_{1}, …, a_{r} ∈ Ã. We set k := r, A := Z^{r} and denote by ψ the natural homomorphism of lattices Z^{r} → Ã which sends the standard basis of Z^{r} into a_{1}, …, a_{r} ∈ Ã, so that Λ̃ is the image of the simplicial cone R_{≥0}^{r} ⊂ R^{r} under the surjective map of vector spaces ψ : R^{r} → A_{R}. Denote by M_{R} the kernel of ψ and set M := Z^{r} ∩ M_{R}. Let s = (s_{1}, ..., s_{r}) be the standard coordinates in C^{r}. Then

X_{Λ}(ψ(s)) = \frac{1}{(2π)^{r−k}|A′|} ∫_{MR} \frac{1}{∏_{j=1,n}(s_{j} + iy_{j})}dy

where dy is the Haar measure on the additive group M_{R} normalized by the lattice M, y_{j} are the coordinates of y in R^{r}, and |A′| is the index of the sublattice in Ã generated by a_{1}, …, a_{r}.

Example 5.8 Consider an example of a non-simplicial convex cone which appears as the cone of effective divisors of the split toric Del Pezzo surface X of anticanonical degree 6. The cone Λ_{eff} has 6 generators corresponding to exceptional curves of the first kind on X. We can construct X as the blow up of 3 points p_{1}, p_{2}, p_{3} in general position in P^{2}. Denote the exceptional curves by C_{1}, C_{2}, C_{3}, C_{12}, C_{13}, C_{23}, where C_{ij} is the proper pullback of the line joining p_{i} and p_{j}. Let s = s_{1}[C_{1}]+s_{2}[C_{2}]+s_{3}[C_{3}]+s_{12}[C_{12}]+s_{13}[C_{13}]+s_{23}[C_{23}] ∈ Λ_{eff}^{∘} be an element in the interior of the cone of effective divisors. The sublattice M ⊂ Z^{6} of rank 2 consisting of principal divisors is generated by γ_{1} = C_{1} + C_{13} − C_{2} − C_{23} and γ_{2} = C_{1} + C_{12} − C_{3} − C_{23} = 0.

In our case, the integral formula in 5.7 is a 2-dimensional integral (r = 6) which can be computed by applying twice the residue theorem to two 1dimensional integrals like the one in 5.6. We obtain the following formula for the characteristic function of Λ_{eff}:

X_{Λ}(ψ(s)) = \frac{s_{1} + s_{2} + s_{3} + s_{12} + s_{13} + s_{23}}{(s_{1} + s_{23})(s_{2} + s_{13})(s_{3} + s_{12})(s_{1} + s_{2} + s_{3})(s_{12} + s_{13} + s_{23})}.

Definition 5.9 Let X be a smooth proper algebraic variety. Consider the triple (Pic(X), Pic(X) ⊗ R, Λ_{eff}) where Λ_{eff} ⊂ Pic(X) ⊗ R is the cone generated by classes of effective divisors on X. Assume that the anticanonical class [K^{−1}] ∈ Pic(X)_{R} is contained in the interior of Λ_{eff}. We define the constant α(X) by

α(X) = X_{Λ_{eff}}([K^{−1}]).

Corollary 5.10 If Λ_{eff} is a finitely generated rational polyhedral cone, then α(X) is a rational number.

6 Some technical statements

Let E be a number field and χ an unramified Hecke character on G_{m}(A_{E}). Its local components χ_{v} for all valuations v are given by:

χ_{v} : G_{m}(E_{v})/G_{m}(O_{v}) → S^{1}
χ_{v}(x_{v}) = |x_{v}|_{v}^{it_{v}}.

Definition 6.1 Let χ be an unramified Hecke character. We set

y(χ) := {t_{v}}_{v∈S_{∞}(E)} ∈ R^{r_{1}+r_{2}},

where r_{1} (resp. r_{2}) is the number of real (resp. pairs of complex) valuations of E. We also set

∥y(χ)∥ := max_{v∈S_{∞}(E)} |t_{v}|.

We will need uniform estimates for Hecke L-functions in vertical strips. They can be deduced using the Phragmen-Lindelöf principle [22].

Theorem 6.2 For any ε > 0 there exists a δ > 0 such that for any 0 < δ_{1} < δ there exists a constant c(ε, δ_{1}, E) > 0 such that the inequality

|L_{E}(s, χ)| ≤ c(ε, δ_{1}, E)(1 + |Im(s)| + ∥y(χ)∥)^{ε}

holds for all s with δ_{1} < |Re(s) − 1| < δ and every Hecke L-function L_{E}(s, χ) corresponding to an unramified Hecke character χ.

Corollary 6.3 For any ε > 0 there exists a δ > 0 such that for any compact K ⊂ C in the domain 0 < |Re (s) − 1| < δ there exists a constant C(K, ε, E) depending on K, ε and E such that

|L_{E}(s, χ)| ≤ C(K, ε, E)(1 + ∥y(χ)∥)^{ε}

for s ∈ K and every unramified character χ.

Let Σ be the Galois-invariant fan defining P_{Σ} and Σ(1) = Σ_{1}(1)∪...∪Σ_{r}(1) the decomposition of the set of one-dimensional generators of Σ into Gorbits. Let e_{j} be a primitive integral generator of σ_{j}, G_{j} ⊂ G the stabilizer of e_{j}. Denote by K_{j} ⊂ E the subfield of G_{j}-fixed elements. Consider the n-dimensional torus

T′ = ∏_{j=1}^{r} R_{K_{j}/K}(G_{m}).

