1 2

RATIONAL POINTS ON TORIC VARIETIES

V.V.BATYREV AND YU.TSCHINKEL

Abstract. We study the distribution of rational points on smooth compactifications of algebraic tori.

1. Introduction

Let X be a smooth Fano variety defined over a number field F and −K_{X} its anticanonical line bundle. Over some finite extension one expects infinitely many rational points on X. One can introduce appropriate counting functions - height functions and study the asymptotic for the number of rational points of bounded height with growing height. In [1] Batyrev and Manin formulated conjectures describing such asymptotics. Let L = (L, ∥ · ∥_{v}) be a metrized very ample line bundle on X. Denote by H_{L} : X(F) → ℝ_{≥⊬} the corresponding height function. Define the height zeta function associated with L as

Z_{L}(s) = ∑_{x∈X(F)} H_{L}(x)^{−s}.

Analytic properties of Z_{L}(s) determine the asymptotic behaviour of the number of F -rational points on X of bounded L-height. Denote this number by N(X, L, B). It is an experimental fact that in order to get a good description of the asymptotics of N(X, L, B) for B → ∞ in geometrical terms one should delete some Zariski closed subsets. Already for del Pezzo surfaces one has to deal with accumulating subvarieties - exceptional curves C are lines in the anticanonical embedding with an asymptotic N(C, −K_{X}|_{C}, B) ∼ B^{2} while the conjecture predicts

N(U, −K_{X}, B) ∼ B(log B)^{rk P ic(X)−1}

for some Zariski open U. One hopes that U can be chosen to be the complement of exceptional curves. In [5] geometric techniques were developed to prove upper bounds for the number of points on surfaces outside exceptional curves. One of the goals of the present paper is to show that it is still possible to get precise results if the accumulating subvarieties are contained in the divisor at infinity in a smooth compactification of an algebraic torus. Acknowledgements. We would like to thank Yu.I.Manin, D.Kazhdan and B. Mazur.

1Supported by the DFG, Forschungsschwerpunkt Komplexe Mannigfaltigkeiten

2Currently Junior Fellow of the Harvard Society of Fellows

Date: June 1994.

2. Geometry

Let us first recall some terminology from the theory of toric varieties over algebraically closed fields [3, 6].

Definition 2.0.1. A linear algebraic group T_{F} defined over a number field F is called a d-dimensional torus if its base extension T = T_{F} ×_{Spec(F)} (Spec(F̄)) is isomorphic to 𝔾_{m}^{d}(F̄).

The group of F -rational points of T_{F} is denoted by T (F). We call T_{F} split if it is isomorphic to 𝔾_{m}^{d}(F) already over F. We call T_{F} anisotropic if it has no nontrivial F -split F -subtori. Examples of anisotropic tori are provided by norm equations. Equivariant compactifications of split tori are described by the following objects:

1. A finitely generated free ℤ-module M of rank d and the dual module N = Hom(M, ℤ).

2. A complete d-dimensional fan Σ.

Definition 2.0.2. A finite set Σ consisting of convex rational polyhedral cones in N_{ℝ} = N ⊗ ℝ is called a complete 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; (iv) N_{R} is the union of cones from Σ.

We denote by Σ(i) the set of all i-dimensional cones in Σ. The toric variety X_{Σ} is obtained by glueing affine schemes

X_{Σ} = ⋃_{σ∈Σ} U_{σ}

where U_{σ} = Spec(F [M ∩ σ̂]) and σ̂ is the dual to σ cone. It contains T_{F} as a Zariski open subset. The toric variety X_{Σ} is smooth iff every cone in Σ is generated by a part of a ℤ-basis of N. There is a parallel theory of compactifications of nonsplit tori. In this case a T and Galois equivariant compactification of T_{F} is described by: M, N, Σ and an action of the Galois group on these objects.

