Height Zeta Functions of Toric Varieties

We investigate analytic properties of height zeta functions of toric varieties. Using the height zeta functions, we prove an asymptotic formula for the number of rational points of bounded height with respect to an arbitrary line bundle whose first Chern class is contained in the interior of the cone of effective divisors


Introduction
Let X be a d-dimensional algebraic variety defined over a number The paper is organized as follows: The technical heart of the paper is contained in Section 2, where we investigate analytic properties of some complex valued functions related to convex cones.
In Section 3, we review basic facts from harmonic analysis on the adele group of an algebraic torus.
In Section 4, we recall the terminology from the theory of toric varieties as well as the definition and main properties of heights on toric varieties.
In Section 5, we give the proof of 1.4. We remark that the most subtle part in the statement of 1.4 is the nonvanishing of the asymptotic constant g(a(L)) = 0.

Technical theorems
Let I and J be two positive integers, R[s, t] (resp. C[s, t]) the ring of polynomials in I + J variables s 1 , . . . , s I , t 1 , . . . , t J with coefficients in R (resp. in C) and C [[s, t]] the ring of formal power series in s 1 , . . . , s I , t 1 , . . . , t J with complex coefficients.
an arbitrary complex vector with l(γ I ) = 0, and g(s, t) := f 1 (s, t)/f 2 (s). Then the multiplicity of the functioñ is not identically zero.
Let Γ ⊂ Z I+J be a sublattice, Γ R ⊂ R I+J (resp. Γ C ⊂ C I+J ) the scalar extension of Γ to a R-subspace (resp. to a C-subspace). We always assume that Γ R ∩ R I+J ≥0 = 0 and Γ R ∩ R J = 0. We set P R := R I+J /Γ R and P C := C I+J /Γ C . Let π I be the natural projection C I+J → C I . Denote by ψ (resp. by ψ I ) the canonical surjective mapping C I+J → P C (resp. C I → C I /π I (Γ C )).
Remark 2.7. By definition, if f (s, t) descends to P C , then there exists an analytic function g on ψ(U) ⊂ P C such that f = g • ψ. Using Cauchy-Riemann equations, one immediatelly obtains that f descends to P C if and only if for any vector α ∈ Γ R and any u = (u 1 , . . . , u I+J ) ∈ U such that u + iα ∈ U, one has f (u + iα) = f (u).
is called good with respect to Γ and the set of variables {s 1 , ..., s I } if it satisfies the following conditions: (i) f (s, t) descends to P C ; (ii) There exist pairwise coprime linear homogeneous polynomials and positive integers k 1 , . . . , k p such that for every j ∈ {1, . . . , p} the linear form l j (s) descends to P C , l j (s) does not vanish for Re(s) ∈ R I >0 , and is analytic at 0.
(iii) There exists a nonzero constant C(f ) and a homogeneous polynomial q 0 (s) of degree µ(q) in variables s 1 , . . . , s I such that q(s, t) = q 0 (s) + q 1 (s, t) and where q 1 (s, t) is an analytic function at 0 with µ(q 1 ) > µ(q 0 ), both functions q 0 , q 1 descend to P C , and X Λ(I) is the X -function of the cone Λ(I) = ψ I (R I ≥0 ) ⊂ ψ I (R I ) (see Definition 6.1).
Remark 2.9. Let q(s, t) be an arbitrary analytic at 0 function. Collecting terms in the Taylor expansion of q, we see that there exists a unique homogeneous polynomial q 0 (s, t) and an analytic at 0 function q 1 (s, t) such that q(s, t) = q 0 (s, t) + q 1 (s, t) with µ(q) = µ(q 0 ) < µ(q 1 ). In particular, the polynomial q 0 and the function q 1 in 2.8 are uniquely defined. Definition 2.10. If f (s, t) is good with respect to Γ and the set of variables {s 1 , ..., s I } as above, then the meromorphic function q 0 (s) p j=1 l k j j (s) will be called the principal part of f (s, t) at 0 and the constant C(f ) the principal coefficient of f (s, t) at 0.