Let us recall the exact sequence of Galois-modules from Proposition 1.15:

0 → M^{G} → P L(Σ)^{G} → Pic(P_{Σ}) → H^{1}(G, M) → 0.

It induces a map of tori T′ → T and a homomorphism

a : T′(A_{K})/T′(K) → T (A_{K})/T (K).

So we get a dual homomorphism for characters

a^{∗} : (T (A_{K})/T (K))^{∗} → ∏_{j=1}^{r}(G_{m}(A_{K_{j}})/G_{m}(K_{j}))^{∗}.

Proposition 6.4 [7] The kernel of a^{∗} is dual to the obstruction group to weak approximation A(T) defined in 1.6.

Let χ ∈ (T (A_{K})/T (K))^{∗} be a character. Then χ ∘ a defines r Hecke characters of the idele groups

χ_{j} : G_{m}(A_{K_{j}}) → S^{1} ⊂ C^{∗}, j = 1, …, r.

If χ is trivial on K_{T}, then all characters χ_{j} (j = 1, …, r) are trivial on the maximal compact subgroups in G_{m}(A_{K_{j}}). We denote by L_{K_{j}}(s, χ_{j}) the Hecke L-function corresponding to the unramified character χ_{j}.

Proposition 6.5 Let χ = (χ_{v}) be a character and Ĥ_{Σ,v}(χ_{v}, −s) the local Fourier transform of the complex local height function H_{Σ,v}(x_{v}, −s). For any compact K ⊂ C^{r} contained in the domain Re(s) ∈ R_{>1/2}^{r} there exists a constant c(K) such that

∏_{v∉S} Ĥ_{Σ,v}(χ_{v}, −s) · ∏_{i=1}^{r} L_{K_{j}}^{−1}(s_{j}, χ_{j}) ≤ c(K)

for all characters χ ∈ (T (A_{K})/T (K))^{∗}.

The proof follows from explicit computations of local Fourier transforms 3.9 and is almost identical with the proof of Proposition 3.1.3 in [2].

Proposition 6.6 There exists an ε > 0 such that for any open U ⊂ C^{r} contained in the domain 0 < |Re(s_{j}) − 1| < ε for j = 1, ..., r the integral

∫_{(T (A_{K})/T (K))^{∗}} Ĥ_{Σ}(χ, −s)dχ

converges absolutely and uniformly to a holomorphic function for Re(s) ∈ U.

Proof. Using uniform estimates of Fourier transforms for non-archimedian places of bad reduction (3.11) and the proposition above we need only to consider the following integral

∫_{(T (A_{K})/T (K))^{∗}} Ĥ_{Σ,∞}(χ, −s) ∏_{j=1}^{r} L_{K_{j}}(s_{j}, χ_{j})dχ.

Observe that there exist constants c_{1} > 0 and c_{2} > 0 such that we have the following inequalities:

c_{1}∥y(χ)∥ ≤ ∑_{j=1}^{r} ∥y(χ_{j})∥ ≤ c_{2}∥y(χ)∥.

Here we denoted by ∥y(χ)∥ the norm of y(χ) ∈ M_{R,∞}. Recall that since we only consider χ which are trivial on the maximal compact subgroup K_{T}, all characters χ_{j} are unramified. To conclude, we apply uniform estimates of Hecke L-functions from Corollary 6.3 and the Corollary 4.6. □

The rest of this section is devoted to the proof of our main technical result.

Let R[s] (resp. C[s]) be the ring of polynomials in s_{1}, …, s_{r} with coefficients in R (resp. in C), C[[s]] the ring of formal power series in s_{1}, …, s_{r} with complex coefficients.

Definition 6.7 Two elements f(s), g(s) ∈ C[[s]] will be called coprime, if g.c.d.(f(s), g(s)) = 1.

Definition 6.8 Let f(s) be an element of C[[s]]. By the order of a monomial s_{1}^{α_{1}}...s_{r}^{α_{r}} we mean the sum of the exponents α_{1} + ... + α_{r}. By multiplicity μ(f(s)) of f(s) at 0 = (0, …, 0) we always mean the minimal order of nonzero monomials appearing in the Taylor expansion of f(s) at 0 .

Definition 6.9 Let f(s) be a meromorphic at 0 function. Define the multiplicity μ(f(s)) of f(s) at 0 as

μ(f(s)) = μ(g_{1}(s)) − μ(g_{2}(s))

where g_{1}(s) and g_{2}(s) are two coprime elements in C[[s]] such that f = g_{1}/g_{2}.

Remark 6.10 It is easy to show that for any two meromorphic at 0 functions f_{1}(s) and f_{2}(s), one has

(i) μ(f_{1} · f_{2}) = μ(f_{1}) + μ(f_{2}) (in particular, one can omit ”coprime” in Definition 6.9);

(ii) μ(f_{1} + f_{2}) ≥ min{μ(f_{1}), μ(f_{2})};

(iii) μ(f_{1} + f_{2}) = μ(f_{1}) if μ(f_{2}) > μ(f_{1}).

Using the properties 6.10(i)-(ii), one immediately obtains from Definition 6.8 the following:

Proposition 6.11 Let f_{1}(s) and f_{2}(s) be two analytic at 0 functions, l(s) a homogeneous linear function, γ = (γ_{1}, …, γ_{r}) ∈ C^{r} an arbitrary complex vector with l(γ) ≠ 0, and g(s) := f_{1}(s)/f_{2}(s). Then the multiplicity of the function

g̃(s) := (\frac{∂}{∂z} )^{k} g(s + z · γ)|_{z=−l(s)/l(γ)}

at 0 is at least μ(g) − k, if

f_{2}(s + z · γ)|_{z=−l(s)/l(γ)}

is not identically zero.