Definition 2.0.3. A continuous function φ : N_{ℝ} → ℝ is called Σpiecewise linear if the restriction of φ to every cone σ ∈ Σ is a linear function. It is called integral if φ(N) ⊂ ℤ. We denote by m_{σ,φ} the restriction of φ to σ considered as an element in M. The group of integral piecewise linear functions is denoted by P L(Σ).

Proposition 2.0.4. The Picard group Pic(X_{Σ}) is isomorphic to P L(Σ)/M where elements of M are considered as globally linear integral functions on N_{ℝ}.

Denote by e_{1}, …, e_{n} the primitive integral generators of all 1-dimensional cones in Σ. They define T -invariant Weil divisors D_{1}, …, D_{n} on X_{Σ}.

Proposition 2.0.5. The piecewise linear function φ ∈ P L(Σ) such that φ(e_{i}) = 1 for all i (if it exists) corresponds to the anticanonical class −K_{Σ} in P ic(X_{Σ}).

Proposition 2.0.6. The cone of effective divisors N_{eff}^{1}(X_{Σ}) is generated by the classes [D_{1}], …, [D_{n}].

3. Heights

In this section we introduce metrizations on line bundles on toric varieties. The idea is that instead of studying Z_{L}(s) for some fixed metrized line bundle, one should consider the height zeta function as a function on the whole complexified Picard group. Therefore we need canonical simultaneous metrization on all line bundles. We don’t know how to achieve this on arbitrary Fano varieties. We know how to do it in two examples: for flag varieties [4] and for toric varieties.

Let F be a global field and 𝔸_{F} the adele ring of F. Denote by |·|_{v} : F_{v} → ℝ the standard norms on v-adic completions of F satisfying the product formula ∏_{v} |a|_{v} = 1 for all a ∈ F^{∗}. Let X = G/P be generalized flag variety. Here G is a semisimple linear algebraic group defined over F and P a parabolic subgroup containing a Borel subgroup B. Denote by π : G → G/P the natural projection. Denote by X^{∗}(P) the characters of P defined over F. There is an embedding of finite index

\begin{matrix}X^{∗}(P) & → & P ic(G/P) \\ χ & ↦ & L_{χ} \end{matrix}

where the line bundle L_{χ} is defined by

Γ(U, L_{χ}) = {f ∈ Γ(π^{−1}(U), O_{G}) | f(pg) = χ(p)f(g) ∀p ∈ P ∀g ∈ G}.

Fix a compact subgroup K = ∏_{v} K_{v} ⊂ G(𝔸_{F}) such that G(𝔸_{F}) = B(𝔸_{F})K. Let k = k(x) ∈ K be given by π(k) = x ∈ G/P (F). For a section s ∈ Γ(L_{χ}) nonvanishing in x put ∥s(x)∥_{v} = |f(k)|_{v} where f is a function on G nonvanishing in x and satisfying f(pg) = χ(p)f(g) as above. The height function is defined as

H_{L_{χ}}(x) = ∏_{v} ∥s(x)∥_{v}^{−1}.

The corresponding height zeta function is an Eisenstein series and its analytical properties can be summarized as follows: it is holomorphic in the domain ρ_{P} +𝔞_{𝔓}^{+} where ρ_{P} is half of the sum of roots counted with their multiplicities and 𝔞_{P}^{+} is the positive Weyl chamber. In geometric language it translates into holomorphy of the height zeta function inside the domain −K_{X} + N_{eff}^{1}(X) for X = G/P with simple poles at faces of the polyhedral cone −K_{X} + ∂N_{eff}^{1}(X). The multiplicity of the pole at the vertex −K_{X} equals rk P ic(X). There are no accumulating subvarieties. Let X = X_{Σ} be a complete split toric variety obtained by glueing affine schemes U_{σ} = Spec(F [M ∩ σ̌]). One can define a canonical covering of X_{Σ}(F_{v}) by compact subsets K_{σ,v} ⊂ U_{σ}(F_{v}) as follows:

K_{σ,v} = {x_{v} ∈ U_{σ}(F_{v}) | |m(x_{v})|_{v} ≤ 1 for all m ∈ M ∩ σ̌}.