The following easy statement will be helpful in the sequel: 11. Let f (s, t) be an analytic at 0 function, l(s) ∈ R[s] a homogeneous linear function such that l(γ I ) = 0. Assume that f (s, t) and l(s) descend to P C . Theñ Theorem 2.12. Let f (s, t) be a good function with respect to Γ and the set of variables {s 1 , ..., s I } as above, the product of those linear forms l j (s) (j ∈ {1, . . . , p}) which vanish on γ I . Assume that the following statements hold: converges absolutely and uniformly to a holomorphic function on any compact in the domain Re(s) converges absolutely and uniformly in an open neighborhood of 0. Moreover, the multiplicity of the meromorphic functioñ at 0 is at least 1 − dimψ I (R I ); (iii) For any Re(s) ∈ R I >0 and Re(t) ∈ R J >−δ 0 , one has lim λ→+∞ sup 0≤Re(z)≤δ, |Im(z)|=λ |f (s + z · γ I , t + z · γ J )| = 0.
Let U be the intersection of R I+J >0 with an open neighborhood of 0 such that Φ(s) ·f δ (s, t) is analytic for all (s, t) ∈ U. By the property (i), both functionsf δ (s, t) andf (s, t) are analytic in U. Moreover, the integral formulas forf δ (s, t) andf (s, t) show that the equalities f δ (u + iy · γ) =f δ (u) andf (u + iy · γ) =f (u) hold for any y ∈ R and u, u + iy · γ ∈ U. Therefore, both functionsf δ (s, t) andf (s, t) descend toP C (see Remark 2.7). Using the properties (i)-(iii), we can apply the residue theorem and obtaiñ Res z=z j f (s + z · γ I , t + z · γ J ) for s, t ∈ U. We denote by U(γ) the open subset of U which is defined by the inequalities The open set U(γ) is non-empty, since we assume that g.c.d.(l j , l m ) = 1 for j = m. Moreover, for (s, t) ∈ U(γ), we have and where q 0 (s) is a uniquely determined homogeneous polynomial (see Remark 2.9), q 0 (s, t) is an analytic at 0 function with µ(q) = µ(q 0 ) < µ(q 1 ) and X Λ(I) (ψ I (s)) is the X -function of the cone Λ(I) = ψ I (R I ≥0 ). We set . Then Res z=z j R 1 (s + z · γ I , t + z · γ J ).
Moving the contour of integration Re(z) = δ (δ → +∞), by residue theorem, we obtain On the other hand, Consider the decomposition off into the sum:

By our assumption in
and the analytic in the domain U(γ) functions It is clear that Res are analytic at 0. Let us define the set {l 1 (s), . . . ,lp(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 , . . . , np and a representation of the meromorphic functionsf (s, t) andR 0 (s) as quotients whereq(s, t) is analytic at 0,q 0 (s) is a homogeneous polynomial, and none of the formsl 1 (s), . . . ,l q (s) vanishes for (s, t) ∈ R I+J >0 (the last property can be achieved, because both functionsf (s, t) andR 0 (s, t) are analytic in U and the closure of U is equal to R I+J ≥0 ). Defineq whereq 0 (s, t) is a homogeneous polynomial andq 1 (s, t) is an analytic at 0 function such that µ(q) = µ(q 0 ) < µ(q 1 ). Moreover, i.e.,f is good. ✷ Definition 2.13. For any finite dimensional Banach space V over R we denote by · a representative in the class of equivalent norms on V . For y = (y 1 , ..., y r ) ∈ R r we will set The following lemma is elementary: be a complex valued function on V satisfying the inequality Assume that there exist constants ε, ε 0 > 0 such that the following holds: (ii) For all ε 1 (with 0 < ε 1 < ε) there exist a constant C(ε 1 ) > 0 and an estimate which holds for all s such that one of the two inequalities −ε < Re(s j ) < ε 1 or Re(s j ) > ε 1 is satisfied for every j = 1, ..., I.