Let Γ ⊂ Z^{r} be a sublattice, Γ_{R} ⊂ R^{r} (resp. Γ_{C} ⊂ C^{r}) the scalar extension of Γ to a R-subspace (resp. to a C-subspace). We always assume that Γ_{R} ∩ R_{≥0}^{r} = 0. We set V_{R} := R^{r}/Γ_{R} and V_{C} := C^{r}/Γ_{C}. Denote by ψ the canonical C-linear projection C^{r} → V_{C}.

Definition 6.12 A complex analytic function f(s) = f(s_{1}, …, s_{r}) : U → C defined on an open subset U ⊂ C^{r} is said to descend to V_{C} if for any vector α ∈ Γ_{C} and any u = (u_{1}, …, u_{r}) ∈ U one has

f(u + z · α) = f(u) for all {z ∈ C : u + z · α ∈ U}.

Remark 6.13 By definition, if f(s) descends to V_{C}, then there exists an analytic function g on ψ(U) ⊂ V_{C} such that f = g ∘ ψ. Using Cauchy-Riemann equations, one immediately obtains that f descends to V_{C} if and only if for any vector α ∈ Γ_{R} and any u = (u_{1}, …, u_{r}) ∈ U, one has

f(u + iy · α) = f(u) for all {y ∈ R : u + iy · α ∈ U}.

Definition 6.14 Let U be an open subset in R^{r}. By a tube domain U_{C} over U we mean the set

U_{C} := {z ∈ C^{r} : Re(z) ∈ U}.

The following statement can be found in [17] (Prop. 6 on p. 122):

Theorem 6.15 Let U ⊂ R^{r} (with r ≥ 2) be a connected open subset. Then any function f(z) which is analytic in U_{C} extends to an analytic function in Û_{C}, where Û_{C} is the tube domain over the convex hull Û of U.

Definition 6.16 An analytic function W (s) in the domain Re(s) ∈ R_{>0}^{r} is called good with respect to Γ if it satisfies the following conditions:

(i) W (s) descends to V_{C};

(ii) There exist pairwise coprime linear homogeneous polynomials

l_{1}(s), …, l_{p}(s) ∈ R[s]

and positive integers k_{1}, …, k_{p} such that for every j ∈ {1, …, p} the linear form l_{j}(s) descends to V_{C}, l_{j}(s) does not vanish for s ∈ R_{>0}^{n}, and

P (s) = W (s) · ∏_{j=1}^{p} l_{j}^{k_{j}}(s)

is analytic for Re(s) ∈ R_{>−δ_{0}}^{r} for some δ_{0} > 0.

(iii) There exist a non-zero complex number C(W) and a decomposition of P (s) into the sum

P (s) = P_{0}(s) + P_{1}(s)

so that P_{0}(s) is a homogeneous polynomial of degree μ(P), P_{1}(s) is an analytic function in the domain Re(s) ∈ R_{>−δ_{0}}^{r} with μ(P_{1}) > μ(P_{0}), both functions P_{0}, P_{1} descend to V_{C}, and

\frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)} = C(W) · X_{Λ}(ψ(s)),

where X_{Λ} is the X -function of the cone Λ = ψ(R_{≥0}^{r}) ⊂ V_{R}; Definition 6.17 If W (s) is a good with respect to Γ as above, then the meromorphic function

\frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}

will be called the principal part of W (s) at 0 and the non-zero constant C(W) the principal coefficient of W (s) at 0.

Suppose that rk Γ = t < r − 1. Let γ ∈ Z^{r} be an element which is not contained in Γ, Γ̃ := Γ ⊕ Z < γ >, Γ̃_{R} := Γ_{R} ⊕ R < γ >, Ṽ_{R} := R^{r}/Γ̃_{R} and Ṽ_{C} := C^{r}/Γ̃_{C}.

The following easy statement will be helpful in the sequel:

Proposition 6.18 Let f(s) be an analytic at 0 function, l(s) a homogeneous linear function such that l(γ) ≠ 0. Assume that f(s) and l(s) descend to V_{C}. Then

f̃(s) := f (s − \frac{l(s)}{l(γ)} · γ)

descends to Ṽ_{C}.

Theorem 6.19 Let W (s) be a good function with respect to Γ as above,

Φ(s) = ∏_{j : l_{j}(γ)=0} l_{j}^{k_{j}}(s)

the product of those linear forms l_{j}(s) (j ∈ {1, …, p}) which vanish on γ. Assume that Γ̃_{R} ∩ R_{≥0}^{r} = 0 and the following statements hold:

(i) The integral

W̃ (s) := ∫_{Re(z)=0} W (s + z · γ)dz, z ∈ C

converges absolutely and uniformly on any compact K ⊂ C^{r} in the domain Re(s) ∈ R_{>0}^{r};

(ii) There exists a δ > 0 such that the function W (s + z · γ) is defined and analytic in a tube domain U_{C} over an open neighborhood U of 0 for all z with Re(z) = δ and the integral

∫_{Re(z)=δ} Φ(s) · W (s + z · γ)dz

converges absolutely and uniformly in any compact which is contained in the same tube domain U_{C}. Moreover, the multiplicity of the meromorphic function

W̃_{δ}(s) := ∫_{Re(z)=δ} W (s + z · γ)dz

at 0 is at least 1 + rk Γ̃ − r; (iii) For s with Re(s) ∈ R_{>0}^{r}, the function

ϕ(N, s) = sup_{0≤Re(z)≤δ, |Im(z)|=N} |W (s + z · γ)|

tends to 0 as N → +∞, uniformly in any compact K contained in the tube domain over R_{>0}^{r}.