Here we have used the canonical embedding T (F_{v})/T (O_{v}) ↪ N (resp. N ⊗ ℝ for archimedian v). For σ, τ ∈ Σ one has K_{σ,v} ∩ K_{τ,v}. Let L_{φ} be a line bundle on X_{Σ} given by a piecewise linear function φ = {m_{σ,φ}} ∈ P L(Σ) and s ∈ Γ(L_{φ}) a local section of L_{φ} nonvanishing in x ∈ T (F) ⊂ X_{Σ}(F). Define the v-norm of s at x_{v} ∈ K_{σ,v} by

∥s(x_{v})∥_{v} = |s(x_{v})/m_{σ,φ}(x_{v})|_{v}.

This gives a T (O_{v})-invariant v-adic metric on L_{φ} ⊗ F_{v}. The height function is defined as

H_{L_{φ}}(x) = ∏_{v} ∥s(x_{v})∥_{v}^{−1}.

Example 3.1. On X_{Σ} = ℙ^{⊮} the group of piecewise linear functions P L(Σ) is 2- dimensional, φ_{s} = (s_{1}, s_{2}) given by φ(e_{1}) = s_{1}, φ(e_{2}) = s_{2}. The corresponding local height function is

H_{v}(x, s) = \{ \begin{matrix}|x|_{v}^{s_{1}} & if & |x|_{v} ≥ 1 \\ |x|_{v}^{−s_{2}} & if & |x|_{v} ≤ 1 \end{matrix}

Here we have x ∈ T (F) = 𝔾_{⋗}(𝔽) ↪ 𝔽 and | · |_{v} is the standard valuation.

Let us point out that the introduced metrizations on line bundles for flag and for toric varieties are quite different. For ℙ^{⊮} considered as a flag variety the associated height zeta function is

Z(s) = ∑_{(n,m)=1} \frac{1}{(m^{2} + n^{2})^{s/2}}

For ℙ^{⊮} considered as a compactification of 𝔾_{⋗} the height zeta function is

Z_{Σ}(s) = 4 ∑_{n} \frac{ϕ(n)}{n^{s}} − 2

where ϕ(m) is the Euler function. We want to extend the height function to the complexified Picard group P ic(X_{Σ}) ⊗ ℂ. For a piecewise linear function φ_{s} ∈ P L(Σ) ⊗ ℂ given by φ(e_{i}) = s_{i} for s = (s_{1}, …, s_{n}) define the complex local height function as

H_{Σ,v}(x, s) = e^{φ_{s}(x̄_{v}) log q_{v}}.

Here q_{v} equals the cardinality of the residue field of F_{v} for non-archimedian v and log q_{v} = 1 for archimedian valuations and x̄_{v} is the image of x_{v} in N_{v} = N. The global complex height function defined as a product

H_{Σ}(x, s) = ∏_{v} H_{Σ,v}(x, s)

extends naturally to T_{Σ}(𝔸_{F}).

Proposition 3.0.7. For s ∈ ℤ^{⋉} and x ∈ X_{Σ}(F) the function ∏_{v} H_{Σ,v}(x, s) coincides with the (classical) height function H_{L_{φs}}(x) defined through the canonical compact covering.

Summing up: we have defined a function on the Zariski open subset T ⊂ X_{Σ}

H_{Σ}(x, s) : T (𝔸_{F}) × P ic(X_{Σ})_{ℂ} → ℂ

invariant under the compact subgroup K ⊂ T (𝔸_{F}). The corresponding height zeta function

Z_{Σ}(t, s) = ∑_{x∈T (F)} H_{Σ}(tx, −s)

is invariant under T (F). It is not difficult to see that the series converges absolutely and uniformly in the domain Re(s) > 1 = {Re(s_{j}) > 1 for j = 1, …, n}. The next step is to use harmonic analysis on the homogeneous space T (𝔸_{F})/T (F) and to try to establish the analytic properties of Z_{Σ}(t, s). The accumulating subvarieties are contained in the complement X_{Σ} − T. All geometrical information is now encoded in the height H_{Σ}. Here is a precise version of Manin’s conjecture in the case of toric varieties.