Then the integral is a good function with respect to Γ and the set of variables {s 1 , ..., s I }, and C(f ) is its principal coefficient.
Proof. Without loss of generality we can assume that Γ is not contained in any of I coordinate hyperplanes s j = 0 (j = 1, ..., I), otherwise we reduce the problem to a smaller value of I. Therefore, we can choose a basis γ 1 , ..., γ t of Γ such that all first I coordinates of γ u = (γ u I , γ u J ) ∈ Z I+J are not equal to 0 for every u = 1, ..., t.
For any non-negative integer u ≤ t we define a subgroup Γ (u) ⊂ Γ of rank u as follows: We introduce some auxiliary functions We prove by induction that f (u) (s, t) is good with respect to Γ (u) ⊂ Z I+J and {s 1 , ..., s I }. By (i), f (0) (s, t) is good. By induction assumption, we know that f (u−1) (s, t) is good with respect to Γ (u−1) and {s 1 , ..., s I }. Moreover, we have Choose δ u > 0 in such a way that for every j = 1, ..., I one of the following two inequalities is satisfied: converges absolutely and uniformly in an open neighborhood of 0, i.e. the multiplicity of f Hence it is holomorphic at 0 and satisfies assumption (ii) of 2.12. By lemma 2.14, the property 2.12 (iii) holds. Applying theorem 2.12, we conclude that f (u) (s, t) is a good function with the principal coefficient g(0). ✷

Fourier analysis on algebraic tori
Let X F be an algebraic variety over a number field F and E/F a finite extension of number fields. We shall denote by X E the E-variety obtained by base change from X F and by X(E) the set of E-rational points of X F . Sometimes we omit the subscript in X E if the field is clear from the context.
We call the minimal E with this property the splitting field of T . Denote by G = Gal(E/F ) the Galois group of E over F . For every G-module A, A G stands for the submodule of elements fixed by G. For any field E we denote byT E the G-module Hom(T E , G m ) of E-rational characters of T . If E is the splitting field of T , we put M :=T E and N := Hom(M, Z) the dual G-module. We denote by t the rank of the lattice M G .
Let T be an algebraic torus over a number field F . Denote by Val(F ) the set of valuations of F and by Val ∞ (F ) the set of archimedian valuations. Let F v be the completion of F with respect to v ∈ Val(F ), V an extension of v to E, the decomposition group at v, T (F v ) the group of F v -rational points of T and T (O v ) its maximal compact subgroup. We have the canonical embeddings for all non-archimedian v ∈ Val(F ) and We call a valuation v ∈ Val(F ) good, if the mapping π v is an isomorphism. We denote by S a finite subset in Val(F ) containing Val ∞ (F ) and all valuations v ∈ Val(F ) which are not good.
Let us recall some basic arithmetic properties of algebraic tori over the ring of adeles A F . Define   Define the group U T as: We call the characters χ ∈ D T discrete and χ ∈ U T unramified.