Then W̃ (s) is a good function with respect to Γ̃ and C(W̃) = 2πi · C(W).

Proof. Assume that l_{j}(γ) < 0 for j = 1, …, p_{1}, l_{j}(γ) = 0 for j = p_{1}+1, …, p_{2}, and l_{j}(γ) > 0 for j = p_{2} + 1, …, p. In particular, one has

Φ(s) = ∏_{j=p_{1}+1}^{p_{2}} l_{j}^{k_{j}}(s).

Denote by z_{j} the solution of the equation

l_{j}(s) + zl_{j}(γ) = 0, j = 1, …, p_{1}.

Let U^{+} be the intersection of R_{>0}^{r} with an open neighborhood U of 0, such that

Φ(s) · W̃_{δ}(s)

is analytic in U_{C}. Then both functions W̃_{δ}(s) and W̃ (s) are analytic in UC^{+}. Moreover, the integral formulas for W̃_{δ}(s) and W̃ (s) show that the equalities W̃_{δ}(u + iy · γ) = W̃_{δ}(u) and W̃ (u + iy · γ) = W̃ (u) hold for any y ∈ R and u, u + iy · γ ∈ UC^{+}. Therefore, both functions W̃_{δ}(s) and W̃ (s) descend to Ṽ_{C} (see Remark 6.13).

For N ≫ 0, we can apply the residue theorem and obtain

∫_{−N}^{N} W (s + it · γ)dt + ∫_{0}^{δ} W (s + (iN + y)γ)dy−

− ∫_{−N}^{N} W (s + (it + δ) · γ)dt − ∫_{0}^{δ} W (s + (−iN + y)γ)dy =

= −2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}W (s + z · γ)

for all s ∈ UC^{+}. By (iii), we have

lim_{N→+∞} ∫_{0}^{δ} W (s + (iN + y)γ)dy = 0

and

lim_{N→+∞} ∫_{0}^{δ} W (s + (−iN + y)γ)dy = 0

uniformly for all s contained in any compact K ⊂ UC^{+}. By (i) and (ii), we have

W̃ (s) = lim_{N→+∞} ∫_{−N}^{N} W (s + it · γ)dt

and

W̃_{δ}(s) = lim_{N→+∞} ∫_{−N}^{N} W (s + (it + δ) · γ)dt

uniformly for all s contained in a compact K ⊂ UC^{+}. Therefore,

W̃ (s) − W̃_{δ}(s) = −2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}W (s + z · γ)

for all s ∈ UC^{+}.

We denote by U(γ)_{C} the open subset of U_{C} defined by the inequalities

\frac{l_{j}(s)}{l_{j}(γ)} ≠ \frac{l_{m}(s)}{l_{m}(γ)} for all j ≠ m, j, m ∈ \{1, …, p\}.

The open set U(γ)_{C} is non-empty, since we assume that g.c.d.(l_{j}, l_{m}) = 1 for j ≠ m. For s ∈ U(γ)_{C}, we have

Res_{z=z_{j}}W (s + z · γ) = \frac{1}{(k_{j} − 1)!} (\frac{∂}{∂z} )^{k_{j}−1} \frac{l_{j}(s + z · γ)^{k_{j}}P (s + z · γ)}{l_{j}^{k_{j}}(γ) · ∏_{m=1}^{p} l_{m}^{k_{m}}(s + z · γ)}|_{z=z_{j}},

where

z_{j} = −\frac{l_{j}(s)}{l_{j}(γ)}.

Let

W (s) · ∏_{j=1}^{p} l_{j}^{k_{j}}(s) = P (s) = P_{0}(s) + P_{1}(s),

where P_{0}(s) is a uniquely determined homogeneous polynomial and P_{1}(s) is an analytic in the tube domain over R_{>−δ_{0}}^{r} function such that μ(P) = μ(P_{0}) < μ(P_{1}) and

\frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)} = C(W) · X_{Λ}(s)

(X_{Λ}(s) is the X -function of the cone Λ = ψ(R_{≥0}^{r})). We set

R_{0}(s) := \frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}, R_{1}(s) := \frac{P_{1}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}.

Then μ(W) = μ(R_{0}) < μ(R_{1}). Moreover, μ(W) = −dimV_{R} = r − rk Γ. Define

R̃_{0}(s) := −2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}R_{0}(s + z · γ)

and

R̃_{1}(s) := −2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}R_{1}(s + z · γ).

By Proposition 6.11, we have μ(R̃_{1}) ≥ 1 + μ(R_{1}) ≥ 2 + μ(R_{0}) = 1 + rk Γ̃ − r. We claim that

R̃_{0}(s) = 2πi · C(W)XΛ̃(ψ̃(s)).

(in particular, μ(R̃_{0}) = μ(R_{0})+1 = rk Γ̃−r). Indeed, repeating for X_{Λ}(ψ(s)) the same arguments as for W (s), we obtain

∫_{Re(z)=0} X_{Λ}(ψ(s + z · γ))dz − ∫_{Re(z)=δ} X_{Λ}(ψ(s + z · γ))dz

= −2πi · ∑_{j=1}^{k_{1}} Res_{z=z_{j}}X_{Λ}(ψ(s + z_{j} · γ)).

Moving the contour of integration Re(z) = δ (δ → +∞), by residue theorem, we obtain

∫_{Re(z)=δ} X_{Λ}(ψ(s + z · γ))dz = 0.

On the other hand,

XΛ̃(ψ̃(s)) = \frac{1}{2πi} ∫_{Re(z)=0} X_{Λ}(ψ(s + z · γ))dz

(see Theorem 5.6).