Conjecture 3.0.8. The height zeta function Z_{Σ}(t, s) is a function on \frac{P L(Σ)}{M} ⊗ ℂ, it is holomorphic in the domain Re(s) ∈ −K_{Σ} + N_{eff}^{1}(X_{Σ}) and has simple poles on codimension 1 faces of this cone.

In general, we cannot hope to prove a functional equation and analytic continuation to the whole plane.

4. Fourier Analysis

4.1. Adelization. Denote by T (𝔸_{F}) the adele group of T_{F}, by T^{1}(𝔸_{F}) the subgroup of norm 1 adeles, by KT(F) the smallest closed subgroup of T^{1}(𝔸_{F}) containing T (F) and K = ∏_{v} T (O_{v}). Let us formulate analogs of well known properties of ideles.

1. T (𝔸_{F})/T^{1}(𝔸_{F}) ≃ ℝ^{∖} where r equals the rank of the group of F -rational characters of T_{F},

2. T^{1}(𝔸_{F})/KT(F) is isomorphic to a product of a finite abelian group (analog of the idele class group) and a connected compact abelian group,

3. ∏_{v} T (O_{v}) ∩ T (F) is a finite group of torsion elements in T (F) (this is the analog of the group of roots of unity).

4.2. Poisson formula. For a locally compact abelian group G denote by G^{∗} its dual group of characters. For a function f ∈ L^{1}(G) define the Fourier Transform

f̂(y) = ∫_{G} f(x) < x, y > dx.

Let H ⊂ G be a closed subgroup. Define the orthogonal subgroup in G^{∗} by

H^{⊥} = {y ∈ G^{∗} |< x, y >= 0 ∀x ∈ H}.

Given Haar measures on G and H there exists a unique measure dh^{⊥} on H^{⊥} such that for all f ∈ L^{1}(G) with f̂ ∈ L^{1}(G^{∗}) the Poisson formula holds:

∫_{H} f(x)dx = ∫_{H^{⊥}} f̂(y)dh^{⊥}.

Example 4.1. Let P L^{+} ⊆ ℝ^{⋉} = (∼_{⊮}, …, ∼_{⋉}) be the simplicial cone given by s_{j} ≥ 0 for all j = 1, …, n. Let N_{eff}^{1} be its n − d dimensional image under a linear surjective map of vector spaces. We have an exact sequence

0 → M → ℝ^{⋉} → ℝ^{⋉−} → ⊬.

Then the function

Λ(N_{eff}^{1}, s) = ∫_{M} \frac{dm}{∏_{j=1}^{n}(s_{j} + \sqrt{−1}m_{j})}

is a rational function on N_{eff}^{1} ⊗ ℝ = ℝ^{⋉−} with simple poles on codimension 1 faces of N_{eff}^{1}. Here m_{j} are the coordinates of m in the fixed basis (s_{1}, ..., s_{n}) of ℝ^{⋉}. This follows from Poisson formula applied to the Laplace transform of the characteristic function of the dual cone N̂_{eff}^{1}.

4.3. An integral representation. Recall that for Re(s) > 1 we have defined the height zeta function by the series

Z_{Σ}(t, s) = ∑_{x∈T (F)} H_{Σ}(tx, −s).

In the following let us assume t = 1 and denote Z_{Σ}(1, s) by Z_{Σ}(s). The K invariance of the height H_{Σ} implies the following Proposition 4.3.1. Let dx be a Haar measure on T (𝔸_{F}). There is an explicit constant c_{0} such that

Z_{Σ}(s) = c_{0} ∫_{KT(F)} H_{Σ}(x, −s)dx.