Using 3.2 (i), we see that a choice of a splitting of the exact sequence and M 1 R,∞ is the minimal R-subspace in M R,∞ containing the image of U T under the canonical mapping ¿From now on we fix such a non-canonical splitting. This allows to consider U T as a subgroup of H T . By 3.2, we have: where M T is the image of the canonical projection of U T to M 1 R,∞ and cl * (T ) is a finite abelian group dual to cl(T ). We see from 3.4 that a character χ ∈ M G R ⊕ U T is determined by its archimedian component which is an element in M R,∞ up to a finite choice. Denote by y We define the canonical measure on the group T (A F ) For archimedian valuations the Haar measure dµ v is the pullback of the Lebesgue measure on N Gv R under the logarithmic map Let dx be the Lebesgue measure on T (A F )/T 1 (A F ). There exists a unique Haar measure ω 1 on T 1 (A F ) such that ω = ω 1 dx. We define For any L 1 -function f on T (A F ) and any topological character χ we denote byf (χ) its global Fourier transform with respect to ω and bŷ f v (χ v ) the local Fourier transforms. We will use the following version of the Poisson formula: Theorem 3.5. Let G be a locally compact abelian group with Haar measure dg, G 0 ⊂ G a closed subgroup with Haar measure dg 0 . The factor group G/G 0 has a unique Haar measure dx normalized by the condition dg = dx · dg 0 . Let f : G → C be an L 1 -function on G and f its Fourier transform with respect to dg. Suppose thatf is also an L 1 -function on G ⊥ 0 , where G ⊥ 0 is the group of topological characters χ which are trivial on G 0 . Then where dχ is the orthogonal Haar measure on G ⊥ 0 with respect to the Haar measure dx on G/G 0 .
We will apply this formula with G = T (A F ), G 0 = T (F ), dg = ω and dg 0 is the discrete measure on T (F ). The Haar measure dχ induces the Lebesgue measure on M G R normalized by the lattice M G ⊂ M G R and the discrete measure on D T .  (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; (iv) N R is the union of cones from Σ;

Geometry of toric varieties
(v) every cone σ ∈ Σ is generated by a part of a Z-basis of N; (vi) For any g ∈ G and any σ ∈ Σ, one has g(σ) ∈ Σ.
A complete regular d-dimensional fan Σ defines a smooth toric variety X Σ,E as follows: whereσ ⊂ M R is the dual to σ cone. We can see that T E ⊂ U σ for all σ ∈ Σ and that U 0 = T .

Theorem 4.2. [14]
Let Σ be a complete regular G-invariant fan in N R . Assume that the complete toric variety X Σ,E defined over the splitting field E by Σ is projective. Then there exists a unique complete algebraic variety X Σ,F over F such that its base extension to E is isomorphic to X Σ,E .
Denote by Σ(j) the subset of j-dimensional cones in Σ and by N σ,R ⊂ N R the minimal linear subspace containing σ. Let {e 1 , ..., e n } be the set of 1-dimensional generators of Σ. Denote by P L(Σ) the lattice of piecewise linear integral functions on N. By definition, a function ϕ ∈ P L(Σ) iff ϕ(N) ⊂ Z and the restriction of ϕ to every cone σ ∈ Σ is a linear function; equivalently, there exist elements m σ ∈ M such that the restriction of ϕ to σ is given by < ·, m σ > where < ·, · > is induced from the pairing between N and M. The G-action on M (and N) induces a G-action on the free abelian group P L(Σ). Let be the decomposition of Σ(1) into a union of G-orbits. A G-invariant piecewise linear function ϕ ∈ P L(Σ) G is determined by the vector u = (u 1 , ..., u r ), where u i is the value of ϕ on the generator of some 1-dimensional cone in the G-orbit Σ i (1), (i = 1, ..., r). It will be convenient for us to consider complex valued piecewise linear functions and to identify ϕ = ϕ u ∈ P L(Σ) G C with its complex coordinates u = (u 1 , ..., u r ) ∈ P L(Σ) G C . Theorem 4.3. The toric variety X Σ has the following properties: (i) There is a representation of X Σ,E as a disjoint union of split algebraic tori T σ,E of dimension dim T σ,E = d − dim σ. For each jdimensional cone σ ∈ Σ(j) we denote by T σ,E the kernel of a homomorphism T E → G j m,E defined by a Z-basis of the sublattice N ∩ N σ,R . (ii) The closures of (d − 1)-dimensional tori corresponding to the 1dimensional cones R ≥0 e 1 , ..., R ≥0 e n ∈ Σ(1) define divisors T 1 , ..., T n . We can identify the lattices P L(Σ) = ⊕ n j=1 Z[T j ]. (iii) There is an exact sequence of G-modules 0 → M → P L(Σ) → Pic(X Σ,E ) → 0, moreover, we have Pic(X Σ,F ) = Pic(X Σ,E ) G ; (iv) The cone of effective divisors Λ eff (X Σ,F ) ⊂ Pic(X Σ,F ) R is generated by the classes of G-invariant divisors Let v be an archimedian valuation. The complex local height function is defined as Remark 4.6. This provides a piecewise smooth metrization of line bundles on the toric variety X Σ . One can show that this metrization is, in a sense, "canonical". Namely, an algebraic torus admits a morphism to itself (n-th power morphism), which extends to a compactification. Using the construction of Tate one can obtain a metrization on a line bundle by a limiting process. This metrization coincides with ours.