By Proposition 6.11, we have μ(R̃_{1}) > 1+μ(R_{1}) ≥ 2+μ(R_{0}) = 1+rk Γ̃−r. By 6.9(iii), using the decomposition

W̃ (s) = W̃_{δ}(s) + R̃_{0}(s) + R̃_{1}(s)

and our assumption μ(W_{δ}) ≥ 1 + rk Γ̃ − r in (ii), we obtain that μ(W̃) = μ(R̃_{0}) = rk Γ̃ − r.

By 6.18, the linear forms

h_{m,j}(s) := l_{m}(s + z_{j} · γ) = l_{m}(s) − \frac{l_{j}(s)}{l_{j}(γ)}l_{m}(γ)

and the analytic in the domain U(γ)_{C} functions

Res_{z=z_{j}}W (s + z · γ), Res_{z=z_{j}}R_{0}(s + z · γ)

descend to Ṽ_{C}. For any j ∈ {1, …, p_{1}}, let us denote

Q_{j}(s) = ∏_{m≠j,m=1}^{p} h_{m,j}^{k_{m}}(s).

It is clear that

Q_{j}^{k_{j}}(s) · Res_{z=z_{j}}W (s + z · γ) and Q_{j}^{k_{j}}(s) · Res_{z=z_{j}}R_{0}(s + z · γ)

are analytic at 0 and the polynomial Φ(s) divides each Q_{j}(s). So we obtain that

W̃ (s) ∏_{j=1}^{p_{1}} Q_{j}^{k_{j}}(s) = (W̃_{δ}(s) − 2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}W (s + z · γ)) ∏_{j=1}^{p_{1}} Q_{j}^{k_{j}}(s)

and

R̃_{0}(s) ∏_{j=1}^{p_{1}} Q_{j}^{k_{j}}(s) = (−2πi · ∑_{j=1}^{p_{1}} Res_{z=z_{j}}R_{0}(s + z · γ)) ∏_{j=1}^{p_{1}} Q_{j}^{k_{j}}(s)

are analytic in U_{C}.

Define the set {l̃_{1}(s), …, l̃_{q}(s)} as a subset of pairwise coprime elements in the set of homogeneous linear forms {h_{m,j}(s)} (m ∈ {1, …, p}, j ∈ {1, …, p_{1}}) such that there exist positive integers n_{1}, …, n_{q} and a representation of the meromorphic functions W̃ (s) and R̃_{0}(s) as quotients

W̃ (s) = \frac{P̃(s)}{∏_{j=1}^{q} l̃_{j}^{n_{j}}(s)}, R̃_{0}(s) = \frac{P̃_{0}(s)}{∏_{j=1}^{q} l̃_{j}^{n_{j}}(s)},

where P̃(s) is analytic at 0, P̃_{0}(s) is a homogeneous polynomial and none of the forms l̃_{1}(s), …, l̃_{q}(s) vanishes for s ∈ R_{>0}^{r} (the last property can be achieved, because both functions W̃ (s) and R̃_{0}(s) are analytic in UC^{+}).

Define

P̃_{1}(s) = (W̃_{δ}(s) + R̃_{1}(s)) · ∏_{j=1}^{q} l̃_{j}^{n_{j}}(s).

Then

P̃(s) = P̃_{0}(s) + P̃_{1}(s)

where P̃_{0}(s) is a homogeneous polynomial and P̃_{1}(s) is an analytic in U_{C} function such that μ(P̃) = μ(P̃_{0}) < μ(P̃_{1}) and

\frac{P̃_{0}(s)}{∏_{j=1}^{q} l̃_{j}^{n_{j}}(s)} = 2πi · C(W) · XΛ̃(ψ̃(s)).

For sufficiently small positive δ_{0}, the domain R_{>−δ_{0}}^{r} is contained in the convex hull of U ∪ R_{>0}^{r}. By 6.15,

∏_{j=1}^{q} l̃_{j}^{n_{j}}(s)W̃ (s)

is analytic in Re(s) ∈ R_{−δ_{0}}^{r}. This proves that W̃ (s) is a good function with respect to Γ̃. □

7 Main theorem

Let us set

W_{Σ}(s) := Z_{Σ}(φ_{s} + φ_{Σ}) = Z_{Σ}(s_{1} + 1, …, s_{r} + 1).

By Theorem 4.2, W_{Σ}(s) is an analytic function in the domain Re(s) ∈ R_{>0}^{r}.

Theorem 7.1 The analytic function W_{Σ}(s) is good with respect to the lattice M^{G} ⊂ P L(Σ)^{G} = Z^{r}.

Proof. By Theorem 4.4, we have the following integral representation for Z_{Σ}(s) in the domain Re(s) ∈ R_{>1}^{r}

Z_{Σ}(s) = \frac{1}{(2π)^{t}b_{S}(T)} ∫_{(T (A_{K})/T (K))^{∗}} Ĥ_{Σ}(χ, −s)dχ

We need only to consider characters χ which are trivial on the maximal compact subgroup K_{T} ⊂ T^{1}(A_{K}), because for all other characters the Fourier transform Ĥ_{Σ}(χ, −s) vanishes. Choosing a non-canonical splitting of characters corresponding to some splitting of the sequence

0 → T^{1}(A_{K}) → T (A_{K}) → T (A_{K})/T^{1}(A_{K}) → 0

we obtain

Z_{Σ}(s) = \frac{1}{(2π)^{t}b_{S}(T)} ∫_{(T (A_{K})/T^{1}(A_{K}))^{∗}} dχ_{y} ∫_{(T^{1}(A_{K})/T (K))^{∗}} Ĥ_{Σ}(χ, −s)dχ_{l}