Theorem 4.3.2. The following formula holds for Re(s) > 1

∫_{KT(F)} H_{Σ}(x, −s)dx = ∫_{(T (𝔸_{F})/KT(F))^{∗}} dy ∫_{T (𝔸_{F})} H_{Σ}(x, −s)χ_{y}(x)dx.

Here χ_{y} : T (𝔸_{F})/KT(F) → ℂ is a continuous character and dx, dy are appropriate Haar measures. 4.4. Reductions. The character χ_{y} ∈ (T (𝔸_{F})/KT(F)) splits through the section of the map T^{1}(𝔸_{F}) → T (𝔸_{F}) into a product

χ_{y} = χ_{m} × χ_{u}

such that χ_{m} is trivial on T^{1}(𝔸_{F}) and χ_{u} ∈ (T^{1}(𝔸_{F})/KT(F))^{∗}. Notice that for split tori T (𝔸_{F})/T^{1}(𝔸_{F}) ≃ ℝ ≃ 𝕄 and that χ_{m,v} = |x|_{v}^{\sqrt{−1}m}.

We can rewrite the integral above

Z_{Σ}(s) = ∫_{(T (𝔸_{F})/T^{1}(𝔸_{F}))^{∗}} dm ∫_{(T^{1}(𝔸_{F})/KT(F))^{∗}} du ∫_{T (𝔸_{F})} H_{Σ}(x, −s)χ_{m}(x)χ_{u}(x)dx.

Consider

Z′_{Σ}(s) = ∫_{(T (𝔸_{F})/T^{1}(𝔸_{F}))^{∗}} dm ∫_{T (𝔸_{F})} H_{Σ}(x, −s)χ_{m}(x)dx.

A variant of Poisson formula implies that Z_{Σ}(s) and Z′_{Σ}(s) have the same analytic properties in the domain Re(s) > 1 − δ for some small positive δ. This is enough to prove the Theorem 4.4.1. Let X_{Σ} be a compactification of an anisotropic torus. Then the height zeta function Z_{Σ}(s) satisfies Manin’s conjecture (3.0.8).

4.5. Further reductions. Just like ℙ^{⊮} is seen to be a factor of 𝔾_{⅁}^{⊭}\{(⊬, ⊬)} by the diagonal action of 𝔾_{⋗} one can represent split toric varieties as factors of some open dense U(Σ) ↪ 𝔾_{⅁}^{⋉} by an action of a torus 𝔾_{⋗}^{⋉−}. We have a diagramm

0 → 𝔾_{⋗}^{⋉−} → U(Σ) → X_{Σ} → 0
∥ ∪ ∪
0 → 𝔾_{⋗}^{⋉−} → 𝔾_{⋗}^{⋉} → T → 0

and an induced embedding of characters

T (𝔸_{F})^{∗} ↪ 𝔾_{⋗}^{⋉}(𝔸_{𝔽})^{∗}.

Definition 4.5.1. For x = (x_{1}, …, x_{n}) ∈ 𝔾_{⋗}^{⋉}(𝔸_{𝔽}) define f_{Σ}(x) = ⊗_{v}f_{Σ,v}(x) by

f_{Σ,v}(x) = ∑_{k=0}^{d}(−1)^{d−k} ∑_{σ∈Σ(k)} f_{σ,v}(x)

where f_{σ,v}(x) = ∏_{j=1}^{n} f_{σ,v}^{j}(x) and

f_{σ,v}^{j}(x) = f_{σ,v}^{j}(x_{j}) = \{ \begin{matrix}δ_{1,v}(x_{j}) & if & σ ∈ Σ(d) & and & e_{j} ∉ σ \\ χ_{1,v}(x_{j}) & if & σ ∈ Σ(k), k < d & and & e_{j} ∉ σ \\ χ_{≤1,v}(x_{j}) & otherwise \end{matrix}