be a rational point. The global height function is defined by
By the product formula, the function H Σ (x, ϕ) as a function on T (F ) descends to the complexified Picard group Pic(X Σ ) C . Moreover, we have the following It induces a surjective map of tori a : r j=1 R F j /F (G m ) → T and a surjective homomorphism Every character χ ∈ H T defines r Hecke characters χ 1 , ..., χ r of the groups G m (A F,j )/G m (F j ) by χ • a. It is known [6], that Coker(a) is isomorphic to the obstruction group to weak approximation A(T ) (see 3.6). Similarly, every local character χ v defines local characters χ 1,v , ...χ r,v . If χ is trivial on K T then all χ j are trivial on the maximal compact subgroups in in G m (A F j ), in other words, all χ j are unramified. Their local components for all valuations are given by In the remaining part of this section we recall some estimates which will be used it the study of analytic properties of the height zeta function (see [2]).
Let T be an algebraic torus and χ ∈ U T an unramified character. Denote by χ v its local components and by χ 1 , ..., χ r the induced unramified Hecke characters of G m (A F j ).
where for every field F j we denoted by L F j (χ j , u) the standard Hecke L-function of F j . For any χ ∈ D T we define Proposition 4.11.
[2] Let χ ∈ U T be an unramified character and y(χ) its image in M R,∞ . For all δ 0 > 0 there exists a constant c(δ 0 ) such that for any u in the domain Re(u) ∈ R r >1/2+δ 0 we have the following estimate where ρ + t is the dimension of the real vector space M R,∞ .
Corollary 4.13. For any δ 0 > 0, there exists a constant c(δ 0 ) such that for any χ ∈ U T and any u in the domain Re(u) ∈ R r >1/2+δ 0 we have the following estimate:

Analytic properties of height zeta functions
Definition 5.1. Let X Σ be a smooth projective toric variety. Let ϕ = ϕ u ∈ P L(Σ) G C be a complexified piecewise linear function. Let Y ⊂ X Σ be a locally closed subset. The height zeta function with respect to Y is defined as Let us formulate the first main result.

Theorem 5.2. [2]
The height zeta function Z Σ (T, u) as a function on P L(Σ) G C is holomorphic for Re(u) ∈ R r >1 . Moreover, it descends to Pic(X Σ ) C and is holomorphic for Re(u) contained in the open cone we have the following formula: The integral converges absolutely and uniformly to a holomorphic function in u in any compact in the domain Re(u) ∈ R r >1 .
Let L be a line bundle on X Σ metrized as above, such that its class [L] is contained in the interior of the cone of effective divisors Λ eff (X Σ ) ⊂ Pic(X Σ ). We have defined a(L) as where λ j ∈ Q >0 . Therefore, Fix these λ j and choose ε > 0 such that 2ε < min j∈J(L) λ j . We denote by ϕ L the piecewise linear function from P L(Σ) G R such that a(L)ϕ L (e i ) = 1 for i = 1, . . . , I and a(L)ϕ L (e j ) = 1 + λ j for j ∈ J(L).
Here e i are generators of one-dimensional cones R ≥0 e i in the G-orbits Σ i (1).