We have an isomorphism MR^{G} ≃ (T (A_{K})/T^{1}(A_{K}))^{∗} and the measure dχ_{y} coincides with the usual Lebesgue measure on MR^{G}. Recall that a character χ ∈ (T (A_{K})/T (K))^{∗} defines r Hecke characters χ_{1}, ..., χ_{r} of the idele groups G_{m}(A_{K_{j}}). In particular, we get r characters χ_{1,y}, ..., χ_{r,y}. We have an embedding M^{G} ⊂ P L(Σ)^{G}, which together with explicit formulas for Fourier transforms of local heights shows that the integral

A_{Σ}(s, χ_{y}) := \frac{1}{b_{S}(T)} ∫_{(T^{1}(A_{K})/T (K))^{∗}} Ĥ_{Σ}(χ, −(s + 1))dχ_{l}

is a function on P L(Σ)C^{G} and we have

A_{Σ}(s, χ_{y}) = A_{Σ}(s + iy) = A_{Σ}(s_{1} + iy_{1}, ..., s_{r} + iy_{r}).

Denote by Γ := M^{G} the lattice of K-rational characters of T. Let t be the rank of Γ. The case t = 0 corresponds to an anisotropic torus T. It has been considered in [2]. So we assume t > 0.

For any element γ ∈ Γ ⊂ Z^{r} we denote by l(γ) the number of its coordinates which are zero (0 ≤ l(γ) ≤ r). Let l(Γ) be the minimum of l(γ) among γ ∈ Γ. Notice that l(Γ) ≤ r − t − 1. Indeed, if we had l(Γ) ≥ r − t, then M^{G} would be contained in the intersection of r − t linear coordinate hyperplanes s_{j} = 0 (the latter contradicts the condition MR^{G} ∩ R_{≥0}^{r} = 0). We can always choose a Z-basis γ^{1}, …, γ^{t} of Γ in such a way that l(Γ) = l(γ^{u}) (u = 1, …, t). Without loss of generality we assume that Γ is contained in the intersection of coordinate hyperplanes s_{j} = 0, j ∈ {1, …, l(Γ)}. We set

Φ(s) := ∏_{j=1}^{l(Γ)} s_{j}.

For any u ≤ t we define a subgroup Γ^{(u)} ⊂ Γ of rank u as

Γ^{(u)} := ⨁_{j=1}^{u} Z < γ^{j} > .

We introduce some auxiliary functions

W_{Σ}^{(u)}(s) = ∫_{ΓR^{(u)}} A_{Σ}(s + iy^{(u)})dy^{(u)}

where dy^{(u)} is the induced Lebesgue measure on ΓR^{(u)} ⊂ R^{r}, normalized by Γ^{(u)}. Denote VC^{(u)} = C^{r}/ΓC^{(u)}. We prove by induction that W_{Σ}^{(u)}(s) is good with respect to Γ^{(u)} ⊂ Z^{r}.

By 4.6, W_{Σ}^{(u)}(s) is an analytic function in the domain Re(s) ∈ R_{>0}^{r}.

There exist δ_{1}, ..., δ_{t} > 0 such that the integral

∫_{Re(z)=δ_{u}} Φ(s) · W_{Σ}^{(u−1)}(s + z · γ^{u})dz (u = 1, 2, …, t)

converges absolutely and uniformly in any compact contained in a tube domain U_{C} over an open neighborhood U of 0. This can be seen as follows: For any ε with 0 < ε < 1/rd′, where d′ = dim M_{R,∞}, we can choose a ball B_{e_{1}} ⊂ R^{r} of radius e_{1} around 0 such that for any ball B_{e_{2}} ⊂ B_{e_{1}} of radius e_{2} (0 < e_{2} < e_{1}) around 0 we can uniformly bound the Hecke L-functions L_{K_{j}}(s_{j} + 1, χ_{j}) appearing in Ĥ_{Σ}(χ, s) by

c_{j}(e_{2})(∥y(χ_{j})∥ + |Im(s_{j})| + 1)^{ε}

with some constants c_{j}(e_{2}) for all s in the domain Re(s_{j}) ∈ B_{e_{1}}\B_{e_{2}} for j = 1, ..., r (see 6.2). By 4.6, this assures the absolute and uniform convergence of the integral

∫_{ΓR^{(u)}} A_{Σ}(s + iy^{(u)})dy^{(u)}

for all s contained in a compact in C^{r} such that Re(s_{j}) ∈ B_{ε_{1}}\B_{ε_{2}} for j = 1, ...r. We know that the coordinates γ_{j}^{u} of the vectors γ^{u} = (γ_{1}^{u}, ..., γ_{r}^{u}) ∈ Z^{r} are not equal to zero for l(Γ) < j ≤ r. Therefore, we can now choose such real δ_{u} > 0 that δ_{u}γ_{j}^{u} are all contained in the open ball B_{e_{1}}. So there must exist some e_{2} > 0 such that δ_{u}γ_{j}^{u} ∉ B_{e_{2}} for all u = 1, ..., t and all l(Γ) < j ≤ r. It follows that there exists an open neighborhood of 0, such that for all s contained in this neighborhood we have Re(s_{j} + δ_{u}γ_{j}^{u}) ∈ B_{e_{1}}\B_{e_{2}} for all l(Γ) < j ≤ r. Since we remove the remaining poles by multiplying with Φ(s) we obtain the absolute and uniform convergence of W_{Σ}^{(u)}(s) to a holomorphic function in s in this neighborhood.