Here δ_{1,v}(t) is defined by

∫_{𝔾_{⋗}(𝔽_{≿})} δ_{1,v}(t)|x|^{s}χ(t)dt = 1

for all χ ∈ (T (𝔸_{F})/KT (F))^{∗} with Re(χ) > 0 and all s with Re(s) > 0. The other functions are defined as

\begin{matrix}χ_{1,v}(t) = 1 & for |t|_{v} = 1 & and & 0 otherwise \\ χ_{≤1,v}(t) = 1 & for |t|_{v} ≤ 1 & and & 0 otherwise. \end{matrix}

The above definition needs some explanation. At each non-archimedian place f_{Σ,v}(x) is a smooth function. At archimedian places f_{Σ,v}(x) is a distribution. We don’t want to define a product of delta functions. The notation δ_{1,v}(t_{1}) × δ_{1,v}(t_{2}) stands for δ_{(1,1)}(t_{1}, t_{2}).

Theorem 4.5.2. For Re(s) > 1 we have the following identity:

∫_{T (𝔸_{F})} H_{Σ}(x, −s)χ_{y}(x)dx = ∫_{𝔾_{⋗}^{⋉}(𝔸_{𝔽})} f_{Σ}(x)|x|^{s}χ_{y}(x)dx

Here χ_{y} is considered as an element of (𝔾_{⋗}^{⋉}(𝔸_{𝔽}))^{∗} and |x|^{s} = |x_{1}|^{s_{1}} … |x_{n}|^{s_{n}}. The integrals converge absolutely.

Proof. To prove the theorem we compare local factors. Using

1. T (O)_{v} invariance of H_{Σ,v} and χ_{y,v}, 2. the embedding T (F_{v})/T (O_{v}) ↪ N (resp. N ⊗ ℝ for archimedian

v),

3. the explicit description of the height

H_{Σ,v}(x_{v}) = e^{φ_{s}(x̄_{v}) log q_{v}}

where φ_{s} ∈ P L(Σ) is a piecewise linear function on N (resp. N⊗ℝ ).

we deduce our claim. □ We have expressed the height zeta function as a Mellin transform of a (generalized) function on a bigger group. The next step is to isolate the poles of the adelic integral

I_{Σ}(s, χ_{y}) = ∫_{𝔾_{⋗}^{⋉}(𝔸_{𝔽})} f_{Σ}(x)|x|^{s}χ_{y}(x)dx,

and to study to integral

Z_{Σ}(s) = ∫ I_{Σ}(s, χ_{y})dy.

4.6. Examples. Assume F = ℚ. In this section we consider split toric varieties. Recall that in this case (T (𝔸_{ℚ})/T^{1}(𝔸_{ℚ}))^{∗} ≃ ℝ ≃ 𝕄 and χ_{m}(x) = |x|^{\sqrt{−1}m}. It follows that I_{Σ}(s, χ_{m}) = I_{Σ}(s + \sqrt{−1}m); our notations are in accordance with example (4.1). Let us introduce more notations: For G = ∏ 𝔾_{⋗} define

J_{Σ}(s) = ∫_{|x|≥1,x∈G(𝔸_{F})} f̂_{Σ}(x)|x|^{1−s}dx

where f̂_{Σ}(x) is the additive Fourier transform of f_{Σ}(x). The integral converges uniformly for Re(s) > 1 − δ for some δ > 0.

1. Here is yet another expression for the height zeta function of ℙ^{⊮}

Z_{ℙ^{⊮}}(s_{1}, s_{2}) =

∫_{ℝ^{⊮}} \frac{ζ_{ℚ}(s_{1} + \sqrt{−1}m)ζ_{ℚ}(s_{2} − \sqrt{−1}m)}{ζ_{ℚ}(s_{1} + s_{2})}(\frac{1}{(s_{1} + \sqrt{−1}m)}+\frac{1}{(s_{2} − \sqrt{−1}m)})dm

where ζ_{ℚ}(s) is the Riemann zeta function.