Using the explicit computation of the Fourier transform of local height functions and the absolute convergence of the integral in the domain Re(u) ∈ R r >1 , we have because the local height functions are invariant under the maximal compact subgroups T (O v ) ⊂ T (F v ) andĤ Σ (χ, −u) = 0 for all χ which are not trivial on the maximal compact subgroup K T . By 4.10, we have:Ĥ where χ 1 , ..., χ r are unramified Hecke characters of G 1 m (A F j ) induced from a character χ ∈ U T , and ζ ∞ (χ, u) is a function in u which is holomorphic in the domain Re(u) ∈ R r >1/2+δ 0 (for all δ 0 > 0). We have dy I is the Lebesgue measure on M G I,R and dy J the Lebesgue measure on M G J,R . Using the estimates 2.14, 4.12, 4.9, 4.13, we see that the sums and integrals above converge absolutely and uniformly to an analytic function in any compact in the domain Re(s) ∈ R I >0 and Re(t) ∈ R J >−δ 0 for some δ 0 > 0 (δ 0 < ε). Now the fact that the function Z Σ (T, s, t) is good with respect to the lattice M G I ⊂ Z I and the variables (s 1 , ..., s I ) follows from 2.15 and the following statement: exists and is not equal to zero.
We divide the proof of Theorem 5.5 into a sequence of lemmas: Lemma 5.6. Let U T (I) be the subgroup of U T consisting of characters χ ∈ U T such that the corresponding Hecke characters χ i (i = 1, . . . , I) are trivial. Denote Then lim s→0 s 1 · · · s I f Σ (s, 0) = lim s→0 s 1 · · · s I f I Σ (s, 0).
where the subgroup A ⊂ T (A F ) is defined as

Proof.
By definition of f I Σ (s, t), we conclude that this function equals to the integral of the Fourier transform of the adelic height function over the subgroup of characters χ of T (A F ) which are trivial on T (F ) and such that the induced Hecke characters χ i are trivial for i ∈ I(L). It follows from the diagram that the common kernel of all such characters is T (F )T I (F ) (here we used the isomorphism A(T I ) = T I (A F )/T I (F )). The proof of the absolute convergence of the integral over A in the domain Re(s, t) ∈ R I >0 × R J >−δ 0 is analogous to the proof of theorem 4.2 in [3]. ✷ Lemma 5.8. The function s 1 · · · s I f I Σ (s, t) extends to an analytic function in the domain Re(s, t) ∈ R r >−δ 0 .
Proof. The proof is similar to the proof of theorem 4.2 in [3]. The integral A H Σ (x, (−s, −t))dα can be estimated from above by an Euler product which is absolutely convergent in the domain Re(s, t) ∈ R r >−δ 0 times a product of zeta functions I i=1 ζ F i (s i + 1). ✷ For (s, t) ∈ R r the function H Σ (x, (−s, −t)) has values in positive real numbers. Therefore, to prove the non-vanishing of the constant, it suffices to show the following: Hence, Here we denoted by dα S and dα v the Haar measures induced from dα I . We claim that is an absolutely convergent Euler product for Re(s, t) ∈ R r >−δ 0 . This statement follows from the explicit calculation of the local integrals (see 5.10).
for some ε 0 > 0 and all Re(s, t) ∈ R r >−δ 0 Proof. Denote by N R (I) the minimal R-subspace of N R spanned by all e with R ≥0 e contained in the set of 1-dimensional cones in ∪ i∈I(L) Σ i (1). Let Σ(L) be the complete G-invariant fan of cones in N R (I) which consists of intersections of cones in Σ ⊂ N R with the subspace N R (I). Since Σ(L) is not necessary a regular fan, we construct a new G-invariant fanΣ(L) by subdivision of cones in Σ(L) into regular ones using the method of Brylinski [4]. This reduces the computation of the local intergral to the one made for local height functions on smooth toric varieties in [2], theorem 2.2.6.