Moreover, the multiplicity of the meromorphic function

W̃_{δ_{u}}^{(u)}(s) := ∫_{Re(z)=δ_{u}} W_{Σ}^{(u−1)}(s + z · γ_{u})dz

at 0 is at least

μ(Φ(s)) = 1 + t − r = 1 + rk Γ − r ≥ 1 + rk Γ^{(u)} − r.

We apply Theorem 6.19. It is clear that

∑_{χ∈(T^{1}(A_{K}/T (K)K_{T})^{∗}} Ĥ_{Σ}(χ, −(s_{1} − 1), …, −(s_{r} − 1))

is good with respect to the trivial lattice Γ = 0 (The property (iii) follows from estimates 6.3 and 4.6). This concludes the proof.

□

Theorem 7.2 Denote by Ĥ_{Σ,S}(χ, −s) the multiplicative Fourier transform of the height function with respect to the measure ω_{Ω,S} (see 2.2). The principal coefficient C(Σ) of

A_{Σ}(s) = \frac{1}{b_{S}(T)} ∫_{(T^{1}(A_{K}/T (K)K_{T})^{∗}} Ĥ_{Σ,S}(χ_{l}, −s)dχ_{l}

at s_{1} = ... = s_{r} = 1 is equal to β(P_{Σ})τ_{K}(P_{Σ}).

Proof. We follow closely the exposition of the proof of theorem 3.4.6 in [2]. Since M^{G} ↪ P L(Σ)^{G} we have an embedding of characters

(T (A_{K})/T^{1}(A_{K}))^{∗} = MR^{G} ↪ ∏_{j=1}^{r}(G_{m}(A_{K_{j}})/G_{m}^{1}(A_{K_{j}}))^{∗}.

Recall that the kernel of

a^{∗} : (T (A_{K})/T (A_{K}))^{∗} → ∏_{j=1}^{r}(G_{m}(A_{K_{j}})/G_{m}(A_{K_{j}}))^{∗}

is dual to the obstruction group to weak approximation A(T) = T (A_{K})/\overline{T (K)}. We have a splitting

\overline{T (K)} = \overline{T (K)}_{S} × T (A_{K,S}).

Here we denoted by \overline{T (K)}_{S} the image of \overline{T (K)} in ∏_{v∈S} T (K_{v}) and T (A_{K,S}) = T (A_{K}) ∩ ∏_{v∉S} T (K_{v}). The pole of the highest order r of Ĥ_{Σ,S}(χ_{l}, −s) at s_{1} = ... = s_{r} = 1 appears from characters χ_{l} such that the corresponding χ_{1}, ..., χ_{r} are trivial characters of the groups G_{m}(A_{K_{j}})/G_{m}(K_{j}), i.e., χ_{l} is a character of the finite group A(T) = ∏_{v∈S} T (K_{v})/\overline{T (K)}_{S}, and is trivial on the group T (A_{K,S}).

For s ∈ R_{>1}^{r} we can again apply the Poisson formula to A(T). By 1.18, the order of A(T) equals β(P_{Σ})/i(T). We obtain

\frac{1}{b_{S}(T)} ∑_{χ∈(A(T))^{∗}} Ĥ_{Σ,S}(χ_{l}, −s) = \frac{β(P_{Σ})}{i(T)b_{S}(T)} ∫\overline{T (K)} H_{Σ}(x, −s)ω_{Ω,S}

(see 1.18). We restrict to the line s_{1} = ... = s_{r} = s and we want to compute the limit

lim_{s→1}(s − 1)^{r} ∫\overline{T (K)} H_{Σ}(x, −s)ω_{Ω,S}.

We have

∫\overline{T (K)} H_{Σ}(x, −s)ω_{Ω,S} =  (1)

= ∫_{\overline{T (K)}_{S}} ∏_{v∈S} H_{Σ,v}(x_{v}, −s)ω_{Ω,v} · ∏_{v∉S} ∫_{T (K_{v})} H_{Σ,v}(x_{v}, −s)dμ_{v}

(recall that ω_{Ω,v} = ∏_{v∈Val(K)} dμ_{v} and dμ_{v} = L_{v}(1, T; E/K)ω_{Ω,v} for all v and L_{v}(1, T; E/K) = 1 for v ∈ S).

From our calculations of the Fourier transform of local height functions for v ∉ S (3.10), we have

∏_{v∉S} ∫_{T (K_{v})} H_{Σ,v}(x_{v}, −s)dμ_{v} =  (2)

= L_{S}(s, T; E/K) · L_{S}(s, T_{NS}; E/K) ∏_{v∉S} Q_{Σ}(q_{v}^{−s}, …, q_{v}^{−s}).

By 3.7,

∏_{v∉S} Q_{Σ}(q_{v}^{−s}, …, q_{v}^{−s})

is an absolutely convergent Euler product for s = 1. Moreover, the limits

lim_{s→1}(s − 1)^{t}L_{S}(s, T; E/K)

lim_{s→1}(s − 1)^{(r−t)}L_{S}(s, T_{NS}; E/K)

exist and equal the non-zero constants l_{S}(T) and l_{S}^{−1}(P_{Σ}) (r = t + k). By 3.11,

∫_{\overline{T (K)}_{S}} ∏_{v∈S} H_{Σ,v}(x_{v}, −s)ω_{Ω,v}

is absolutely convergent for s_{1} = ... = s_{r} = 1. Using (1) and (2), we obtain:

lim_{s→1}(s − 1)^{r} ∫\overline{T (K)} H_{Σ}(x, −s)ω_{Ω,S} =  (3)

= \frac{l_{S}(T)}{l_{S}(P_{Σ})} ∫_{\overline{T (K)}_{S}} ∏_{v∈S} H_{Σ,v}(x_{v}, −s)ω_{Ω,v} · ∏_{v∉S} Q_{Σ}(q_{v}^{−1}, …, q_{v}^{−1}).