2. Consider a smooth split del Pezzo surface X = X_{Σ} of degree 6. It is a toric variety obtained by blowing up ℙ^{⊭} in 3 points or by compactifying 𝔾_{⋗}^{⊭} with 6 rational curves. The cone N_{eff}^{1}(X) is generated by the divisors at infinity (exceptional curves) D_{1}, …, D_{6}. From the discussion above we see that

Z_{Σ}(s) = ∫_{ℝ^{⊭}} I_{Σ}(s + \sqrt{−1}m)dm

Put s_{−1} = s_{6}.

Claim.

I_{Σ}(s) = J_{Σ}(s) + \frac{c_{Σ}}{∏_{j=1}^{6}(s_{j} − 1)}+

+c_{1} ∑_{j=1}^{6} \frac{J_{ℙ^{⊮}}(s_{j+1}, s_{j−1})}{s_{j}(s_{j} − 1)} + c_{2} ∑_{j=1}^{6} \frac{1}{s_{j}s_{j+1}} +

+c_{3} ∑_{j=1}^{6} \frac{1}{(s_{j} − 1)(s_{j+1} − 1)} + c_{4} ∑_{j=1}^{6} \frac{1}{(s_{j} − 1)s_{j}(s_{j+1} − 1)}

−c_{5} ∑_{j=1}^{6} \frac{1}{s_{j}(s_{j+1} − 1)} − c_{6} ∑_{j=1}^{6} \frac{1}{s_{j}(s_{j−1} − 1)}.

Notice that

∫ \frac{1}{∏_{j=1}^{6}(s_{j} − 1 + \sqrt{−1}m_{j})}dm

coincides with the function

Λ(N_{eff}^{1}, s − 1) = Λ(−K_{Σ} + N_{eff}^{1}, s)

from example (4.1).

3. The constant c_{Σ} in the example (2) involves an Euler product. Let us give a general expression for a local factor:

c_{Σ,p} = ∑_{k=0}^{d}(−1)^{d−k} ∑_{σ∈Σ(k)}(1 − \frac{1}{p})^{n−k}.

From an easy combinatorial identity on the fan it follows that 1/p terms cancel and that the Euler product converges. For ℙ^{⊮} we get \frac{1}{ζ(2)} as expected, for ℙ^{⊭} blown up in 3 points we get

c_{Σ,p} = (1 − \frac{9}{p^{2}} + \frac{16}{p^{3}} − \frac{9}{p^{4}} + \frac{1}{p^{6}} ).

References

[1] V.V. Batyrev, Yu. I. Manin, Sur le nombre des points rationnels de hauteur bornée des variétés algébriques, Math. Ann., 286, 1990, 27-43.

[2] V.V.Batyrev, Yu.Tschinkel, Height functions on compactifications of algebraic tori and Fourier analysis, work in progress.

[3] V.I.Danilov, The geometry of toric varieties. Russian Math. Surveys, 33(2), 1978, 97-154.

[4] J. Franke, Yu. I. Manin, Yu. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95, 1989, 425-435.

[5] Yu. I. Manin, Yu. Tschinkel, Rational points of bounded height on Del Pezzo surfaces, Compositio Math., 85(3), 1993, 315-332.

[6] T. Oda, Toric Varieties and Convex Geometry. Springer [7] J. Tate, Fourier Analysis in Number Fields and Hecke’s Zeta-Functions, in

Algebraic Number Theory (eds. Cassels, Frölich) 305-347.

Universität-GHS-Essen, Fachbereich 6, Mathematik Universitätsstr. 3,, 45141 Essen, Federal Republic of Germany

E-mail address: matf∅∅@vm.hrz.uni-essen.de

Department of Mathematics, Harvard University, Cambridge, MA 02138

E-mail address: tschink@math.harvard.edu