Let σ 1 , ..., σñ be the set of representatives of G v -orbits in the set of 1-dimensional cones inΣ(L) ⊂ N R (I). We obtain where l σ (s, t) are linear forms which are ≥ 1 − ε 0 in the domain Re(s, t) ∈ R r >−δ 0 , and QΣ (L) (z) is a polynomial in the variables z = (z 1 , ..., zñ) such that all monomials in QΣ (L) (z) − 1 have degree ≥ 2. Now we notice that l σ (0, 0) = 1 iff σ is a 1-dimensional cone in Σ and therefore, the cone R ≥0 e i for some i ∈ I(L) is contained in the G v -orbit of σ (see 5.11). ✷ Lemma 5.11. The set of lattice vectors e ∈ N such that a(L)ϕ L (e) = 1 coincides with the set of lattice vectors e i ∈ N R (I) with R ≥0 e i ∈ Σ(1) and a(L)ϕ L (e i ) = 1.
Proof. Let e be a lattice point in N. Since Σ is complete, there exists a d-dimensional cone σ ∈ Σ such that e ∈ σ. We claim that the property a(L)ϕ L (e) = 1 implies that e is a generator of a 1-dimensional face of σ. Indeed, we have a(L)ϕ L (x) ≥ ϕ Σ (x) for all x ∈ N R . On the other hand, σ is generated by a basis of N and ϕ Σ has value 1 on these generators. Hence, e must be one of the generators of σ.
It remains to show that the property a(L)ϕ L (e i ) = 1 for some generator e i of a 1-dimensional cone R ≥0 e i ∈ Σ implies that e i ∈ N R (L). But this follows from the definition of N R (L) as the subspace in N R generated by all elements e i ∈ N such that R ≥0 e i ∈ Σ and a(L)ϕ L (e i ) = 1. 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 finitely generated polyhedral cone Λ ⊂ A R such that Λ∩−Λ = 0 ∈ A R . Denote by Λ • the interior of Λ and by Λ • C = Λ • +iA R the complex tube domain over Λ • . Let (A * , A * R , Λ * ) be the triple consisting of the dual abelian group A * = Hom(A, Z), the dual real vector space A * R = Hom(A R , R), and the dual cone Λ * ⊂ A * R . We normalize the Haar measure dy on A * R by the condition: vol(A * R /A * ) = 1. 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. In particular, if (A, A R , Λ) = (Z k , R k , R k ≥0 ), then X Λ (s) = 1 s 1 · · · s k . Proposition 6.3. [3] Let (A, A R , Λ) and (Ã,Ã R ,Λ) be two triples as above, k = rk A andk = rkÃ, and ψ : A →Ã a homomorphism of free abelian groups with a finite cokernel Coker(ψ) (i.e., the corresponding linear mapping of real vector spaces ψ : A R →Ã R is surjective), and ψ(Λ) =Λ. Let Γ = Ker ψ ⊂ A, dy the Haar measure on Γ R = Γ ⊗ R normalized by the condition vol(Γ R /Γ) = 1. Then for all s with Re(s) ∈ Λ • the following formula holds: XΛ(ψ(s)) = 1 (2π) k−k |Coker(ψ)| Γ R X Λ (s + iy)dy, where |Coker(ψ)| is the order of the finite abelian group Coker(ψ).
Assume that ak-dimensional rational finite polyhedral coneΛ ⊂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 thatΛ is the image of the simplicial cone R r ≥0 ⊂ R r under the surjective map of vector spaces ψ : R r → A R . Denote by Γ the kernel of ψ. By 6.3 we obtain the following: Corollary 6.4. Let s = (s 1 , ..., s r ) be the standard coordinates in C r . Then where dy is the Haar measure on the additive group Γ R normalized by the lattice Γ, y j are the coordinates of y in R r , and |Coker(ψ)| is the index of the sublattice inÃ generated by a 1 , . . . , a r .