Now recall (3.10), that for v ∉ S we have

Q_{Σ}(q_{v}^{−1}, …, q_{v}^{−1}) = ∫_{T (K_{v})} L_{v}^{−1}(1, T_{NS}; E/K)H_{Σ,v}(x_{v}, −1)ω_{Ω,v}.

It was proved in [2] Proposition 3.4.4 that the restriction of the v-adic measure ω_{K,v} to T (K_{v}) ⊂ P_{Σ}(K_{v}) coincides with the measure

H_{Σ,v}(x, −1)ω_{Ω,v}.

Here K is the canonical sheaf on the toric variety P_{Σ} metrized as above.

We also have

∫_{\overline{T (K)}_{S}} ∏_{v∈S} H_{Σ,v}(x_{v}, −1)ω_{Ω,v} = ∫_{\overline{T (K)}_{S}} ∏_{v∈S} ω_{K,v}.  (4)

Using the splitting \overline{T (K)} = \overline{T (K)}_{S} × T (A_{K,S}) and multiplying the above equations we get

∫\overline{T (K)} ω_{K,S} = ∫_{\overline{T (K)}_{S}} ∏_{v∈S} ω_{K,v} · ∏_{v∉S} ∫_{T (K_{v})} L_{v}^{−1}(1, T_{NS}; E/K)ω_{K,v}.

On the other hand, it was proved in [2], Proposition 3.4.5 that we have

∫\overline{T (K)} ω_{K,S} = ∫\overline{P_{Σ}(K)} ω_{K,S} = b_{S}(P_{Σ}).

Therefore,

b_{S}(P_{Σ}) = ∫_{\overline{T (K)}_{S}} ∏_{v∈S} H_{Σ}(x, −φ_{Σ})ω_{Ω,v} · ∏_{v∉S} Q_{Σ}(q_{v}^{−1}, …, q_{v}^{−1}).

Collecting terms, we obtain

C(Σ) = \frac{β(P_{Σ})}{i(T)b_{S}(T)} · \frac{l_{S}(T)}{l_{S}(P_{Σ})} · b_{S}(P_{Σ}).

By 2.4 and 2.9, we have the following equality

i(T)b_{S}(T) = h(T)l_{S}(T).

It remains to notice that we have an exact sequence of lattices

0 → M^{G} → P L(Σ)^{G} → Pic(P_{Σ}) → H^{1}(G, M) → 0

and that the number h(T) = |H^{1}(G, M)| appears in the integral formula for the X -function of the cone Λ_{eff} ⊂ Pic(P_{Σ}). We apply Theorem 5.5 and obtain that

W_{Σ}(s) = \frac{1}{(2π)^{t}b_{S}(T)} ∫_{MR^{G}} A_{Σ}(s + iy)dy

is good with respect to the lattice M^{G} and that

C(Σ) = β(P_{Σ})τ_{K}(P_{Σ})

is the principal coefficient of W_{Σ}(s) at 0. □

Theorem 7.3 There exists a δ > 0 such that the height zeta-function ζ_{Σ}(s) obtained by restiction of the zeta-function Z_{Σ}(s) to the complex line s_{j} = φ(e_{j}) = s for all j = 1, ..., r has a representation of the form

ζ_{Σ}(s) = \frac{Θ(Σ)}{(s − 1)^{k}} + \frac{g(s)}{(s − 1)^{k−1}}

with k = r − t = rk Pic(P_{Σ}) and some holomorphic function g(s) in the domain Re(s) > 1 − δ. Moreover, we have

Θ(Σ) = α(P_{Σ})β(P_{Σ})τ_{K}(P_{Σ}).

Proof. Since W_{Σ}(s) is good with respect to the lattice M^{G} ⊂ Z^{r}, we have the following representation of W_{Σ}(s) in a small open neighborhood of 0:

W_{Σ}(s) = \frac{P (s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}

where P (s) = P_{0}(s) + P_{1}(s), μ(P_{1}) > μ(P_{0}) and

\frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)} = β(P_{Σ})τ_{K}(P_{Σ}) · X_{Λ_{eff}}(ψ(s)),

where X_{Λ_{eff}} is the X -function of the cone Λ_{eff} = ψ(R_{≥0}^{r}) ⊂ Pic(P_{Σ})_{R}.

If we restrict

\frac{P_{0}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}

to the line s_{j} = s − 1 (j = 1, …, r), then we get the meromorphic function Θ(s − 1)^{−k} with Θ = α(P_{Σ})β(P_{Σ})τ_{K}(P_{Σ}). Moreover, the order of the pole at s = 1 of the restriction of

\frac{P_{1}(s)}{∏_{j=1}^{p} l_{j}^{k_{j}}(s)}

to the line s_{j} = s − 1 (j = 1, …, r) is less than k. Therefore, this restriction can be written as g(s)/(s − 1)^{k−1} for some analytic at s = 1 function g(s).

Corollary 7.4 Let T be an algebraic torus and P_{Σ} its smooth projective compactification. Let k be the rank of Pic(P_{Σ}). Then the number of Krational points x ∈ T (K) having the anticanonical height H_{K^{−1}}(x) ≤ B has the asymptotic

N(T, K^{−1}, B) = \frac{Θ(Σ)}{(k − 1)!} · B(log B)^{k−1}(1 + o(1)), B → ∞.

Proof. We apply a Tauberian theorem to ζ_{Σ}(s). □

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